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\leftheadtext{ANDREW JOHN WILES}\rightheadtext{MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM}
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\ \vskip-0.5in\centerline{\eightpoint{Annals of Mathematics, {\bf 141} (1995), 443-551}}
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{\fiverm\hskip-0.08in Pierre de Fermat\hskip3.04in Andrew John Wiles\vskip-1.49in}
\vskip0.3in
\centerline{\boldeighteenpoint{Modular elliptic curves}}\vskip4pt
\centerline{\boldeighteenpoint{and}}\vskip4pt
\centerline{\boldeighteenpoint{Fermat's Last Theorem}}
\vskip4pt\centerline{By {\smc Andrew John Wiles}\footnote"*"{\eightpoint{The work on this paper was supported by an NSF
grant.}}}\vskip4pt
\centerline{\it For Nada, Claire, Kate and Olivia}\vskip6pt
\
\hskip2pt{\it Cubum autem in duos cubos, aut quadratoquadratum in duos quadra-
\hskip2pt toquadratos, \ et \ generaliter \ nullam \ in \ infinitum \ ultra \ quadratum
\hskip2pt potestatum \ in \ duos \ ejusdem \ nominis \ fas \ est \ \!dividere$:$ \ cujes \ \!rei
\hskip2pt demonstrationem \ mirabilem \ sane \ \!detexi. \ \!Hanc \ \!marginis \ \!exiguitas
\hskip2pt non caperet.
\
\hskip2pt - Pierre de Fermat $\sim$ 1637}
\
\noindent{\eightpoint{\bf Abstract.} When Andrew John Wiles was 10 years old, he read Eric Temple Bell's {\it The Last
Problem} and was so impressed by it that he decided that he would be the first person to prove Fermat's
Last Theorem. This theorem states that there are no
nonzero integers
$a,b,c,n$ with
$n\gt2$ such that $a^n+b^n=c^n$. The object of this paper is to prove that all semistable elliptic curves over the set of
rational numbers are modular. Fermat's Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.}
\
\centerline{\bf Introduction}\vskip6pt
An elliptic curve over $\bold Q$ is said
to be modular if it has a finite covering by a modular curve of the form
$X_0(N).$ Any such elliptic curve has the property that its Hasse-Weil zeta function has an analytic continuation and
satisfies a functional equation of the standard type. If an elliptic curve over $\bold Q$ with a given $j$-invariant is
modular then it is easy to see that all elliptic curves with the same $j$-invariant are modular (in which case we say that
the
$j$-invariant\linebreak is modular). A well-known conjecture which grew out of the work of Shimura and Taniyama in the
1950's and 1960's asserts that every elliptic curve over $\bold Q$ is modular. However, it only became widely known through
its publication in a paper of Weil in 1967 [We] (as an exercise for the interested reader!), in which,\linebreak moreover,
Weil gave conceptual evidence for the conjecture. Although it had been numerically verified in many cases, prior to the
results described in this paper it had only been known that finitely many $j$-invariants were modular.
In 1985 Frey made the remarkable observation that this conjecture should imply Fermat's Last Theorem. The precise mechanism
relating the two was formulated by Serre as the $\varepsilon$-conjecture and this was then proved by Ribet in the summer of
1986. Ribet's result only requires one to prove the conjecture for semistable elliptic curves in order to deduce Fermat's
Last Theorem.
\eject\pageno=444
Our approach to the study of elliptic curves is via their associated Galois representations. Suppose that $\rho_p$
is the representation of $\roman{Gal}(\bar{\bold Q}/\bold Q)$ on the $p$-division points of an elliptic curve over $\bold Q$,
and suppose for the moment that $\rho_3$ is irreducible. The choice of 3 is critical because a crucial theorem of Lang-lands
and Tunnell shows that if $\rho_3$ is irreducible then it is also modular. We then proceed by showing that under the
hypothesis that $\rho_3$ is semistable at 3, together with some milder restrictions on the ramification of $\rho_3$ at the
other primes, every suitable lifting of $\rho_3$ is modular. To do this we link the problem, via some novel arguments from
commutative algebra, to a class number prob-lem of a well-known type. This we then solve with the help of the paper [TW].
This suffices to prove the modularity of $E$ as it is known that $E$ is modular if and only if the associated 3-adic
representation is modular.
The key development in the proof is a new and surprising link between two\linebreak strong but distinct traditions in number
theory, the relationship between Galois\linebreak representations and modular forms on the one hand and the interpretation of
special values of $L$-functions on the other. The former tradition is of course more recent. Following the original results
of Eichler and Shimura in the\linebreak 1950's and 1960's the other main theorems were proved by Deligne, Serre and Langlands
in the period up to 1980. This included the construction of Galois representations associated to modular forms, the
refinements of Langlands and Deligne (later completed by Carayol), and the crucial application by Langlands of base change
methods to give converse results in weight one. However with the exception of the rather special weight one case, including
the extension by \linebreak Tunnell of Langlands' original theorem, there was no progress in the direction of associating
modular forms to Galois representations. From the mid 1980's the main impetus to the field was given by the conjectures of
Serre which\linebreak elaborated on the $\varepsilon$-conjecture alluded to before. Besides the work of Ribet and others on
this problem we draw on some of the more specialized developments\linebreak of the 1980's, notably those of Hida and Mazur.
The second tradition goes back to the famous analytic class number formula of Dirichlet, but owes its modern revival to the
conjecture of Birch and Swinnerton-Dyer. In practice however, it is the ideas of Iwasawa in this field on which we attempt
to draw, and which to a large extent we have to replace. The principles of Galois cohomology, and in particular the
fundamental theorems of Poitou and Tate, also play an important role here.
The restriction that $\rho_3$ be irreducible at 3 is bypassed by means of an intriguing argument with families of elliptic
curves which share a common\linebreak $\rho_5$. Using this, we complete the proof that all semistable elliptic curves are
modular. In particular, this finally yields a proof of Fermat's Last Theorem. In\linebreak addition, this method seems well
suited to establishing that all elliptic curves over $\bold Q$ are modular and to generalization to other totally real number
fields.
Now we present our methods and results in more detail.
\eject
Let $f$ be an eigenform associated to the congruence subgroup $\Gamma_1(N)$ of $\roman{SL}_2(\bold Z)$ of weight $k\ge2$ and
character $\chi.$ Thus if $T_n$ is the Hecke operator associated to an integer $n$ there is an algebraic integer $c(n,f)$
such that $T_nf=c(n,f)f$ for each $n$. We let $K_f$ be the number field generated over $\bold Q$ by the\linebreak
$\{c(n,f)\}$ together with the values of $\chi$ and let $\Cal O_f$ be its ring of integers.\linebreak For any prime
$\lambda$ of $\Cal O_f$ let
$\Cal O_{f,\lambda}$ be the completion of $\Cal O_f$ at $\lambda$. The following theorem is due to Eichler and Shimura (for
$k=2$) and Deligne (for $k\gt2$). The analogous result when $k=1$ is a celebrated theorem of Serre and Deligne but is more
naturally stated in terms of complex representations. The image in that case is finite and a converse is known in many
cases.
\
{\smc Theorem} 0.1. {\it For each prime $p\in\bold Z$ and each prime $\lambda|p$ of $\Cal O_f$ there\linebreak is a
continuous representation $$\rho_{f,\lambda}:\roman{Gal}(\bar\bold Q/\bold Q)\longrightarrow\roman{GL}_2(\Cal
O_{f,\lambda})$$ which is unramified outside the primes dividing $Np$ and such that for all primes $q\nmid Np$,
$$\roman{trace}\ \rho_{f,\lambda}(\roman{Frob}\ q)=c(q,f),\ \ \ \roman{det}\ \rho_{f,\lambda}(\roman{Frob}\
q)=\chi(q)q^{k-1}.$$}
We will be concerned with trying to prove results in the opposite direction, that is to say, with establishing criteria
under which a $\lambda$-adic representation arises in this way from a modular form. We have not found any advantage in
assuming that the representation is part of a compatible system of $\lambda$-adic representations except that the proof may
be easier for some $\lambda$ than for others.
Assume $$\rho_0:\roman{Gal}(\bar\bold Q/\bold Q)\longrightarrow\roman{GL}_2(\bar\bold F_p)$$ is a continuous representation
with values in the algebraic closure of a finite field of characteristic $p$ and that $\roman{det}\ \rho_0$ is odd. We say
that $\rho_0$ is modular\linebreak if $\rho_0$ and $\rho_{f,\lambda}\mod\lambda$ are isomorphic over $\bar\bold F_p$ for
some $f$ and $\lambda$ and some embedding of $\Cal O_f/\lambda$ in $\bar\bold F_p$. Serre has conjectured that every
irreducible $\rho_0$ of odd determinant is modular. Very little is known about this conjecture except when the image of
$\rho_0$ in $\roman{PGL}_2(\bar\bold F_p)$ is dihedral, $A_4$ or $S_4$. In the dihedral case it is true and due
(essentially) to Hecke, and in the $A_4$ and $S_4$ cases it is again true and due primarily to Langlands, with one important
case due to Tunnell (see Theorem 5.1 for a statement). More precisely these theorems actually associate a form of weight one
to the corresponding complex representation but the versions we need are straightforward deductions from the complex case.
Even in the reducible case not much is known about the problem in\linebreak the form we have described it, and in that case
it should be observed that one must also choose the lattice carefully as only the semisimplification of
$\overline{\rho_{f,\lambda}}=\rho_{f,\lambda}\mod\lambda$ is independent of the choice of lattice in $K^2_{f,\lambda}.$
\eject
If $\Cal O$ is the ring of integers of a local field (containing $\bold Q_p$) we will say that $\rho:\roman{Gal}(\bar\bold
Q/\bold Q)\longrightarrow\roman{GL}_2(\Cal O)$ is a lifting of $\rho_0$ if, for a specified embedding of the residue field
of $\Cal O$ in $\bar\bold F_p,\bar\rho$ and $\rho_0$ are isomorphic over $\bar\bold F_p$. Our point of view\linebreak will be
to assume that $\rho_0$ is modular and then to attempt to give conditions under which a representation $\rho$ lifting
$\rho_0$ comes from a modular form in the sense that $\rho\simeq\rho_{f,\lambda}$ over $\overline{K_{f,\lambda}}$ for some
$f,\lambda.$ We will restrict our attention to two cases:
\
\hskip-12pt(I) $\rho_0$ is ordinary (at $p$) by which we mean that there is a one-dimensional\linebreak
${}$\hskip24pt subspace of $\bar\bold F^2_p,$ stable under a decomposition group at $p$ and such that\linebreak
${}$\hskip24pt the action on the quotient space is unramified and distinct from the\linebreak
${}$\hskip24pt action on the subspace.
\
\hskip-15pt(II) $\rho_0$ is flat (at $p$), meaning that as a representation of a decomposition\linebreak
${}$\hskip24pt group at $p,\rho_0$ is equivalent to one that arises from a finite flat group\linebreak
${}$\hskip25pt scheme over $\bold Z_p$, and $\det\rho_0$ restricted to an inertia group at $p$ is the\linebreak
${}$\hskip25pt cyclotomic character.
\
\noindent We say similarly that $\rho$ is ordinary (at $p$), if viewed as a representation to $\bar\bold Q^2_p$, there is a
one-dimensional subspace of $\bar\bold Q^2_p$ stable under a decomposition group at $p$ and such that the action on the
quotient space is unramified.
Let $\varepsilon:\roman{Gal}(\bar\bold Q/\bold Q)\longrightarrow\bold Z^\times_p$ denote the cyclotomic character.
Conjectural converses to Theorem 0.1 have been part of the folklore for many years but have hitherto lacked any evidence.
The critical idea that one might dispense with compatible systems was already observed by Drinfield in the function field
case [Dr]. The idea that one only needs to make a geometric condition on the restriction to the decomposition group at $p$
was first suggested by Fontaine and\linebreak Mazur. The following version is a natural extension of Serre's conjecture
which is convenient for stating our results and is, in a slightly modified form, the one proposed by Fontaine and Mazur. (In
the form stated this incorporates Serre's conjecture. We could instead have made the hypothesis that $\rho_0$ is modular.)
\
{\smc Conjecture.} {\it Suppose that $\rho:\roman{Gal}(\bar\bold Q/\bold Q)\longrightarrow\roman{GL}_2(\Cal O)$ is an
irreducible lifting of $\rho_0$ and that $\rho$ is unramified outside of a finite set of primes. There are two cases}:
\
\hskip-12pt(i) {\it Assume that $\rho_0$ is ordinary. Then if $\rho$ is ordinary and $\det\rho=\varepsilon^{k-1}\chi$
for\linebreak
${}$\hskip22pt some integer $k\ge2$ and some $\chi$ of finite order}, {\it $\rho$ comes from a modular\linebreak
${}$\hskip22pt form.}
\
\hskip-15pt(ii) {\it Assume that $\rho_0$ is flat and that $p$ is odd. Then if $\rho$ restricted to a
de-\linebreak ${}$\hskip22pt composition group at $p$ is equivalent to a representation on a $p$-divisible\linebreak
${}$\hskip22pt group}, {\it again $\rho$ comes from a modular form.}
\eject
In case (ii) it is not hard to see that if the form exists it has to be of\linebreak weight 2; in (i) of course it would
have weight $k$. One can of course enlarge this conjecture in several ways, by weakening the conditions in (i) and (ii),
by\linebreak considering other number fields of $\bold Q$ and by considering groups other\linebreak than $\roman{GL}_2$.
We prove two results concerning this conjecture. The first includes the hypothesis that $\rho_0$ is modular. Here and for
the rest of this paper we will assume that $p$ is an odd prime.
\
{\smc Theorem 0.2.} {\it Suppose that $\rho_0$ is irreducible and satisfies either} (I) {\it or}\linebreak (II) {\it above.
Suppose also that $\rho_0$ is modular and that}
\
\hskip-12pt (i) \hskip1pt$\rho_0$ {\it is absolutely irreducible when restricted to $\bold
Q\bigg(\sqrt{(-1)^{p-1\over2}p}\bigg)$.}
\
\hskip-15pt (ii) {\it If $q\equiv-1\mod p$ is ramified in $\rho_0$ then either $\rho_0|_{D_q}$ is reducible over\linebreak
${}$\hskip22pt the algebraic closure where $D_q$ is a decomposition group at $q$ or $\rho_0|_{I_q}$ is\linebreak
${}$\hskip22pt absolutely irreducible where $I_q$ is an inertia group at $q$.
\
\noindent Then any representation $\rho$ as in the conjecture does indeed come from a mod-ular form.}
\
The only condition which really seems essential to our method is the requirement that $\rho_0$ be modular.
The most interesting case at the moment is when $p=3$ and $\rho_0$ can be de-\linebreak fined over $\bold F_3$. Then since
$\roman{PGL}_2(\bold F_3)\simeq S_4$ every such representation is modular by the theorem of Langlands and Tunnell mentioned
above. In particular, ev-ery representation into $\roman{GL}_2(\bold Z_3)$ whose reduction satisfies the given conditions is
modular. We deduce:
\
{\smc Theorem 0.3.} {\it Suppose that $E$ is an elliptic curve defined over $\bold Q$ and\linebreak that $\rho_0$ is the
Galois action on the $3$-division points. Suppose that $E$ has the following properties}:
\
\hskip-12pt(i) {\it $E$ has good or multiplicative reduction at $3$.}
\
\hskip-14.5pt(ii) {\it $\rho_0$ is absolutely irreducible when restricted to $\bold Q\big(\sqrt{-3}\ \big)$.}
\
\hskip-17pt(iii) {\it For any $q\equiv-1\mod3$ either $\rho_0|_{D_q}$ is reducible over the algebraic
closure\linebreak
${}$\hskip22pt or $\rho_0|{I_q}$ is absolutely irreducible.
\
\noindent Then $E$ should be modular.}
\
We should point out that while the properties of the zeta function follow directly from Theorem 0.2 the stronger version
that $E$ is covered by $X_0(N)$\linebreak
\eject
\noindent requires also the isogeny theorem proved by Faltings (and earlier by Serre when $E$ has nonintegral $j$-invariant,
a case which includes the semistable curves).\linebreak We note that if $E$ is modular then so is any twist of $E$, so we
could relax condition (i) somewhat.
The important class of semistable curves, i.e., those with square-free conductor, satisfies (i) and (iii) but not
necessarily (ii). If (ii) fails then in fact $\rho_0$ is reducible. Rather surprisingly, Theorem 0.2 can often be applied in
this case\linebreak also by showing that the representation on the 5-division points also occurs for another elliptic curve
which Theorem 0.3 has already proved modular. Thus Theorem 0.2 is applied this time with $p=5$. This argument, which is
explained in Chapter 5, is the only part of the paper which really uses deformations of the elliptic curve rather than
deformations of the Galois representation. The argument works more generally than the semistable case but in this
setting\linebreak we obtain the following theorem:
\
{\smc Theorem 0.4.} {\it Suppose that $E$ is a semistable elliptic curve defined over $\bold Q$. Then $E$ is modular.}
\
\noindent More general families of elliptic curves which are modular are given in Chap-\linebreak ter 5.
In 1986, stimulated by an ingenious idea of Frey [Fr], Serre conjectured and Ribet proved (in [Ri1]) a property of the
Galois representation associated to modular forms which enabled Ribet to show that Theorem 0.4 implies `Fer-\linebreak mat's
Last Theorem'. Frey's suggestion, in the notation of the following theorem, was to show that the (hypothetical) elliptic
curve $y^2=x(x+u^p)(x-v^p)$ could not be modular. Such elliptic curves had already been studied in [He] but without the
connection with modular forms. Serre made precise the idea of Frey by proposing a conjecture on modular forms which meant
that the rep-\linebreak resentation on the $p$-division points of this particular elliptic curve, if modular, would be
associated to a form of conductor 2. This, by a simple inspection, could not exist. Serre's conjecture was then proved by
Ribet in the summer\linebreak of 1986. However, one still needed to know that the curve in question would have to be
modular, and this is accomplished by Theorem 0.4. We have then (finally!):
\
{\smc Theorem 0.5.} {\it Suppose that $u^p+v^p+w^p=0$ with $u,v,w\in\bold Q$ and $p\ge3,$ then $uvw=0$.} ({\it
Equivalently - there are no nonzero integers $a,b,c,n$ with $n\gt2$ such that $a^n+b^n=c^n$.})
\
The second result we prove about the conjecture does not require the assumption that $\rho_0$ be modular (since it is
already known in this case).
\eject
{\smc Theorem 0.6.} {\it Suppose that $\rho_0$ is irreducible and satisfies the hypothesis of the conjecture, including} (I)
{\it above. Suppose further that}
\
\hskip-12pt(i) $\rho_0=\roman{Ind}^\bold Q_L\kappa_0$ {\it for a character $\kappa_0$ of an imaginary quadratic extension
$L$\linebreak${}$\hskip22pt of $\bold Q$ which is unramified at $p$.}
\
\hskip-15pt(ii) $\det\rho_0|_{I_p}=\omega${\it .
\
\noindent Then a representation $\rho$ as in the conjecture does indeed come from a modular form.}
\
This theorem can also be used to prove that certain families of elliptic curves are modular. In this summary we have only
described the principal theorems associated to Galois representations and elliptic curves. Our results concerning
generalized class groups are described in Theorem 3.3.
The following is an account of the origins of this work and of the more specialized developments of the 1980's that affected
it. I began working on these problems in the late summer of 1986 immediately on learning of Ribet's result. For several
years I had been working on the Iwasawa conjecture for totally real fields and some applications of it. In the process, I
had been using and developing results on $\ell$-adic representations associated to Hilbert modular forms. It was therefore
natural for me to consider the problem of modularity from the point of view of $\ell$-adic representations. I began with the
assumption that the reduction of a given ordinary $\ell$-adic representation was reducible and tried to prove under this
hypothesis that the representation itself would have to be modular. I hoped rather naively that in this situation I could
apply the techniques of Iwasawa theory. Even more optimistically I hoped that the case $\ell=2$ would be tractable as this
would suffice for the study of the curves used by Frey. From now on and in the main text, we write $p$ for $\ell$ because of
the connections with Iwasawa theory.
After several months studying the 2-adic representation, I made the first real breakthrough in realizing that I could use
the 3-adic representation instead: the Langlands-Tunnell theorem meant that $\rho_3$, the mod 3 representation of any given
elliptic curve over $\bold Q$, would necessarily be modular. This enabled me\linebreak to try inductively to prove
that the
$\roman{GL}_2(\bold Z/3^n\bold Z)$ representation would be\linebreak modular for each $n$. At this time I considered only the
ordinary case. This led quickly to the study of $H^i(\roman{Gal}(F_\infty/\bold Q),W_f)$ for $i=1$ and 2, where $F_\infty$
is the\linebreak splitting field of the $\frak m$-adic torsion on the Jacobian of a suitable modular curve, $\frak m$ being
the maximal ideal of a Hecke ring associated to $\rho_3$ and $W_f$ the module associated to a modular form $f$ described in
Chapter 1. More specifically, I needed to compare this cohomology with the cohomology of $\roman{Gal}(\bold Q_\Sigma/\bold
Q)$ acting on the same module.
I tried to apply some ideas from Iwasawa theory to this problem. In my solution to the Iwasawa conjecture for totally real
fields [Wi4], I had introduced
\eject
\noindent a new technique in order to deal with the trivial zeroes. It involved replacing the standard Iwasawa theory method
of considering the fields in the cyclotomic $\bold Z_p$-extension by a similar analysis based on a choice of infinitely many
distinct primes $q_i\equiv1\mod p^{n_i}$ with $n_i\rightarrow\infty$ as $i\rightarrow\infty.$ Some aspects of this method
suggested that an alternative to the standard technique of Iwasawa theory, which seemed problematic in the study of $W_f$,
might be to make a comparison between the cohomology groups as $\Sigma$ varies but with the field $\bold Q$ fixed. The new
principle said roughly that the unramified cohomology classes are trapped by the tamely ramified ones. After reading the
paper [Gre1]. I realized that the\linebreak duality theorems in Galois cohomology of Poitou and Tate would be useful for
this. The crucial extract from this latter theory is in Section 2 of Chapter 1.
In order to put ideas into practice I developed in a naive form the\linebreak
techniques of the first two sections of Chapter 2. This drew in particular on\linebreak
a detailed study of all the congruences between $f$ and other modular forms\linebreak
of differing levels, a theory that had been initiated by Hida and Ribet. The outcome was that I could estimate the first
cohomology group well under two assumptions, first that a certain subgroup of the second cohomology group vanished and
second that the form $f$ was chosen at the minimal level for $\frak m$. These assumptions were much too restrictive to be
really effective but at least they pointed in the right direction. Some of these arguments are to be found in the second
section of Chapter 1 and some form the first weak approximation to the argument in Chapter 3. At that time, however, I used
auxiliary primes $q\equiv-1\mod p$ when varying $\Sigma$ as the geometric techniques I worked with did not apply in general
for primes $q\equiv1\mod p$. (This was for much the same\linebreak reason that the reduction of level argument in [Ri1] is
much more difficult\linebreak when $q\equiv1\mod p.$) In all this work I used the more general assumption that $\rho_p$ was
modular rather than the assumption that $p=-3$.
In the late 1980's, I translated these ideas into ring-theoretic language. A few years previously Hida had constructed some
explicit one-parameter fam-ilies of Galois representations. In an attempt to understand this, Mazur had been developing the
language of deformations of Galois representations. Moreover, Mazur realized that the universal deformation rings he found
should be given by Hecke ings, at least in certain special cases. This critical conjecture refined the expectation that all
ordinary liftings of modular representations should be modular. In making the translation to this ring-theoretic language I
realized that the vanishing assumption on the subgroup of $H^2$ which I had needed should be replaced by the stronger
condition that the Hecke rings were complete intersections. This fitted well with their being deformation rings where one
could estimate the number of generators and relations and so made the original assumption more plausible.
To be of use, the deformation theory required some development. Apart from some special examples examined by Boston and
Mazur there had been\linebreak
\eject
\noindent little work on it. I checked that one could make the appropriate adjustments to\linebreak the theory in order to
describe deformation theories at the minimal level. In the\linebreak fall of 1989, I set Ramakrishna, then a student of
mine at Princeton, the task of proving the existence of a deformation theory associated to representations arising from
finite flat group schemes over $\bold Z_p.$ This was needed in order to remove the restriction to the ordinary case. These
developments are described in the first section of Chapter 1 although the work of Ramakrishna was not completed until the
fall of 1991. For a long time the ring-theoretic version\linebreak of the problem, although more natural, did not look any
simpler. The usual methods of Iwasawa theory when translated into the ring-theoretic language seemed to require unknown
principles of base change. One needed to know the exact relations between the Hecke rings for different fields in the
cyclotomic $\bold Z_p$-extension of $\bold Q$, and not just the relations up to torsion.
The turning point in this and indeed in the whole proof came in the\linebreak spring of 1991. In searching for a clue from
commutative algebra I had been particularly struck some years earlier by a paper of Kunz [Ku2]. I had already needed to
verify that the Hecke rings were Gorenstein in order to compute the congruences developed in Chapter 2. This property had
first been proved by Mazur in the case of prime level and his argument had already been extended by other authors as the
need arose. Kunz's paper suggested the use of an invariant (the $\eta$-invariant of the appendix) which I saw could be used
to test for isomorphisms between Gorenstein rings. A different invariant (the $\frak p/\frak p^2$-invariant of the
appendix) I had already observed could be used to test for isomorphisms between complete intersections. It was only on
reading Section 6\linebreak of [Ti2] that I learned that it followed from Tate's account of Grothendieck duality theory for
complete intersections that these two invariants were equal for such rings. Not long afterwards I realized that, unlike
though it seemed at first, the equality of these invariants was actually a criterion for a Gorenstein ring to be a complete
intersection. These arguments are given in the appendix.
The impact of this result on the main problem was enormous. Firstly, the relationship between the Hecke rings and the
deformation rings could be tested just using these two invariants. In particular I could provide the inductive ar-gument of
section 3 of Chapter 2 to show that if all liftings with restricted ramification are modular then all liftings are modular.
This I had been trying to do for a long time but without success until the breakthrough in commuta-tive algebra. Secondly,
by means of a calculation of Hida summarized in [Hi2] the main problem could be transformed into a problem about class
numbers of a type well-known in Iwasawa theory. In particular, I could check this in\linebreak the ordinary CM case using
the recent theorems of Rubin and Kolyvagin. This is the content of Chapter 4. Thirdly, it meant that for the first time it
could be verified that infinitely many $j$-invariants were modular. Finally, it meant that I could focus on the minimal
level where the estimates given by me earlier\linebreak
\eject
\noindent Galois cohomology calculations looked more promising. Here I was also using the work of Ribet and others on
Serre's conjecture (the same work of Ribet that had linked Fermat's Last Theorem to modular forms in the first place) to
know that there was a minimal level.
The class number problem was of a type well-known in Iwasawa theory and in the ordinary case had already been conjectured by
Coates and Schmidt. However, the traditional methods of Iwasawa theory did not seem quite suf-ficient in this case and, as
explained earlier, when translated into the ring-theoretic language seemed to require unknown principles of base change. So
instead I developed further the idea of using auxiliary primes to replace the change of field that is used in Iwasawa
theory. The Galois cohomology esti-mates described in Chapter 3 were now much stronger, although at that time I was still
using primes $q\equiv-1\mod p$ for the argument. The main difficulty was that although I knew how the $\eta$-invariant
changed as one passed to an auxiliary level from the results of Chapter 2, I did not know how to estimate the change in the
$\frak p/\frak p^2$-invariant precisely. However, the method did give the right bound for the generalised class group, or
Selmer group as it often called in this context, under the additional assumption that the minimal Hecke ring was a complete
intersection.
I had earlier realized that ideally what I needed in this method of auxiliary primes was a replacement for the power series
ring construction one obtains in the more natural approach based on Iwasawa theory. In this more usual setting,\linebreak
the projective limit of the Hecke rings for the varying fields in a cyclotomic tower would be expected to be a power series
ring, at least if one assumed\linebreak the vanishing of the $\mu$-invariant. However, in the setting with auxiliary primes
where one would change the level but not the field, the natural limiting process did not appear to be helpful, with the
exception of the closely related and very important construction of Hida [Hi1]. This method of Hida often gave one step
towards a power series ring in the ordinary case. There were also tenuous hints of a patching argument in Iwasawa theory
([Scho], [Wi4, \S10]), but I searched without success for the key.
Then, in August, 1991, I learned of a new construction of Flach [Fl] and quickly became convinced that an extension of his
method was more plausi-\linebreak ble. Flach's approach seemed to be the first step towards the construction of an Euler
system, an approach which would give the precise upper bound for the size of the Selmer group if it could be completed. By
the fall of 1992, I believed I had achieved this and begun then to consider the remaining case where the mod 3
representation was assumed reducible. For several months I tried simply to repeat the methods using deformation rings and
Hecke rings. Then unexpectedly in May 1993, on reading of a construction of twisted forms of modular curves in a paper of
Mazur [Ma3], I made a crucial and surprising\linebreak breakthrough: I found the argument using families of elliptic curves
with a\linebreak
\eject
\noindent common $\rho_5$ which is given in Chapter 5. Believing now that the proof was complete, I sketched the whole
theory in three lectures in Cambridge, England on June 21-23. However, it became clear to me in the fall of 1993 that the
con-\linebreak struction of the Euler system used to extend Flach's method was incomplete and possibly flawed.
Chapter 3 follows the original approach I had taken to the problem of bounding the Selmer group but had abandoned on
learning of Flach's paper. Darmon encouraged me in February, 1994, to explain the reduction to the com-plete intersection
property, as it gave a quick way to exhibit infinite families\linebreak of modular $j$-invariants. In presenting it in a
lecture at Princeton, I made, almost unconsciously, critical switch to the special primes used in Chapter 3 as auxiliary
primes. I had only observed the existence and importance of these primes in the fall of 1992 while trying to extend Flach's
work. Previously, I had\linebreak only used primes $q\equiv-1\mod p$ as auxiliary primes. In hindsight this change was
crucial because of a development due to de Shalit. As explained before, I had realized earlier that Hida's theory often
provided one step towards a power series ring at least in the ordinary case. At the Cambridge conference de Shalit had
explained to me that for primes $q\equiv1\mod p$ he had obtained a version of\linebreak Hida's results. But excerpt for
explaining the complete intersection argument in the lecture at Princeton, I still did not give any thought to my initial
approach, which I had put aside since the summer of 1991, since I continued to\linebreak believe that the Euler system
approach was the correct one.
Meanwhile in January, 1994, R. Taylor had joined me in the attempt to repair the Euler system argument. Then in the spring
of 1994, frustrated in the efforts to repair the Euler system argument, I begun to work with Taylor on an attempt to devise
a new argument using $p=2.$ The attempt to use $p=2$ reached an impasse at the end of August. As Taylor was still not
convinced that\linebreak the Euler system argument was irreparable, I decided in September to take one last look at my
attempt to generalise Flach, if only to formulate more precisely the obstruction. In doing this I came suddenly to a
marvelous revelation: I saw in a flash on September 19th, 1994, that de Shalit's theory, if generalised, could be used
together with duality to glue the Hecke rings at suitable auxiliary levels into a power series ring. I had unexpectedly
found the missing key to my\linebreak old abandoned approach. It was the old idea of picking $q_i$'s with $q_i\equiv1$mod
$p^{n_i}$\linebreak and $n_i\rightarrow\infty$ as $i\rightarrow\infty$ that I used to achieve the limiting process. The
switch to the special primes of Chapter 3 had made all this possible.
After I communicated the argument to Taylor, we spent the next few days making sure of the details. the full argument,
together with the deduction of the complete intersection property, is given in [TW].
In conclusion the key breakthrough in the proof had been the realization in the spring of 1991 that the two invariants
introduced in the appendix could be used to relate the deformation rings and the Hecke rings. In effect the
$\eta$-\linebreak
\eject
\noindent invariant could be used to count Galois representations. The last step after the\linebreak June, 1993,
announcement, though elusive, was but the conclusion of a long process whose purpose was to replace, in the ring-theoretic
setting, the methods based on Iwasawa theory by methods based on the use of auxiliary primes.
One improvement that I have not included but which might be used to simplify some of Chapter 2 is the observation of Lenstra
that the criterion for Gorenstein rings to be complete intersections can be extended to more general rings which are finite
and free as $\bold Z_p$-modules. Faltings has pointed out an improvement, also not included, which simplifies the argument
in Chapter 3 and [TW]. This is however explained in the appendix to [TW].
It is a pleasure to thank those who read carefully a first draft of some of this\linebreak paper after the Cambridge
conference and particularly N. Katz who patiently answered many questions in the course of my work on Euler systems, and
together with Illusie read critically the Euler system argument. Their questions led to my discovery of the problem with it.
Katz also listened critically to my first attempts to correct it in the fall of 1993. I am grateful also to Taylor for his
assistance in analyzing in depth the Euler system argument. I am indebted to F. Diamond for his generous assistance in the
preparation of the final version of this paper. In addition to his many valuable suggestions, several others also made
helpful comments and suggestions especially Conrad, de Shalit, Faltings, Ribet, Rubin, Skinner and Taylor.I am most grateful
to H. Darmon for his encouragement to reconsider my old argument. Although I paid no heed to his advice at the time, it
surely left its mark.
\
\centerline{\bf Table of Contents}
\
\noindent Chapter 1\ \ 1.\ \ Deformations of Galois representations
\hskip0.43in 2.\ \ Some computations of cohomology groups
\hskip0.43in 3.\ \ Some results on subgroups of $\roman{GL}_2(k)$
\noindent Chapter 2\ \ 1.\ \ The Gorenstein property
\hskip0.43in 2.\ \ Congruences between Hecke rings
\hskip0.43in 3.\ \ The main conjectures
\noindent Chapter 3\hskip0.3in Estimates for the Selmer group
\noindent Chapter 4\ \ 1.\ \ The ordinary CM case
\hskip0.43in 2.\ \ Calculation of $\eta$
\noindent Chapter 5\hskip0.3in Application to elliptic curves
\noindent Appendix
\noindent References
\eject
\centerline{\bf Chapter 1}
\
This chapter is devoted to the study of certain Galois representations.\linebreak In the first section we introduce and
study Mazur's deformation theory and discuss various refinements of it. These refinements will be needed later to make
precise the correspondence between the universal deformation rings and the Hecke rings in Chapter 2. The main results needed
are Proposition 1.2 which is used to interpret various generalized cotangent spaces as Selmer groups and (1.7) which later
will be used to study them. At the end of the section we relate these Selmer groups to ones used in the Bloch-Kato
conjecture, but this connection is not needed for the proofs of our main results.
In the second section we extract from the results of Poitou and Tate on Galois cohomology certain general relations between
Selmer groups as $\Sigma$ varies, as well as between Selmer groups and their duals. The most important observation of the
third section is Lemma 1.10(i) which guarantees the existence of\linebreak the special primes used in Chapter 3 and [TW].
\
\centerline{\bf 1. Deformations of Galois representations}
\
Let $p$ be an odd prime. Let $\Sigma$ be a finite set of primes including $p$ and\linebreak let $\bold Q_\Sigma$ be the
maximal extension of $\bold Q$ unramified outside this set and $\infty$. Throughout we fix an embedding of $\overline{\bold
Q}$, and so also of $\bold Q_\Sigma$, in $\bold C$. We will also fix a choice of decomposition group $D_q$ for all primes $q$
in $\bold Z$. Suppose that $k$ is a finite field characteristic $p$ and that $$\rho_0:\roman{Gal}(\bold Q_\Sigma/\bold
Q)\rightarrow\roman{GL}_2(k)\leqno(1.1)$$ is an irreducible representation. In contrast to the introduction we will assume
in the rest of the paper that $\rho_0$ comes with its field of definition $k$. Suppose further that $\det\rho_0$ is odd. In
particular this implies that the smallest field of definition for $\rho_0$ is given by the field $k_0$ generated by the
traces but we will not assume that $k=k_0$. It also implies that $\rho_0$ is absolutely irreducible. We con-\linebreak sider
the deformation $[\rho]$ to $\roman{GL}_2(A)$ of $\rho_0$ in the sense of Mazur [Ma1]. Thus if $W(k)$ is the ring of Witt
vectors of $k,A$ is to be a complete Noeterian local $W(k)$-algebra with residue field $k$ and maximal ideal $m,$ and a
deformation $[\rho]$ is just a strict equivalence class of homomorphisms $\rho:\roman{Gal}(\bold Q_\Sigma/\bold
Q)\rightarrow\roman{GL}_2(A)$ such that $\rho\mod m=\rho_0,$ two such homomorphisms being called strictly
equiv-\linebreak alent if one can be brought to the other by conjugation by an element of
$\ker:\roman{GL}_2(A)\rightarrow\roman{GL}_2(k).$ We often simply write $\rho$ instead of $[\rho]$ for the\linebreak
equivalent class.
\eject
We will restrict our choice of $\rho_0$ further by assuming that either:
\hskip-12pt(i) $\rho_0$ is {\it ordinary}; viz., the restriction of $\rho_0$ to the decomposition group $D_p$\linebreak
${}$\hskip22pt has (for a suitable choice of basis) the form $$\rho_0|_{D_p}\approx\pmatrix\chi_1&*\\
0&\chi_2\endpmatrix\leqno(1.2)$$
${}$\hskip22pt where $\chi_1$ and $\chi_2$ are homomorphisms from $D_p$ to $k^*$ with $\chi_2$ unramified.\linebreak
${}$\hskip22pt Moreover we require that $\chi_1\ne\chi_2$. We do allow here that $\rho_0|_{D_p}$ be\linebreak
${}$\hskip22pt semisimple. (If $\chi_1$ and $\chi_2$ are both unramified and $\rho_0|_{D_p}$ is semisimple\linebreak
${}$\hskip22pt then we fix our choices of $\chi_1$ and $\chi_2$ once and for all.)
\
\hskip-15pt(ii) $\rho_0$ is {\it flat} at $p$ but not ordinary (cf. [Se1] where the terminology {\it finite} is\linebreak
${}$\hskip22pt used); viz., $\rho_0|_{D_p}$ is the representation associated to a finite flat group\linebreak
${}$\hskip23pt scheme over $\bold Z_p$ but is not ordinary in the sense of (i). (In general when we\linebreak
${}$\hskip23pt refer to the flat case we will mean that $\rho_0$ is assumed not to be ordinary\linebreak
${}$\hskip22pt unless we specify otherwise.) We will assume also that $\det\rho_0|_{I_p}=\omega$\linebreak
${}$\hskip22pt where $I_p$ is an inertia group at $p$ and $\omega$ is the Teichm\"uller character\linebreak
${}$\hskip23pt giving the action on $p^\roman{th}$ roots of unity.
\
In case (ii) it follows from results of Raynaud that $\rho_0|_{D_p}$ is absolutely\linebreak irreducible and one can describe
$\rho_0|_{I_p}$ explicitly. For extending a Jordan-H\"older series for the representation space (as an $I_p$-module) to one
for finite flat group schemes (cf. \![Ray 1]) we observe first that the trivial character does not occur on\linebreak a
subquotient, as otherwise (using the classification of Oort-Tate or Raynaud) the group scheme would be ordinary. So we find by
Raynaud's results, that $\rho_0|_{I_p}\mathop{\otimes}\limits_k\bar k\simeq\psi_1\oplus\psi_2$ where $\psi_1$ and $\psi_2$ are
the two fundamental characters of degree 2 (cf. Corollary 3.4.4 of [Ray1]). Since $\psi_1$ and $\psi_2$ do not extend to
characters of $\roman{Gal}(\bar\bold Q_p/\bold Q_p),\rho_0|_{D_p}$ must be absolutely irreducible.
We sometimes wish to make one of the following restrictions on the\linebreak deformations we allow:
\
\noindent\hskip-5pt(i) (a) {\it Selmer deformations.} In this case we assume that $\rho_0$ is ordinary, with no-\linebreak
${}$\hskip24pt tion as above, and that the deformation has a representative\linebreak
${}$\hskip24pt$\rho:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(A)$ with the property that (for a suitable
choice\linebreak
${}$\hskip24pt of basis) $$\rho|_{D_p}\approx\pmatrix\tilde\chi_1&*\\0&\tilde\chi_2\endpmatrix$$
${}$\hskip24pt with $\tilde\chi_2$ unramified, $\tilde\chi\equiv\chi_2\mod m$, and
$\det\rho|_{I_p}=\varepsilon\omega^{-1}\chi_1\chi_2$ where\linebreak
${}$\hskip24pt$\varepsilon$ is the cyclotomic character, $\varepsilon:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\bold
Z^*_p,$ giving the action\linebreak
${}$\hskip24pt on all $p$-power roots of unity, $\omega$ is of order prime to $p$ satisfying
$\omega\equiv\varepsilon\mathbreak{}\hskip17pt\mod p,$ and $\chi_1$ and $\chi_2$ are the characters of (i) viewed as taking
values in\linebreak
${}$\hskip24pt $k^*\hookrightarrow A^*.$
\eject
\noindent\hskip-5pt(i) (b) {\it Ordinary deformations.} The same as in (i)(a) but with no condition on\linebreak
${}$\hskip28pt the determinant.
\
\noindent\hskip-5pt(i) (c) \hskip2pt{\it Strict deformations.} This is a variant on (i) (a) which we only use when\linebreak
${}$\hskip26pt$\rho_0|_{D_p}$ is not semisimple and not flat (i.e. not associated to a finite flat\linebreak
${}$\hskip26pt group scheme). We also assume that $\chi_1\chi_2^{-1}=\omega$ in this case. Then a\linebreak
${}$\hskip26pt strict deformation is as in (i)(a) except that we assume in addition that\linebreak
${}$\hskip26pt$(\tilde\chi_1/\tilde\chi_2)|_{D_p}=\varepsilon.$
\
\hskip-11pt(ii) {\it Flat} ({\it at $p$}) {\it deformations.} We assume that each deformation $\rho$ to
$\roman{GL}_2(A)$\linebreak
${}$\hskip26pt has the property that for any quotient $A/\frak a$ of finite order $\rho|_{D_p}\mod\frak a$\linebreak
${}$\hskip26pt is the Galois representation associated to the $\bar\bold Q_p$-points of a finite flat\linebreak
${}$\hskip26pt group scheme over $\bold Z_p.$
\
In each of these four cases, as well as in the unrestricted case (in which we\linebreak impose \!no \!local \!restriction \!at
\!$p$) \!one \!can \!verify \!that \!Mazur's \!use \!of \!Schlessinger's\linebreak criteria [Sch] proves the existence of a
universal deformation $$\rho:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(R).$$
In the ordinary and restricted
case this was proved by Mazur and in the\linebreak flat case by Ramakrishna [Ram]. The other cases require minor modifications
of Mazur's argument. We denote the universal ring $R_\Sigma$ in the unrestricted case and
$R^\roman{se}_\Sigma,R^\roman{ord}_\Sigma,R^\roman{str}_\Sigma,R^\roman f_\Sigma$ in the other four cases. We often omit the
$\Sigma$ if the context makes it clear.
There are certain generalizations to all of the above which we will also need. The first is that instead of considering
$W(k)$-algebras $A$ we may consider $\Cal O$-algebras for $\Cal O$ the ring of integers of any local field with residue field
$k$. If we need to record which $\Cal O$ we are using we will write $R_{\Sigma,\Cal O}$ etc. It is easy to see that the
natural local map of local $\Cal O$-algebras $$R_{\Sigma,\Cal O}\rightarrow R_\Sigma\mathop{\otimes}\limits_{W(k)}\Cal O$$ is
an isomorphism because for functorial reasons the map has a natural section which induces an isomorphism on Zariski tangent
spaces at closed points, and one can then use Nakayama's lemma. Note, however, hat if we change the residue field via
$i:\hookrightarrow k'$ then we have a new deformation problem associated to the representation $\rho'_0=i\circ\rho_0.$ There
is again a natural map of $W(k')$-\linebreak algebras $$R(\rho'_0)\rightarrow R\mathop{\otimes}\limits_{W(k)}W(k')$$ which is
an isomorphism on Zariski tangent spaces. One can check that this\linebreak is again an isomorphism by considering the subring
$R_1$ of
$R(\rho'_0)$ defined as the\linebreak subring of all elements whose reduction modulo the maximal ideal lies in $k$. Since
$R(\rho'_0)$ is a finite $R_1$-module, $R_1$ is also a complete local Noetherian ring\linebreak
\eject
\noindent with residue field $k$. The universal representation associated to $\rho'_0$ is defined over $R_1$ and the
universal property of $R$ then defines a map $R\rightarrow R_1.$ So we obtain a section to the map $R(\rho'_0)\rightarrow
R\mathop{\otimes}\limits_{W(k)}W(k')$ and the map is therefore an isomorphism. (I am grateful to Faltings for this
observation.) We will also need to extend the consideration of $\Cal O$-algebras tp the restricted cases. In each case we can
require $A$ to be an $\Cal O$-algebra and again it is easy to see that $R^\cdot_{\Sigma,\Cal O}\simeq
R^\cdot_\Sigma\mathop{\otimes}\limits_{W(k)}\Cal O$ in each case.
The second generalization concerns primes $q\ne p$ which are ramified in $\rho_0$. We distinguish three special cases (types
(A) and (C) need not be disjoint):
\vskip6pt\hskip-12pt(A) $\rho_0|_{D_q}=({\chi_1\atop{}}{*\atop\chi_2})$ for a suitable choice of basis, with $\chi_1$ and
$\chi_2$ unramified,\linebreak
${}$\hskip26pt$\chi_1\chi_2^{-1}=\omega$ and the fixed space of $I_q$ of dimension 1,
\vskip6pt\hskip-12pt(B) $\rho_0|_{I_q}=({\chi_q\atop0}{0\atop1}),\chi_q\ne1,$ for a suitable choice of basis,
\vskip6pt\hskip-12pt(C) $H^1(\bold Q_q,W_\lambda)=0$ where $W_\lambda$ is as defined in (1.6).
\vskip6pt
Then in each case we can define a suitable deformation theory by imposing additional restrictions on those we have already
considered, namely:
\vskip6pt
\hskip-12pt(A) $\rho|_{D_q}=({\psi_1\atop{}}{*\atop\psi_2})$ for a suitable choice of basis of $A^2$ with $\psi_1$ and
$\psi_2$ un-\linebreak
${}$\hskip27pt ramified and $\psi_1\psi_2^{-1}=\varepsilon;$
\vskip6pt\hskip-11pt(B) $\rho|_{I_q}=({\chi_q\atop0}{0\atop1})$ for a suitable choice of basis ($\chi_q$ of order prime to
$p$, so the\linebreak
${}$\hskip26pt {\it same} character as above);
\vskip6pt
\hskip-12pt(C) $\det\rho|_{I_q}=\det\rho_0|_{I_q},$ i.e., of order prime to $p.$
\vskip6pt
\noindent Thus if $\Cal M$ is a set of primes in $\Sigma$ distinct from $p$ and each satisfying one of (A), (B) or (C) for
$\rho_0$, we will impose the corresponding restriction at each prime in $\Cal M$.
Thus to each set of data $\Cal D=\{\cdot,\Sigma,\Cal O,\Cal M\}$ where $\cdot$ is Se, str, ord, flat or unrestricted, we can
associate a deformation theory to $\rho_0$ provided $$\rho_0:\roman{Gal}(\bold Q_\Sigma/\bold
Q)\rightarrow\roman{GL}_2(k)\leqno(1.3)$$ is itself of type $\Cal D$ and $\Cal O$ is the ring of integers of a totally
ramified extension of $W(k);\rho_0$ is ordinary if $\cdot$ is Se or ord, strict if $\cdot$ is strict and flat if $\cdot$ is
fl (meaning flat); $\rho_0$ is of type $\Cal M$, i.e., of type (A), (B) or (C) at each ramified primes $q\ne p,q\in\Cal M.$
We allow different types at different $q$'s. We will refer\linebreak to these as the standard deformation theories and write
$R_\Cal D$ for the universal ring associated to $\Cal D$ and $\rho_\Cal D$ for the universal deformation (or even $\rho$ if
$\Cal D$ is clear from the context).
We note here that if $\Cal D=(\roman{ord},\Sigma,\Cal O,\Cal M)$ and $\Cal D'=(\roman{Se},\Sigma,\Cal O,\Cal M)$ then there is
a simple relation between $R_\Cal D$ and $R_{\Cal D'}.$ Indeed there is a natural map\linebreak
\eject
\noindent$R_\Cal D\rightarrow R_{\Cal D'}$ by the universal property of $R_\Cal D$, and its kernel is a principal ideal
generated by $T=\varepsilon^{-1}(\gamma)\det\rho_\Cal D(\gamma)-1$ where $\gamma\in\roman{Gal}(\bold Q_\Sigma/\bold Q)$ is
any element whose restriction to $\roman{Gal}(\bold Q_\infty/\bold Q)$ is a generator (where $\bold Q_\infty$ is the $\bold
Z_p$-extension of $\bold Q$) and whose restriction to $\roman{Gal}(\bold Q(\zeta_{N_p})/\bold Q)$ is trivial for any $N$
prime to $p$ with $\zeta_N\in\bold Q_\Sigma,\zeta_N$ being a primitive $N^\roman{th}$ root of 1: $$R_\Cal
D/T\simeq R_\Cal D'.\leqno(1.4)$$
It turns out that under the hypothesis that $\rho_0$ is strict, i.e. that $\rho_0|_{D_p}$ is not associated to a finite flat
group scheme, the deformation problems in (i)(a) and (i)(c) are the same; i.e., every Selmer deformation is already a strict
deformation. This was observed by Diamond. the argument is local, so the decomposition group $D_p$ could be replaced by
$\roman{Gal}(\bar\bold Q_p/\bold Q).$
\
{\smc Proposition 1.1} (Diamond). {\it Suppose that $\pi:D_p\rightarrow\roman{GL}_2(A)$ is a continuous representation where
$A$ is an Artinian local ring with residue field $k$, a finite field of characteristic $p.$ Suppose
$\pi\approx({\chi_1\varepsilon\atop0}{*\atop\chi_2})$ with $\chi_1$ and $\chi_2$ unramified and $\chi_1\ne\chi_2$. Then the
residual representation $\bar\pi$ is associated to a finite flat group scheme over $\bold Z_p$.\vskip6pt
Proof} (taken from [Dia, Prop. 6.1]). We may replace $\pi$ by $\pi\otimes\chi_2^{-1}$ and we let $\varphi=\chi_1\chi_2^{-1}.$
Then $\pi\cong({\varphi\varepsilon\atop0}{t\atop1})$ determines a cocycle $t:D_p\rightarrow M(1)$ where $M$ is a free
$A$-module of rank one on which $D_p$ acts via $\varphi$. Let $u$ denote the cohomology class in $H^1(D_p,M(1))$ defined by
$t$, and let $u_0$ denote its image\linebreak in $H^1(D_p,M_0(1))$ where $M_0=M/\frak mM.$ Let $G=\ker\varphi$ and let $F$ be
the fixed field of $G$ (so $F$ is a finite unramified extension of $\bold Q_p$). Choose $n$ so that $p^nA\mathbreak=0.$ Since
$H^2(G,\mu_{p^r}\rightarrow H^2(G,\mu_{p^s})$ is injective for $r\le s,$ we see that the natural map of $A[D_p/G]$-modules
$H^1(G,\mu_{p^n}\otimes_{\bold Z_p}M)\rightarrow H^1(G,M(1))$ is an isomorphism. By Kummer theory, we have $H^1(G,M(1))\cong F
^\times/(F^\times)^{p^n}\otimes_{\bold Z_p}M$ as $D_p$-modules. Now consider the commutative diagram
$$\matrix \hskip-6pt H^1(G,M(1))^{D_p}&\hskip-10pt\mathop{\hbox to
32pt{\rightarrowfill}}\limits^\sim&\hskip-10pt((F^\times/(F^\times)^{p^n}\otimes_{\bold Z_p}M)^{D_p}&\hskip-10pt{\hbox to
32pt{\rightarrowfill}}&\hskip-10pt M^{D_p}\\
\hskip-6pt\Bigg\downarrow&&\hskip-10pt\Bigg\downarrow&&\hskip-10pt\ \ \Bigg\downarrow\ \ ,\\
\hskip-6pt H^1(G,M_0(1))&\hskip-10pt\mathop{\hbox to
32pt{\rightarrowfill}}\limits^\sim&\hskip-10pt(F^\times/(F^\times)^p)\otimes_{\bold F_p}M_0&\hskip-10pt{\hbox to
32pt{\rightarrowfill}}&\hskip-10pt M_0\endmatrix$$
where the right-hand horizontal maps are induced by $v_p:F^\times\rightarrow\bold Z.$ If $\varphi\ne1,$ then
$M^{D_p}\subset\frak mM,$ so that the element res \!$u_0$ of $H^1(G,M_0(1))$ is in the image of $(\Cal O^\times_F/(\Cal
O^\times_F)^p)\otimes_{\bold F_p}M_0.$ But this means that $\bar\pi$ is ``peu ramifi\'e'' in the sense of [Se] and therefore
$\bar\pi$ comes from a finite flat group scheme. (See [E1, (8.20].)
\
{\it Remark.} Diamond also observes that essentially the same proof shows that if $\pi:\roman{Gal}(\bar\bold Q_q/\bold
Q_q)\rightarrow\roman{GL}_2(A),$ where $A$ is a complete local Noetherian\linebreak
\eject
\noindent ring with residue field $k$, has the form $\pi|_{I_q}\cong({1\atop0}{*\atop1})$ with $\bar\pi$ ramified then $\pi$
is of type (A).
\
Globally, Proposition 1.1 says that if $\rho_0$ is strict and if $\Cal D=(\roman{Se},\Sigma,\Cal O,\Cal M)$ and $\Cal
D'=(\roman{str}, \Sigma,\Cal O,\Cal M)$ then the natural map $R_\Cal D\rightarrow R_{\Cal D'}$ is an isomorphism.
In each case the tangent space of $R_\Cal D$ may be computed as in [Ma1]. Let
$\lambda$
be a uniformizer for $\Cal O$ and let $U_\lambda\simeq k^2$ be the representation space for $\rho_0.$ (The motivation for
the subscript $\lambda$ will become apparent later.) Let $V_\lambda$ be the\linebreak representation space of
$\roman{Gal}(\bold Q_\Sigma/\bold Q)$ on $\roman{Ad}\rho_0=\roman{Hom}_k(U_\lambda,U_\lambda)\simeq M_2(k).$ Then there is an
isomorphism of $k$-vector spaces (cf. the proof of Prop. 1.2 below) $$\roman{Hom}_k(m_\Cal D/(m^2_\Cal
D,\lambda),k)\simeq H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_\lambda)\leqno(1.5)$$ where $H^1_\Cal
D(\roman{Q}_\Sigma/\roman{Q},V_\lambda)$ is a subspace of $H^1(\bold Q_\Sigma/\bold Q,V_\lambda)$ which we now describe and
$m_\Cal D$ is the maximal ideal of $R_Cal D$. It consists of the cohomology classes which satisfy certain local restrictions
at $p$ and at the primes in $\Cal M$. We call $m_\Cal D/(m^2_\Cal D,\lambda)$ the reduced cotangent space of $R_\Cal D$.
We begin with $p$. First we may write (since $p\ne2$), as $k[\roman{Gal}(\bold Q_\Sigma/\bold Q)]$-modules,
$$\leqalignno{V_\lambda=W_\lambda\oplus k,\ \roman{where}\
W_\lambda&=\{f\in\roman{Hom}_k(U_\lambda,U_\lambda):\roman{trace}f=0\}&(1.6)\cr&\simeq
(\roman{Sym}^2\otimes\roman{det}^{-1})\rho_0}$$ and $k$ is the one-dimensional subspace of scalar multiplications. Then if
$\rho_0$\linebreak is ordinary the action of $D_p$ on $U_\lambda$ induces a filtration of $U_\lambda$ and also on $W_\lambda$
and $V_\lambda$. Suppose we write these $0\subset U^0_\lambda\subset U_\lambda,\ 0\subset W^0_\lambda\subset
W^1_\lambda\subset W_\lambda$ and $0\subset V^0_\lambda\subset V^1_\lambda\subset V_\lambda.$ Thus $U^0_\lambda$ is defined
by the requirement that $D_p$ act on it via the character $\chi_1$ (cf. (1.2)) and on $U_\lambda/U^0_\lambda$ via $\chi_2$.
For $W_\lambda$ the filtrations are defined by
$$\aligned W^1_\lambda\ \ \ &=\ \ \ \{f\in W_\lambda:f(U^0_\lambda)\subset
U^0_\lambda\},\\ W^0_\lambda\ \ \ &=\ \ \ \{f\in W^1_\lambda:f=0\ \roman{on}\ U^0_\lambda\},\endaligned$$
and the filtrations for $V_\lambda$ are obtained by replacing $W$ by $V$. We note that these filtrations are often
characterized by the action of $D_p.$ Thus the action\linebreak of $D_p$ on $W^0_\lambda$ is via $\chi_1/\chi_2$; on
$W^1_\lambda/W^0_\lambda$ it is trivial and on $Q_\lambda/W^1_\lambda$ it is via $\chi_2/\chi_1$. These determine the
filtration if either $\chi_1/\chi_2$ is not quadratic or $\rho_0|_{D_p}$ is not semisimple. We define the $k$-vector spaces
$$\aligned V^\roman{ord}_\lambda&=\{f\in V^1_\lambda:f=0\ \ \roman{in}\ \
\roman{Hom}(U_\lambda/U^0_\lambda,U_\lambda/U^0_\lambda)\},\\
H^1_\roman{Se}(\bold Q_p,V_\lambda)&=\roman{ker}\{H^1(\bold Q_p,V_\lambda)\rightarrow H^1(\bold
Q^\roman{unr}_p,V_\lambda/W^0_\lambda)\},\\
H^1_\roman{ord}(\bold Q_p,V_\lambda)&=\roman{ker}\{H^1(\bold Q_p,V_\lambda)\rightarrow H^1(\bold
Q^\roman{unr}_p,V_\lambda/V^\roman{ord}_\lambda)\},\\
H^1_\roman{str}(\bold Q_p,V_\lambda)&=\roman{ker}\{H^1(\bold Q_p,V_\lambda)\rightarrow H^1(\bold
Q_p,W_\lambda/W^0_\lambda)\oplus H^1(\bold Q^\roman{unr}_p,k)\}.
\endaligned$$
\eject
In the Selmer case we make an analogous definition for $H^1_\roman{Se}(\bold Q_p,W_\lambda)$ by replacing $V_\lambda$ by
$W_\lambda$, and similarly in the strict case. In the flat case we use the fact that there is a natural isomorphism of
$k$-vector spaces $$H^1(\bold Q_p,V_\lambda)\rightarrow\roman{Ext}^1_{k[D_p]}(U_\lambda,U_\lambda)$$ where the extensions are
computed in the category of $k$-vector spaces with local\linebreak Galois action. Then $H^1_\roman{f}(\bold Q_p,V_\lambda)$ is
defined as the $k$-subspace of $H^1(\bold Q_p,V_\lambda)$ which is the inverse image of $\roman{Ext}^1_\roman{fl}(G,G),$ the
group of extensions in the category of finite flat commutative group schemes over $\bold Z_p$ killed by $p,G$ being the
(unique) finite flat group scheme over $\bold Z_p$ associated to $U_\lambda$. By [Ray1] all such extensions in the inverse
image even correspond to $k$-vector space schemes. For more details and calculations see [Ram].
For $q$ different from $p$ and $q\in\Cal M$ we have three cases (A), (B), (C). In\linebreak case (A) there is a filtration by
$D_q$ entirely analogous to the one for $p$. We write this $0\subset W^{0,q}_\lambda\subset W^{1,q}_\lambda\subset W_\lambda$ and
we set
$$H^1_{D_q}(\bold Q_q,V_\lambda)=\cases\roman{ker}:H^1(\bold Q_q,V_\lambda\\
\hskip0.18in\rightarrow H^1(\bold Q_q,W_\lambda/W^{0,q}_\lambda)\oplus H^1(\bold Q^\roman{unr}_q,k)\hskip0.06in \roman{in\
case\ (A)}\\
\ \\
\roman{ker}:H^1(\bold Q_q,V_\lambda)\\
\hskip0.18in\rightarrow H^1(\bold Q^\roman{unr}_q,V_\lambda)\hskip1.26668in\roman{in\ case\ (B)\ or\ (C).}
\endcases$$
Again we make an analogous definition for $H^1_{D_q}(\bold Q_q,W_\lambda)$ by replacing $V_\lambda$\linebreak by $W_\lambda$
and deleting the last term in case (A). We now define the $k$-vector space $H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_\lambda)$ as
$$\aligned H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_\lambda)=\{\alpha\in H^1(\bold Q_\Sigma/\bold
Q,V_\lambda):&\ \ \alpha_q\in H^1_{D_q}(\bold Q_q, V_\lambda)\ \roman{for\ all}\ q\in\Cal M,\\
&\ \ \alpha_q\in H^1_*(\bold Q_p,V_\lambda)\}\endaligned$$
where $*$ is Se, str, ord, fl or unrestricted according to the type of $\Cal D$. A similar definition applies to $H^1_\Cal
D(\bold Q_\Sigma/\bold Q, W_\lambda)$ if $\cdot$ is Selmer or strict.
Now and for the rest of the section we are going to assume that $\rho_0$ arises from the reduction of the $\lambda$-adic
representation associated to an eigenform. More precisely we assume that there is a normalized eigenform $f$ of weight 2 and
level $N$, divisible only by the primes in $\Sigma$, and that there ia a prime $\lambda$ of $\Cal O_f$ such that
$\rho_0=\rho_{f,\lambda}\mod\lambda.$ Here $\Cal O_f$ is the ring of integers of the field generated by the Fourier
coefficients of $f$ so the fields of definition of the two representations need not be the same. However we assume that
$k\supseteq\Cal O_{f,\lambda}/\lambda$ and we fix such an embedding so the comparison can be made over $k$. It will be
convenient moreover to assume that if we are considering $\rho_0$ as being of type $\Cal D$ then $\Cal D$ is defined using
$\Cal O$-algebras where $\Cal O\supseteq\Cal O_{f,\lambda}$ is an unramified extension whose residue field is $k$. (Although
this condition is unnecessary, it is convenient to use $\lambda$ as the uniformizer for $\Cal O$.) Finally we assume that
$\rho_{f,\lambda}$\linebreak
\eject
\noindent itself is of type $\Cal D$. Again this is a slight abuse of terminology as we are really considering the extension
of scalars $\rho_{f,\lambda}\mathop{\otimes}\limits_{\Cal O_{f,\lambda}}\Cal O$ and not $\rho_{f,\lambda}$ itself, but we
will do this without further mention if the context makes it clear. (The analysis of this section actually applies to any
characteristic zero lifting of $\rho_0$ but in all our applications we will be in the more restrictive context we have
described here.)
With these hypotheses there is a unique local homomorphism $R_\Cal D\rightarrow\Cal O$ of $\Cal O$-algebras which takes the
universal deformation to (the class of) $\rho_{f,\lambda}.$ Let $\frak p_\Cal D=\ker:R_\Cal D\rightarrow\Cal O.$ Let $K$ be
the field of fractions of $\Cal O$ and let $U_f=(K/\Cal O)^2$ with the Galois action taken from $\rho_{f,\lambda}$.
Similarly, let $V_f=\roman{Ad}\rho_{f,\lambda}\otimes_\Cal OK/\Cal O\simeq(K/\Cal O)^4$ with the adjoint representation so
that $$V_f\simeq W_f\oplus K/\Cal O$$ where $W_f$ has Galois action via
$\roman{Sym}^2\rho_{f,\lambda}\otimes\det\rho^{-1}_{f,\lambda}$ and the action on the second factor is trivial. Then if
$\rho_0$ is ordinary the filtration of $U_f$ under the $\roman{Ad}\rho$ action of $D_p$ induces one on $W_f$ which we write
$0\subset W^0_f\subset W^1_f\subset W_f.$ Often to simplify the notation we will drop the index $f$ from $W^1_f,V_f$ etc.
There is also a filtration on $W_{\lambda^n}=\{\ker\lambda^n:W_f\rightarrow W_f\}$ given by
$W^i_{\lambda^n}=W^{\lambda^n}\cap W^i$ (compatible with our previous description for $n=1$). Likewise we write
$V_{\lambda^n}$ for $\{\ker\lambda^n:V_f\rightarrow V_f\}.$
We now explain how to extend the definition of $H^1_\Cal D$ to give meaning to $H^1_\Cal D(\bold Q_\Sigma/\bold
Q,V_{\lambda^n})$ and $H^1_\Cal D(\Cal Q_\Sigma/\bold Q,V)$ and these are $\Cal O/\lambda^n$ and $\Cal O$-modules,
respectively. In the case where $\rho_0$ is ordinary the definitions are the same with $V_{\lambda^n}$ or $V$ replacing
$V_\lambda$ and $\Cal O/\lambda^n$ or $K/\Cal O$ replacing $k$. One checks easily that as $\Cal O$-modules
$$H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_{\lambda^n})\simeq H^1_\Cal D(\bold Q_\Sigma/\bold Q,V)_{\lambda^n},\leqno(1.7)$$
where as usual the subscript $\lambda^n$ denotes the kernel of multiplication by $\lambda^n$. This just uses the divisibility
of $H^0(\bold Q_\Sigma/\bold Q,V)$ and $H^0(\bold Q_p,W/W^0)$ in the strict case. In the Selmer case one checks that for $m\gt
n$ the kernel of
$$H^1(\bold Q^\roman{unr}_p,V_{\lambda^n}/W^0_{\lambda^n})\rightarrow H^1(\bold
Q^\roman{unr}_p,V_{\lambda^m}/W^0_{\lambda^m})$$
has only the zero element fixed under $\roman{Gal}(\bold Q^\roman{unr}_p/\bold Q_p)$ and the ord case is similar. Checking
conditions at $q\in\Cal M$ is dome with similar arguments. In the Selmer and strict cases we make analogous definitions with
$W_{\lambda^n}$ in place of $V_{\lambda^n}$ and $W$ in place of $V$ and the analogue of (1.7) still holds.
We now consider the case where $\rho_0$ is flat (but not ordinary). We claim first that there is a natural map of $\Cal
O$-modules
$$H^1(\bold Q_p,V_{\lambda_n})\rightarrow\roman{Ext}^1_{\Cal O[D_p]}(U_{\lambda^m},U_{\lambda^n})\leqno(1.8)$$
for each $m\ge n$ where the extensions are of $\Cal O$-modules with local Galois\linebreak action. To describe this suppose
that
$\alpha\in H^1(\bold Q_p,V_{\lambda^n}).$ Then we can associate to $\alpha$ a representation
$\rho_\alpha:\roman{Gal}(\bar\bold Q_p/\bold Q_p)\rightarrow\roman{GL}_2(\Cal O_n[\varepsilon])$ (where $\Cal
O_n[\varepsilon]=$\linebreak
\eject
\noindent$\Cal O[\varepsilon]/(\lambda^n\varepsilon,\varepsilon^2))$ which is an $\Cal O$-algebra deformation of $\rho_0$
(see the proof of Proposition 1.1 below). Let $E=\Cal O_n[\varepsilon]^2$ where the Galois action is via $\rho_\alpha.$ Then
there is an exact sequence
$$\matrix0&\longrightarrow&\varepsilon E/\lambda^m&\longrightarrow&
E/\lambda^m&\longrightarrow&(E/\varepsilon)/\lambda^m&\longrightarrow&0\\
\ \\
&&|\wr&&&&|\wr\\
\ \\
&&U_{\lambda^n}&&&&U_{\lambda^m}\endmatrix$$
and hence an extension class in $\roman{Ext}^1(U_{\lambda^m},U_{\lambda^n}).$ One checks now that (1.8) is a map of $\Cal
O$-modules. We define $H^1_f(\bold Q_p,V_{\lambda^n})$ to be the inverse image of
$\roman{Ext}^1_\roman{fl}(U_{\lambda^n},U_{\lambda_n})$ under (1.8), i.e., those extensions which are already extensions in
the category of finite flat group schemes $\bold Z_p.$ Observe that
$\roman{Ext}^1_\roman{fl}(U_{\lambda^n},U_{\lambda^n})\cap\roman{Ext}^1_{\Cal O[D_p]}(U_{\lambda^n},U_{\lambda^n})$ is an
$\Cal O$-module, so $H^1_\roman f(\bold Q_p,V_{\lambda^n})$ is seen to be an $\Cal O$-sub-module of $H^1(\bold
Q_p,V_{\lambda_n}).$ We observe that our definition is equivalent to requiring that the classes in $H^1_\roman f(\bold
Q_p,V_{\lambda^n})$ map under (1.8) to $\roman{Ext}^1_\roman{fl}(U_{\lambda^m},U_{\lambda^n})$ for all $m\ge n.$ For if $e_m$
is the extension class in $\roman{Ext}^1(U_{\lambda^m},U_{\lambda^n})$ then $e_m\hookrightarrow e_n\oplus U_{\lambda^m}$ as
Galois-modules and we can apply results of [Ray1] to see that $e_m$ comes from a finite flat group scheme over $\bold Z_p$ if
$e_n$ does.
In the flat (non-ordinary) case $\rho_0|_{I_p}$ is determined by Raynaud's results as mentioned at the beginning of the
chapter. It follows in particular that, since $\rho_0|_{D_p}$ is absolutely irreducible, $V(\bold Q_p=H^0(\bold Q_p,V)$ is
divisible in this case\linebreak (in fact $V(\bold Q_p)\simeq KT/\Cal O).$ This $H^1(\bold Q_p,V_{\lambda^n})\simeq H^1(\bold
Q_p,V)_{\lambda^n}$ and hence we can\linebreak define
$$H^1_\roman f(\bold Q_p,V)=\bigcup_{n=1}^\infty H^1_\roman f(\bold Q_p,V_{\lambda^n}),$$
and we claim that $H^1_\roman f(\bold Q_p,V)_{\lambda^n}\simeq H^1_\roman f(\bold Q_p,V_{\lambda^n}).$ To see this we have to
compare representations for $m\ge n,$
$$\matrix\rho_{n,m}:\roman{Gal}(\bar\bold Q_p/\bold Q_p)&\longrightarrow&\roman{GL}_2(\Cal O_n[\varepsilon]/\lambda^m)\\
\ \\
\Big\Vert&&\hskip0.25in\Big\downarrow{\scriptstyle\varphi_{m,n}}\\
\ \\
\hskip-0.03in\rho_{m,m}:\roman{Gal}(\bar\bold Q_p/\bold Q_p)&\longrightarrow&\roman{GL}_2(\Cal
O_m[\varepsilon]/\lambda^m)\endmatrix$$
where $\rho_{n,m}$ and $\rho_{m,m}$ are obtained from $\alpha_n\in H^1(\bold Q_p,VX_{\lambda^n})$ and
$\roman{im}(\alpha_n)\in H^1(\bold Q_p,V_{\lambda^m})$ and $\varphi_{m,n}:a+b\varepsilon\rightarrow
a+\lambda^{m-n}b\varepsilon.$ By [Ram, Prop 1.1 and Lemma 2.1] if $\rho_{n,m}$ comes from a finite flat group scheme then so
does $\rho_{m,m}$. Conversely $\varphi_{m,n}$ is injective and so $\rho_{n,m}$ comes from a finite flat group scheme if
$\rho_{m,m}$ does; cf. [Ray1]. The definitions of $H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_{\lambda^n})$ and $H^1_\Cal D(\bold
Q_\Sigma/\bold Q,V)$ now extend to the flat case and we note that (1.7) is also valid in the flat case.
Still in the flat (non-ordinary) case we can again use the determination of $\rho_0|_{I_p}$ to see that $H^1(\bold Q_p,V)$ is
divisible. For it is enough to check that $H^2(\bold Q_p,V_\lambda)=0$ and this follows by duality from the fact that
$H^0(\bold Q_p,V^*_\lambda)=0$\linebreak
\eject
\noindent where $V^*_\lambda=\roman{Hom}(V_\lambda,\boldsymbol\mu_p)$ and $\boldsymbol\mu_p$ is the group of $p^\roman{th}$
roots of unity. (Again this follows from the explicit form of $\rho_0|_{{}_{\scriptstyle{D_p}}}$.) Much
subtler is the fact that
$H^1_\roman f(\bold Q_p,V)$ is divisible. This result is essentially due to Ramakrishna. For, using a local version of
Proposition 1.1 below we have that
$$\roman{Hom}_\Cal O(\frak p_R/\frak p^2_R,K/\Cal O)\simeq H^1_\roman f(\bold Q_p,V)$$
where $R$ is the universal local flat deformation ring for $\rho_0|_{D_p}$ and $\Cal O$-algebras. (This exists by Theorem 1.1
of [Ram] because $\rho_0|_{D_p}$ is absolutely irreducible.) Since $R\simeq R^\roman{fl}\mathop{\otimes}\limits_{W(k)}\Cal O$
where
$R^\roman{fl}$ is the corresponding ring for $W(k)$-algebras the main theorem of [Ram, Th. 4.2] shows that $R$ is a power
series ring and the divisibility of $H^1_\roman f(\bold Q_p,V)$ then follows. We refer to [Ram] for more details about
$R^\roman{fl}.$
Next we need an analogue of (1.5) for $V$. Again this is a variant of standard results in deformation theory and is given (at
least for $\Cal D=(\roman{ord},\Sigma,W(k),\phi)$ with some restriction on $\chi_1,\chi_2$ in i(a)) in [MT, Prop 25].
\
{\smc Proposition 1.2.} {\it Suppose that $\rho_{f,\lambda}$ is a deformation of $\rho_0$ of type\linebreak
$\Cal D=(\cdot,\Sigma,\Cal O,\Cal M)$ with $\Cal O$ an unramified extension of $\Cal O_{f,\lambda}$. Then as $\Cal O$-modules
$$\roman{Hom}_\Cal O(\frak p_\Cal D/\frak p^2_\Cal D,K/\Cal O)\simeq H^1_\Cal D(\bold Q_\Sigma/\bold Q,V).$$
Remark.} The isomorphism is functorial in an obvious way if one changes $\Cal D$ to a larger $\Cal D'$.
\
{\it Proof.} We will just describe the Selmer case with $\Cal M=\phi$ as the other cases use similar arguments. Suppose that
$\alpha$ is a cocycle which represents a cohomology class in $H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,V_{\lambda^n}).$ Let
$\Cal O_n[\varepsilon]$ denote the ring $\Cal O[\varepsilon]/(\lambda^n\varepsilon,\varepsilon^2).$ We can associate to
$\alpha$ a representation
$$\rho_\alpha:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(\Cal O_n[\varepsilon])$$
as follows: set $\rho_\alpha(g)=\alpha(g)\rho_{f,\lambda}(g)$ where $\rho_{f,\lambda}(g),$ {\it a priori} in
$\roman{GL}_2(\Cal O)$, is viewed\linebreak in $\roman{GL}_2(\Cal O_n[\varepsilon])$ via the natural mapping $\Cal
O\rightarrow\Cal O_n[\varepsilon].$ Here a basis for $\Cal O^2$ is chosen so that the representation $\rho_{f,\lambda}$ on
the decomposition group $D_p\subset\roman{Gal}(\bold Q_\Sigma/\bold Q)$ has the upper triangular form of (i)(a), and then
$\alpha(g)\in V_{\lambda^n}$ is\linebreak viewed in $\roman{GL}_2(\Cal O_n[\varepsilon])$ by identifying
$$V_{\lambda_n}\simeq\bigg\{\pmatrix1+y\varepsilon&x\varepsilon\\
z\varepsilon&1-t\varepsilon\endpmatrix\bigg\}=\{\ker:\roman{GL}_2(\Cal O_n[\varepsilon])\rightarrow\roman{GL}_2(\Cal O)\}.$$
Then
$$W^0_{\lambda^n}=\bigg\{\pmatrix1&x\varepsilon\\
&1\endpmatrix\bigg\},$$
\eject
$$\aligned W^1_{\lambda^n}&=\bigg\{\pmatrix 1+y\varepsilon&x\varepsilon\\
&1-y\varepsilon\endpmatrix\bigg\},\\
W_{\lambda^n}&=\bigg\{\pmatrix1+y\varepsilon&x\varepsilon\\
z\varepsilon&1-y\varepsilon\endpmatrix\bigg\},\endaligned$$
and
$$V^1_{\lambda^n}=\bigg\{\pmatrix1+y\varepsilon&x\varepsilon\\
&1-t\varepsilon\endpmatrix\bigg\}.$$
One checks readily that $\rho_\alpha$ is a continuous homomorphism and that the deformation $[\rho_\alpha]$ is unchanged if we
add a coboundary to $\alpha.$
We need to check that $[\rho_\alpha]$ is a Selmer deformation. Let $\Cal H=\mathbreak\roman{Gal}(\bar\bold Q_p/\bold
Q^\roman{unr}_p)$ and $\Cal G=\roman{Gal}(\bold Q^\roman{unr}_p/\bold Q_p).$ Consider the exact sequence of $\Cal O[\Cal
G]$-modules
$$0\rightarrow(V^1_{\lambda^n}/W^0_{\lambda^n})^\Cal H\rightarrow(V_{\lambda^n}/W^0_{\lambda^n})^\Cal H\rightarrow
X\rightarrow0$$
where $X$ is a submodule of $(V_{\lambda^n}/V^1_{\lambda^n})^\Cal H.$ Since the action of $_p$ on
$V_{\lambda^n}/V^1_{\lambda^n}$ is\linebreak via a character which is nontrivial mod $\lambda$ (it equals $\chi_2\chi_1^{-1}$
mod $\lambda$ and $\chi_1\not\equiv\chi_2),$ we see that $X^\Cal G=0$ and $H^1(\Cal G,X)=0.$ Then we have an exact diagram of
$\Cal O$-modules
$$\matrix0\\
\Bigg\downarrow\\
\hskip9pt H^1(\Cal G,(V^1_{\lambda^n}/W^0_{\lambda^n})^\Cal H)\simeq H^1(\Cal G,(V_{\lambda^n}/W^0_{\lambda^n})^\Cal
H)\\
\Bigg\downarrow\\
H^1(\bold Q_p,V_{\lambda^n}/W^0_{\lambda^n})\\
\Bigg\downarrow\\
H^1(\bold Q^\roman{unr}_p,V_{\lambda^n}/W^0_{\lambda^n})^\Cal G.\endmatrix$$
By hypothesis the image of $\alpha$ is zero in $H^1(\bold Q^\roman{unr}_p,V_{\lambda^n}/W^0_{\lambda^n})^\Cal G.$ Hence
it is in the image of $H^1(\Cal G,(V^1_{\lambda^n}/W^0_{\lambda^n})^\Cal H).$ Thus we can assume that it is
represented in
$H^1(\bold Q_p,V_{\lambda^n}/W^0_{\lambda^n})$ by a cocycle, which maps $\Cal G$ to $V^1_{\lambda^n}/W^0_{\lambda^n};$
i.e.,\linebreak $f(D_p)\subset V^1_{\lambda^n}/W^0_{\lambda^n},f(I_p)=0.$ The difference between $f$ and the image of
$\alpha$ is a coboundary
$\{\sigma\mapsto\sigma\bar\mu-\bar\mu\}$ for some
$u\in V_{\lambda^n}.$ By subtracting the coboundary $\{\sigma\mapsto\sigma u-u\}$ from $\alpha$ globally we get a new
$\alpha$ such that $\alpha= f$ as cocycles mapping $\Cal G$ to $V^1_{\lambda^n}/W^0_{\lambda^n}.$ Thus $\alpha(D_p)\subset
V^1_{\lambda^n},\alpha(I_p)\subset W^0_{\lambda^n}$ and it is now easy to check that $[\rho_\alpha]$ is a Selmer deformation
of $\rho_0$.
Since $[\rho_\alpha]$ is a Selmer deformation there is a unique map of local $\Cal O$-\linebreak algebras
$\varphi_\alpha:R_\Cal D\rightarrow\Cal O_n[\varepsilon]$ inducing it. (If $\Cal M\ne\phi$ we must check the\linebreak
\eject
\noindent other conditions also.) Since $\rho_\alpha\equiv\rho_{f,\lambda}$ mod $\varepsilon$ we see that restricting
$\varphi_\alpha$ to $\frak p_\Cal D$ gives a homomorphism of $\Cal O$-modules,
$$\varphi_\alpha:\frak p_\Cal D\rightarrow\varepsilon.\Cal O/\lambda^n$$
such that $\varphi_\alpha(\frak p^2_\Cal D)=0.$ Thus we have defined a map $\varphi:\alpha\rightarrow\varphi_\alpha,$
$$\varphi:H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,V_{\lambda^n})\rightarrow\roman{Hom}_\Cal O(\frak p_\Cal D/\frak p^2_\Cal
D,\Cal O/\lambda^n).$$
It is straightforward to check that this is a map of $\Cal O$-modules. To check the injectivity of $\varphi$ suppose that
$\varphi_\alpha(\frak p_\Cal D)=0.$ Then $\varphi_\alpha$ factors through $R_\Cal D/\frak p_\Cal D\simeq\Cal O$ and being an
$\Cal O$-algebra homomorphism this determines $\varphi_\alpha.$ Thus $[\rho_{f,\lambda}]=[\rho_\alpha].$ If
$A^{-1}\rho_\alpha A=\rho_{f,\lambda}$ then $A$ mod $\varepsilon$ is seen to be central by Schur's lemma and so may be taken
to be $I$. A simple calculation now shows that $\alpha$ is a coboundary.
To see that $\varphi$ is surjective choose
$$\Psi\in\roman{Hom}_\Cal O(\frak p_\Cal D/\frak p^2_\Cal D,\Cal O/\lambda^n).$$
Then $\rho_\Psi:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(R_\Cal D/(\frak p^2_\Cal D,\ker\Psi))$ is induced
by a representative of the universal deformation (chosen to equal $\rho_{f,\lambda}$ when reduced mod $\frak p_\Cal D$) and we
define a map $\alpha_\Psi:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow V_{\lambda^n}$ by
\vskip12pt
\noindent$\alpha_\Psi(g)=\rho_\Psi(g)\rho_{f,\lambda}(g)^{-1}\in\left\{\matrix1+\frak p_\Cal D/(\frak
p^2_\Cal D,\ker\Psi)&\frak p_\Cal D/(\frak p^2_\Cal D,\ker\Psi)\\ \ \\
\frak p_\Cal D/(\frak p^2_\Cal
D,\ker\Psi)&1+\frak p_\Cal D/(\frak p^2_\Cal D,\ker\Psi)\endmatrix\right\}\subseteq V_{\lambda^n}\mathbreak$
\vskip6pt
\noindent where $\rho_{f,\lambda}(g)$ is viewed in $\roman{GL}_2(R_\Cal D/(\frak p^2_\Cal D,\ker\Psi))$ via the structural
map $\Cal O\rightarrow R_\Cal D$ ($R_\Cal D$ being an $\Cal O$-algebra and the structural map being local because of\linebreak
the existence of a section). The right-hand inclusion comes from
$$\matrix\frak p_D/(\frak p^2_D,\ker\Psi)&\mathop{\hookrightarrow}\limits^\Psi&\Cal
O/\lambda^n&\mathop{\rightarrow}\limits^\sim&(\Cal O/\lambda^n)\cdot\varepsilon\\
&&1&\mapsto&\varepsilon.\endmatrix$$
Then $\alpha_\Psi$ is really seen to be a continuous cocycle whose cohomology class lies in $H^1_\roman{Se}(\bold
Q_\Sigma/\bold Q,V_{\lambda^n}).$ Finally $\varphi(\alpha_\Psi)=\Psi.$ Moreover, the constructions are\linebreak compatible
with change of $n$, i.e., for $V_{\lambda^n}\!\hookrightarrow\!V_{\lambda^{n+1}}$ and $\lambda\!:\!\Cal
O/\lambda^n\hookrightarrow\Cal O/\lambda^{n+1}.\hskip0.125in\square$\linebreak
\
We now relate the local cohomology groups we have defined to the theory of Fontaine and in particular to the groups of
Bloch-Kato [BK]. We will distinguish these by writing $H^1_F$ for the cohomology groups of Bloch-Kato. None of the results
described in the rest of this section are used in the rest of the paper. They serve only to relate the Selmer groups we have
defined (and later compute) to the more standard versions. Using the lattice associated to $\rho_{f,\lambda}$ we obtain also
a lattice $T\simeq\Cal O^4$ with Galois action via $\roman{Ad}\ \rho_{f,\lambda}.$ Let $\Cal V=T\otimes_{\bold Z_p}\bold Q_p$
be associated vector space and identify $V$ with $\Cal V/T.$ Let $\roman{pr}:\Cal V\rightarrow V$ be\linebreak
\eject
\noindent the natural projection and define cohomology modules by
$$\aligned H^1_F(\bold Q_p,\Cal V)&=\ker:H^1(\bold Q_p,\Cal V)\rightarrow H^1(\bold Q_p,\Cal
V\mathop{\otimes}\limits_{\bold Q_p}B_\roman{crys}),\\
H^1_F(\bold Q_p,V)&=\roman{pr}\Big(H^1_F(\bold Q_p,\Cal V)\Big)\subset H^1(\bold Q_p,V),\\
H^1_F(\bold Q_p,V_{\lambda^n})&=(j_n)^{-1}\Big(H^1_F(\bold Q_p,V)\Big)\subset H^1(\bold Q_p,V_{\lambda^n}),\endaligned$$
where $j_n:V_{\lambda^n}\rightarrow V$ is the natural map and the two groups in the definition\linebreak of
$H^1_F(\bold Q_p,\Cal V)$ are defined using continuous cochains. Similar definitions apply to $\Cal V^*=\roman{Hom}_{\bold
Q_p}(\Cal V,\bold Q_p(1))$ and indeed to any finite-dimensional continuous $p$-adic representation space. The reader is
cautioned that the definition of $H^1_F(\bold Q_p,V_{\lambda^n})$ is dependent on the lattice $T$ (or equivalently on $V$).
Under certainly conditions Bloch and Kato show, using the theory of Fontaine and Lafaille, that this is independent of the
lattice (see [BK, Lemmas 4.4 and 4.5]). In any case we will consider in what follows a fixed lattice associated to
$\rho=\rho_{f,\lambda},\roman{Ad}\ \rho,$ etc. Henceforth we will only use the notation $H^1_F(\bold Q_p,-)$ when the
underlying vector space is crystalline.
\
{\smc Proposition 1.3.} (i) {\it If $\rho_0$ is flat but ordinary and $\rho_{f,\lambda}$ is associated\linebreak to a
$p$-divisible group then for all} $n$
$$H^1_\roman f(\bold Q_p,V_{\lambda^n})=H^1_F(\bold Q_p,V_{\lambda^n}).$$
(ii) {\it If $\rho_{f,\lambda}$ is ordinary, $\det\rho_{f,\lambda}\Big|_{I_p}=\varepsilon$ and $\rho_{f,\lambda}$ is
associated to a $p$-divisible group, then for all $n$},
$$H^1_F(\bold Q_p,V_{\lambda^n})\subseteq H^1_\roman{Se}(\bold Q_p,V_{\lambda^n}.$$
{\it Proof.} Beginning with (i), we define $H^1_\roman f(\bold Q_p,\Cal V)=\{\alpha\in H^1(\bold Q_p,\Cal
V):\kappa(\alpha/\lambda^n)\in H^1_\roman f(\bold Q_p,V)$ for all $n\}$ where $\kappa:H^1(\bold Q_p,\Cal V)\rightarrow
H^1(\bold Q_p,V).$ Then\linebreak we see that in case (i), $H^1_\roman f(\bold Q_p,V)$ is divisible. So it is enough to how
that
$$H^1_F(\bold Q_p,\Cal V)=H^1_\roman f(\bold Q_p,\Cal V).$$
We have to compare two constructions associated to a nonzero element $\alpha$ of $H^1(\bold Q_p,\Cal V).$ The first is to
associate an extension
$$0\rightarrow\Cal V\rightarrow E\mathop{\rightarrow}\limits^\delta K\rightarrow0\leqno(1.9)$$
of $K$-vector spaces with commuting continuous Galois action. If we fix an $e$ with $\delta(e)=1$ the action on $e$ is
defined by $\sigma e=e+\hat\alpha(\sigma)$ with $\hat\alpha$ a cocycle representing $\alpha$. The second construction
begins with the image of the subspace $\langle\alpha\rangle$ in $H^1(\bold Q_p,V).$ By the analogue of Proposition 1.2 in
the local case, there is an $\Cal O$-module isomorphism
$$H^1(\bold Q_p,V)\simeq\roman{Hom}_\Cal O(\frak p_R/\frak p^2_R,K/\Cal O)$$
\eject
\noindent where $R$ is the universal deformation ring of $\rho_0$ viewed as a representation\linebreak of
$\roman{Gal}(\bar\bold Q_p/\bold Q)$ on $\Cal O$-algebras and $\frak p_R$ is the ideal of $R$ corresponding to $\frak p_\Cal
D$ (i.e., its inverse image in $R$). Since $\alpha\ne0,$ associated to $\langle\alpha\rangle$ is a quotient $\frak p_R/(\frak
p^2_R,\frak a)$ of $\frak p_R/\frak p^2_R$ which is a free $\Cal O$-module of rank one. We then obtain a homomorphism
$$\rho_\alpha:\roman{Gal}(\bar\bold Q_p/\bold Q_p)\rightarrow\roman{GL}_2\Big(R/(\frak p^2_R,\frak a)\Big)$$
induced from the universal deformation (we pick a representation in the universal class). This is associated to an $\Cal
O$-module of rank 4 which tensored with $K$ gives a $K$-vector space $E'\simeq(K)^4$ which is an extension
$$0\rightarrow\Cal U\rightarrow E'\rightarrow\Cal U\rightarrow0\leqno(1.10)$$
where $\Cal U\simeq K^2$ has the Galis representation $\rho_{f,\lambda}$ (viewed locally).
In the first construction $\alpha\in H^1_F(\bold Q_p,\Cal V)$ if and only if the extension (1.9) is crystalline, as the
extension given in (1.9) is a sum of copies of the more usual extension where $\bold Q_p$ replaces $K$ in (1.9). On the other
hand $\langle\alpha\rangle\subseteq H^1_\roman f(\bold Q_p,\Cal V)$ if\linebreak and only if the second construction can be
made through $R^\roman{fl},$ or equivalently if\linebreak and only if $E'$ is the representation associated to a $p$-divisible
group. {\it A priori}, the representation associated to $\rho_\alpha$ only has the property that on all finite quotients it
comes from a finite flat group scheme. However a theorem of Raynaud [Ray1] says that then $\rho_\alpha$ comes from a
$p$-divisible group. For more details on $R^\roman{fl}$, the universal flat deformation ring of the local representation
$\rho_0$, see [Ram].) Now the extension $E'$ comes from a $p$-divisible group if and only if it is crystalline; cf. [Fo,
\S6]. So we have to show that (1.9) is crystalline if and only if (1.10) is crystalline.
One obtains (1.10) from (1.9) as follows. We view $\Cal V$ as $\roman{Hom}_K(\Cal U,\Cal U)$ and let
$$X=\ker:\{\roman{Hom}_K(\Cal U,\Cal U)\otimes\Cal U\rightarrow\Cal U\}$$
where the map is the natural one $f\otimes w\mapsto f(w).$ (All tensor products in this proof will be as $K$-vector spaces.)
Then as $K[D_p]$-modules
$$E'\simeq(E\otimes\Cal U)/X.$$
To check this, one calculates explicitly with the definition of the action on $E$ (given above on $e$) and on $E'$ (given in
the proof of Proposition 1.1). It follows\linebreak from standard properties of crystalline representations that if $E$ is
crystalline, so is $E\otimes\Cal U$ and also $E'$. Conversely, we can recover $E$ from $E'$ as follows.\linebreak Consider
$E'\otimes\Cal U\simeq(E\otimes\Cal U\otimes\Cal U)/(X\otimes\Cal U).$ Then there is a natural map
$\varphi:E\otimes(\det)\rightarrow E'\otimes\Cal U$ induced by the direct sum decomposition $\Cal U\otimes\Cal
U\simeq(\det)\oplus\roman{Sym}^2\Cal U$. Here det denotes a 1-dimensional vector space over $K$ with Galois action via
$\det\rho_{f,\lambda}.$ Now we claim that $\varphi$ is injective on $\Cal V\otimes(\det).$ For\linebreak
\eject
\noindent if $f\in\Cal V$ then $\varphi(f)=f\otimes(w_1\otimes w_2-w_2\otimes w_1)$ where $w_1,w_2$ are a basis for $\Cal U$
for which $w_1\wedge w_2=1$ in $\det\simeq K.$ So if $\varphi(f)\in X\otimes\Cal U$ then
$$f(w_1)\otimes w_2-f(w_2)\otimes w_1=0\ \roman{in}\ \Cal U\otimes\Cal U.$$
But this is false unless $f(w_1)=f(w_2)=0$ whence $f=0$. So $\varphi$ is injective\linebreak on $\Cal V\otimes\det$ and if
$\varphi$ itself were not injective then $E$ would split contradicting $\alpha\ne0.$ So $\varphi$ is injective and we have
exhibited $E\otimes(\det)$ as a subrepresentation of $E'\otimes\Cal U$ which is crystalline. We deduce that $E$ is
crystalline if $E'$ is. This completes the proof of (i).
To prove (ii) we check first that $H^1_\roman{Se}(\bold Q_p,V_{\lambda^n})=j^{-1}_n\Big(H^1_\roman{Se}(\bold Q_p,V)\Big)$
(this was already used in (1.7)). We next have to show that $H^1_F(\bold Q_p,\!\Cal V)\subseteq H^1_\roman{Se}(\bold
Q_p,\!\Cal V)$ where the latter is defined by
$$H^1_\roman{Se}(\bold Q_p,\Cal V)=\ker:H^1(\bold Q_p,\Cal V)\rightarrow H^1(\bold Q^\roman{unr}_p,\Cal V/\Cal V^0)$$
with $\Cal V^0$ the subspace of $\Cal V$ on which $I_p$ acts via $\varepsilon$. But this follows from the computations in
Corollary 3.8.4 of [BK]. Finally we observe that
$$\roman{pr}\Big(H^1_\roman{Se}(\bold Q_p,\Cal V)\big)\subseteq H^1_\roman{Se}(\bold Q_p,V)$$
although the inclusion may be strict, and
$$\roman{pr}\Big(H^1_F(\bold Q_p,\Cal V)\Big)=H^1_F(\bold Q_p,V)$$
by definition. This completes the proof.\hfill$\square$
\
These groups have the property that for $s\ge r,$
$$H^1(\bold Q_p,V_\lambda^r)\cap j^{-1}_{r,s}\Big( H^1_F(\bold Q_p,V_{\lambda^s})\Big)=H^1_F(\bold
Q_p,V_{\lambda^r})\leqno(1.11)$$
where $j_{r,s}:V_{\lambda^r}\rightarrow V_{\lambda^s}$ is the natural injection. The same holds for $V^*_{\lambda^r}$ and
$V^*_{\lambda^s}$ in place of $V_{\lambda^r}$ and $V_{\lambda^s}$ where $V^*_{\lambda^r}$ is defined by
$$V^*_{\lambda^r}=\roman{Hom}(V_{\lambda^r},\boldsymbol\mu_{p^r})$$
and similarly for $V^*_{\lambda^s}.$ Both results are immediate from the definition (and indeed were part of the motivation
for the definition).
We also give a finite level version of a result of Bloch-Kato which is easily deduced from the vector space version. As
before let $T\subset\Cal V$ be a Galois stable lattice so that $T\simeq\Cal O^4.$ Define
$$H^1_F(\bold Q_p,T)=i^{-1}\Big(H^1_F(\bold Q_p,\Cal V)\Big)$$
under the natural inclusion $i:T\hookrightarrow\Cal V,$ and likewise for the dual lattice $T^*=\mathbreak\roman{Hom}_{\bold
Z_p}(V,(\bold Q_p/\bold Z_p)(1))$ in $\Cal V^*$. (Here $\Cal V^*=\roman{Hom}(\Cal V,\bold Q_p(1));$ throughout this paper we
use $M^*$ to denote a dual of $M$ with a Cartier twist.) Also write\linebreak
\eject
\noindent$\roman{pr}_n:T\rightarrow T/\lambda^n$ for the natural projection map, and for the mapping it\linebreak induces on
cohomology.
\
{\smc Proposition 1.4.} {\it If $\rho_{f,\lambda}$ is associated to a $p$-divisible group} ({\it the ordi-nary case is
allowed}) {\it then}
\
\hskip-12pt(i) $\roman{pr}_n\Big(H^1_F(\bold Q_p,T)\Big)=H^1_F(\bold Q_p,T/\lambda^n)$ {\it and similarly for}
$T^*,T^*/\lambda^n.$
\
\hskip-15pt(ii)\hskip3pt $H^1_F(\bold Q_p,V_{\lambda^n})$ {\it is the orthogonal complement of $H^1_F(\bold
Q_p,V^*_{\lambda^n})$ under Tate\linebreak${}$\hskip22pt local duality between $H^1(\bold Q_p,V_{\lambda^n})$ and $H^1(\bold
Q_p,V^*_{\lambda^n})$ and similarly for $W_{\lambda^n}$\linebreak${}$\hskip22pt and $W^*_{\lambda^n}$ replacing
$V_{\lambda^n}$ and $V^*_{\lambda^n}$.
\
More generally these results hold for any crystalline representation $\Cal V'$ in place of $\Cal V$ and $\lambda'$ a
uniformizer in $K'$ where $K'$ is any finite extension of $\bold Q_p$ with
$K'\subset\roman{End}_{\roman{Gal}(\overline{\bold Q}_p/\bold Q_p)}\Cal V'$.
\
Proof.} We first observe that $\roman{pr}_n(H^1_F(\bold Q_p,T))\subset H^1_F(\bold Q_p,T/\lambda^n).$ Now\linebreak from the
construction we may identify $T/\lambda^n$ with $V_{\lambda^n}.$ A result of Bloch-Kato ([BK, Prop. 3.8]) says that
$H^1_F(\bold Q_p,\Cal V)$ and $H^1_F(\bold Q_p,\Cal V^*)$ are orthogonal\linebreak complements under Tate local duality. It
follows formally that $H^1_F(\bold Q_p,V^*_{\lambda^n})$ and $\roman{pr}_n(H^1_F(\bold Q_p,T))$ are orthogonal complements,
so to prove the proposition it is enough to show that
$$\#H^1_F(\bold Q_p,V^*_{\lambda^n})\#H^1_F(\bold Q_p,V_{\lambda^n})=\#H^1(\bold Q_p,V_{\lambda^n}).\leqno(1.12)$$
Now if $r=\dim_KH^1_F(\bold Q_p,\Cal V)$ and $s=\dim_KH^1_F(\bold Q_p,\Cal V^*)$ then
$$r+s=\dim_KH^0(\bold Q_p,\Cal V)+\dim_KH^0(\bold Q_p,\Cal V^*)+\dim_K\Cal V.\leqno(1.13)$$
>From the definition,
$$\#H^1_F(\bold Q_p,V_{\lambda^n})=\#(\Cal O/\lambda^n)^r\cdot\#\ker\{H^1(\bold Q_p,V_{\lambda^n})\rightarrow
H^1(\bold Q_p,V)\}.\leqno(1.14)$$
The second factor is equal to $\#\{V(\bold Q_p)/\lambda^nV(\bold Q_p)\}.$ When we write $V(\bold Q_p)^\roman{div}$ for the
maximal divisible subgroup of $V(\bold Q_p)$ this is the same as
$$\aligned\#(V(\bold Q_p)/V(\bold Q_p)^\roman{div})/\lambda^n&=\#(V(\bold Q_p)/V(\bold Q_p)^\roman{div})_{\lambda^n}\\
&=\#V(\bold Q_p)_{\lambda^n}/\#(V(\bold Q_p)^\roman{div})_{\lambda^n}.\endaligned$$
Combining this with (1.14) gives
$$\leqalignno{\#H^1_F(\bold Q_p,V_{\lambda^n})&=\#(\Cal O/\lambda^n)^r&(1.15)\cr&\hskip0.15in\cdot\#H^0(\bold
Q_p,V_{\lambda^n})/\#(\Cal O/\lambda^n)^{\roman{dim}_KH^0(\bold Q_p,\Cal V)}.}$$
This, together with an analogous formula for $\#H^1_F(\bold Q_p,V^*_{\lambda^n})$ and (1.13), gives
\vskip6pt
\noindent$\#H^1_F(\bold Q_p,V^{\lambda^n})\#H^1_F(\bold Q_p,V^*_{\lambda^n})=\#(\Cal O/\lambda^n)^4\cdot\#H^0(\bold
Q_p,V_{\lambda^n})\#H^0(\bold Q_p,V^*_{\lambda^n}).$\linebreak
\eject
\noindent As $\#H^0(\bold Q_p,V^*{\lambda^n})=\#H^2(\bold Q_p,V_{\lambda^n})$ the assertion of (1.12) now follows from the
formula for the Euler characteristic of $V_{\lambda^n}$.
The proof for $W_{\lambda^n},$ or indeed more generally for any crystalline representation, is the same. \hfill$\square$
\
We also give a characterization of the orthogonal complements of\linebreak $H^1_\roman{Se}(\bold Q_p,W_{\lambda^n})$ and
$H^1_\roman{Se}(\bold Q_p,V_{\lambda^n}),$ under Tate's local duality. We write these duals as $H^1_\roman{Se^*}(\bold
Q_p,W^*_{\lambda^n})$ and $H^1_\roman{Se^*}(\bold Q_p,V^*_{\lambda^n})$ respectively. Let
$$\varphi_w:H^1(\bold Q_p,W^*_{\lambda^n})\rightarrow(\bold Q_p,W^*_{\lambda^n}/(W^*_{\lambda^n})^0)$$
be the natural map where $(W^*_{\lambda^n})^i$ is the orthogonal complement of $W^{1-i}_{\lambda^n}$ in $W^*_{\lambda^n}$, and
let $X_{n,i}$ be defined as the image under the composite map
$$\aligned X_{n,i}=\roman{im}:\bold Z^\times_p/(\bold Z^\times_p)^{p^n}\otimes\Cal O/\lambda^n&\rightarrow H^1(\bold
Q_p,\boldsymbol\mu_{p^n}\otimes\Cal O/\lambda^n)\\
&\rightarrow H^1(\bold Q_p,W^*_{\lambda^n}/(W^*_{\lambda^n})^0)\endaligned$$
where in the middle term $\boldsymbol\mu_{p^n}\otimes\Cal O/\lambda^n$ is to be identified with
$(W^*_{\lambda^n})^1/(W^*_{\lambda^n})^0.$ Similarly if we replace $W^*_{\lambda^n}$ by $V^*_{\lambda^n}$ we let $Y_{n,i}$ be
the image of $\bold Z^\times_p/(\bold Z^\times_p)^{p^n}\otimes(\Cal O/\lambda^n)^2$ in $H^1(\bold
Q_p,V^*_{\lambda^n}/(W^*_{\lambda^n})^0),$ and we replace $\varphi_w$ by the analogous map $\varphi_v.$
\
{\smc Proposition 1.5.}
\
$$\aligned H^1_\roman{Se^*}(\bold Q_p,W^*_{\lambda^n})&=\varphi^{-1}_w(X_{n,i}),\\
H^1_{\roman{Se}^*}(\bold Q_p,V^*_{\lambda^n})&=\varphi^{-1}_v(Y_{n,i}).\endaligned$$
\
{\it Proof.} This can be checked by dualizing the sequence
$$\aligned0&\rightarrow H^1_\roman{Str}(\bold Q_p,W_{\lambda^n})\rightarrow H^1_\roman{Se}(\bold
Q_p,W_{\lambda^n})\\
&\rightarrow\ker:\{H^1(\bold Q_p,W_{\lambda^n}/(W_{\lambda^n})^0)\rightarrow H^1(\bold
Q^\roman{unr}_p,W_{\lambda^n}/(W_{\lambda^n})^0\},\endaligned$$
where $H^1_\roman{str}(\bold Q_p,W_{\lambda^n})=\ker:H^1(\bold Q_p,W_{\lambda^n})\rightarrow H^1(\bold
Q_p,W_{\lambda^n}/(W_{\lambda^n})^0).$ The first term is orthogonal to $\ker:H^1(\bold Q_p,W^*_{\lambda^n})\rightarrow
H^1(\bold Q_p,W^*_{\lambda^n}/(W^*_{\lambda^n})^1).$ By the naturality of the cup product pairing with respect to quotients
and subgroups the claim then reduces to the well known fact that under the cup product pairing
$$H^1(\bold Q_p,\boldsymbol\mu_{p^n})\times H^1(\bold Q_p,\bold Z/p^n)\rightarrow\bold Z/p^n$$
the orthogonal complement of the unramified homomorphisms is the image of the units $\bold Z^\times_p/(\bold
Z^\times_p)^{p^n}\rightarrow H^1(\bold Q_p,\boldsymbol\mu_{p^n}).$ The proof for $V_{\lambda^n}$ is essentially the
same.\hfill$\square$
\eject
\centerline{\bf 2. Some computations of cohomology groups}
\
We now make some comparisons of orders of cohomology groups using\linebreak the theorems of Poitou and Tate. We retain the
notation and conventions of Section 1 though it will be convenient to state the first two propositions in a more general
context. Suppose that
$$L=\prod L_q\subseteq\prod_{p\in\Sigma}H^1(\bold Q_q,X)$$
is a subgroup, where $X$ is a finite module for $\roman{Gal}(\bold Q_\Sigma/\bold Q)$ of $p$-power order. We define $L^*$ to
be the orthogonal complement of $L$ under the perfect pairing (local Tate duality)
$$\prod_{q\in\Sigma}H^1(\bold Q_q,X)\times\prod_{q\in\Sigma}H^1(\bold Q_q,X^*)\rightarrow\bold Q_p/\bold Z_p$$
where $X^*=\roman{Hom}(X,\boldsymbol\mu_{p^\infty}).$ Let
$$\lambda_X:H^1(\bold Q_\Sigma/\bold Q,X)\rightarrow\prod_{q\in\Sigma}H^1(\bold Q_q,X)$$
be the localization map and similarly $\lambda_{X^*}$ for $X^*$. Then we set
$$H^1_L(\bold Q_\Sigma/\bold Q,X)=\lambda^{-1}_X(L),\ \ H^1_{L^*}(\bold Q_\Sigma/\bold Q,X^*)=\lambda^{-1}_{X^*}(L^*).$$
The following result was suggested by a result of Greenberg (cf. [Gre1]) and is a simple consequence of the theorems of
Poitou and Tate. Recall that $p$ is always assumed odd and that $p\in\Sigma$.
\
{\smc Proposition 1.6.}
$$\#H^1_L(\bold Q_\Sigma/\bold Q,X)/\#H^1_{L^*}(\bold Q_\Sigma/\bold Q,X^*)=h_\infty\prod_{q\in\Sigma}h_q$$
{\it where}
$$\cases h_q&=\#H^0(\bold Q_q,X^*)/[H^1(\bold Q_q,X):L_q]\\
h_\infty&=\#H^0(\bold R,X^*)\#H^0(\bold Q,X)/\#H^0(\bold Q,X^*).\endcases$$
\
{\it Proof.}Adapting \!the \!exact \!sequence \!proof \!of \!Poitou \!and \!Tate(\!cf.[Mi2,\!Th.4.20])\linebreak we get a
seven term exact sequence
$$\matrix0&\longrightarrow&H^1_L(\bold Q_\Sigma/\bold Q,X)&\longrightarrow&H^1(\bold Q_\Sigma/\bold
Q,X)&\longrightarrow&\prod\limits_{q\in\Sigma}H^1(\bold Q_q,X)/L_q\\
&&&&&&\Big\downarrow\\
&&\prod\limits_{q\in\Sigma}H^2(\bold Q_q,X)&\longleftarrow&H^2(\bold Q_\Sigma/\bold
Q,X)&\longleftarrow&H^1_{L^*}(\bold Q_\Sigma/\bold Q,X^*)^\wedge\endmatrix$$
\hskip64pt$|$\vskip-7.15pt\hskip64.8pt$\rightarrow H^0(\bold Q_\Sigma/\bold Q,X^*)^\wedge\longrightarrow0,$
\eject
\noindent where $M^\wedge=\roman{Hom}(M,\bold Q_p/\bold Z_p).$ Now using local duality and global Euler characteristics (cf.
[Mi2, Cor. 2.3 and Th. 5.1]) we easily obtain the formula in the\linebreak proposition. We repeat that in the above
proposition
$X$ can be arbitrary of $p$-power order.\hfill$\square$
\
We wish to apply the proposition to investigate $H^1_\Cal D.$ Let $\Cal D=(\cdot,\Sigma,\Cal O,\Cal M)$ be a standard
deformation theory as in Section 1 and define a corresponding group $L_n=L_{\Cal D,n}$ by setting
$$L_{n,q}=\cases H^1(\bold Q_q,V_{\lambda^n})&\roman{for}\ q\ne p\ \roman{and}\ q\not\in\Cal M\\
H^1_{D_q}(\bold Q_q,V_{\lambda^n})&\roman{for}\ q\ne p\ \roman{and}\ q\in\Cal M\\
{H.}^1(\bold Q_p,V_{\lambda^n})&\roman{for}\ q=p.\endcases$$
Then $H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_{\lambda^n})=H^1_{L^n}(\bold Q_\Sigma/\bold Q,V_{\lambda^n})$ and we also define
$$H^1_{\Cal D^*}(\bold Q_\Sigma/\bold Q,V^*_{\lambda^n})=H^1_{L^*_n}(\bold Q_\Sigma/\bold Q,V^*_{\lambda^n}).$$
We will adopt the convention implicit in the above that if we consider $\Sigma'\supset\Sigma$ then $H^1_\Cal D(\bold
Q_{\Sigma'}/\bold Q,V_{\lambda^n})$ places no local restriction on the cohomology classes at primes $q\in\Sigma'-\Sigma.$
Thus in $H^1_{\Cal D^*}(\bold Q_{\Sigma'}/\bold Q,V^*_{\lambda^n})$ we will require (by duality) that the cohomology class be
locally trivial at $q\in\Sigma'-\Sigma.$
We need now some estimates for the local cohomology groups. First we consider an arbitrary finite $\roman{Gal}(\bold
Q_\Sigma/\bold Q)$-module $X$:
\
{\smc Proposition 1.7.} {\it If} $q\not\in\Sigma$, {\it and $X$ is an arbitrary finite $\roman{Gal}(\bold Q_\Sigma/\bold
Q)$-module of $p$-power order},
$$\#H^1_{L'}(\bold Q_{\Sigma\cup q}/\bold Q,X)/\#H^1_L(\bold Q_\Sigma/\bold Q,X)\le\#H^0(\bold Q_q,X^*)$$
{\it where $L'_\ell=L_\ell$ for $\ell\in\Sigma$ and $L'_q=H'(\bold Q_q,X)$.
\
Proof.} Consider the short exact sequence of inflation-restriction:
\vskip6pt
\noindent\ $0\!\rightarrow\!H^1_L(\bold Q_\Sigma/\bold Q,X)\!\rightarrow\!H^1_{L'}(\bold Q_{\Sigma\cup q}/\bold
Q,X)\!\rightarrow\!\roman{Hom}(\roman{Gal}(\bold Q_{\Sigma\cup q}/\bold Q_\Sigma),X)^{\roman{Gal}(\bold
Q_\Sigma/\bold Q)}$
\
\hskip1.63in$\Bigg\downarrow$\hskip1.44in$\Bigg\downarrow$\vskip-0.5in\hskip3.1965in$\cap$\vskip0.45in
\hskip1.29in$H^1(\bold Q^\roman{unr}_q,X)^{\roman{Gal}(\bold Q^\roman{unr}_q/\bold
Q_q)}\mathop{\rightarrow}\limits^{\sim}\!H^1(\bold Q^\roman{unr}_q,X)^{\roman{Gal}(\bold Q^\roman{unr}_q/\bold Q_q)}$
\
\noindent The proposition follows when we note that
\vskip6pt
\hskip0.6in$\#H^0(\bold Q_q,X^*)\ \ =\ \ \#H^1(\bold Q^\roman{unr}_q,X)^{\roman{Gal}(\bold Q^\roman{unr}_q/\bold
Q_q)}.$\hfill$\square$
\
Now we return to the study of $V_{\lambda^n}$ and $W_{\lambda^n}.$
\
{\smc Proposition 1.8.} {\it If} $q\in\Cal M\ (q\ne p)$ {\it and} $X=V_{\lambda^n}$ {\it then} $h_q=1.$
\eject
{\it Proof.} This is a straightforward calculation. For example if $q$ is of type (A) then we have
$$L_{n,q}=\ker\{H^1(\bold Q_q,V_{\lambda^n})\rightarrow H^1(\bold Q_q,W_{\lambda^n}/W^0_{\lambda^n})\oplus H^1(\bold
Q^\roman{unr}_q,\Cal O/\lambda^n)\}.$$
Using the long exact sequence of cohomology associated to
$$0\rightarrow W^0_{\lambda^n}\rightarrow W_{\lambda^n}\rightarrow W_{\lambda^n}/W^0_{\lambda^n}\rightarrow0$$
one obtains a formula for the order of $L_{n,q}$ in terms of $\#H^1(\bold Q_q,W_{\lambda^n}),\mathbreak\#H^i(\bold
Q_q,W_{\lambda^n}/W^0_{\lambda^n})$ etc. Using local Euler characteristics these are easily re-\linebreak duced to ones
involving
$H^0(\bold Q_q,W^*_{\lambda^n})$ etc. and the result follows easily.\hfill$\square$
\
The calculation of $h_p$ is more delicate. We content ourselves with an inequality in some cases.
\
{\smc Proposition 1.9.} (i) {\it If $X=V_{\lambda^n}$ then
$$h_ph_\infty=\#(\Cal O/\lambda)^{3n}\#H^0(\bold Q_p,V^*_{\lambda^n})/\#H^0(\bold Q,V^*_{\lambda^n})$$
in the unrestricted case.}
(ii) {\it If $X=V_{\lambda^n}$ then
$$h_ph_\infty\le\#(\Cal O/\lambda)^n\#H^0(\bold Q_p,(V^\roman{ord}_{\lambda^n})^*)/\#H^0(\bold Q,W^*_{\lambda^n})$$ in the
ordinary case.}
(iii) {\it If $X=V_{\lambda^n}$ or $W_{\lambda^n}$ {\it then} $h_ph_\infty\le\#H^0(\bold
Q_p,(W^0_{\lambda^n})^*)/\#H^0(\bold Q,W^*_{\lambda^n})$ in the Selmer case.}
(iv) {\it If $X=V_{\lambda^n}$ or $W_{\lambda^n}$ then $h_ph_\infty=1$ in the strict case.}
(v) {\it If $X=V_{\lambda^n}$ then $h_ph_\infty=1$ in the flat case.}
(vi) {\it If $X=V_{\lambda^n}$ or $W_{\lambda^n}$ then $h_ph_\infty=1/\#H^0(\bold Q,V^*_{\lambda^n})$ if
$L_{n,p}=\mathbreak H^1_F(\bold Q_p,X)$ and $\rho_{f,\lambda}$ arises from an ordinary $p$-divisible group.
\
Proof.} Case (i) is trivial. Consider then case (ii) with $X=V_{\lambda^n.}$ We have a long exact sequence of cohomology
associated to the exact sequence:
$$0\rightarrow W^0_{\lambda^n}\rightarrow V_{\lambda^n}\rightarrow V_{\lambda^n}/W^0_{\lambda^n}\rightarrow0.\leqno(1.16)$$
In particular this gives the map $u$ in the diagram
\vskip6pt\hskip1.87in$H^1(\bold Q_p,V_{\lambda^n})$
\vskip0.26in\hskip2.12in$u$ \vskip-0.36in\hskip2.25in$\vert$\vskip-0.02in\hskip2.223in$\Bigg\downarrow$
\vskip-0.6in\hskip2.6in$\diagdown$
\vskip-0.047in\hskip2.715in$\diagdown$ $\delta$
\vskip-0.047in\hskip2.829in$\diagdown$
\vskip-0.047in\hskip2.944in$\diagdown$
\vskip-0.06in\hskip3.039in$\searrow$
\vskip8pt
\noindent$1\!\!\rightarrow\!\!Z\!\!=\!\!H^1\!(\!\bold Q^\roman{unr}_p\!/\!\bold
Q_p,\!(\!V_{\lambda^n}\!/W^0_{\lambda^n}\!)\!^\Cal H)\!\!\rightarrow\!\!H^1\!(\!\bold
Q_p,\!\!V_{\lambda^n}\!/W^0_{\lambda^n}\!)\!\!\rightarrow\!\!H^1\!(\!\bold
Q^\roman{unr}_p\!,\!V_{\lambda^n}\!/W^0_{\lambda^n}\!)^\Cal G\!\!\rightarrow\!\!1\mathbreak$
\
\noindent where $\Cal G=\roman{Gal}(\bold Q^\roman{unr}_p/\bold Q_p),\Cal H=\roman{Gal}(\bar\bold Q_p/\bold
Q^\roman{unr}_p)$ and $\delta$ is defined to make the triangle commute. Then writing $h_i(M)$ for $\#H^1(\bold Q_p,M)$ we
have that
$\#Z=$\linebreak
\eject
\noindent$h_0(V_{\lambda^n}/W^0_{\lambda^n})$ and $\#\roman{im}\ \delta\ge(\#\roman{im}\ u)/(\#Z).$ A simple calculation
using the\linebreak long exact sequence associated to (1.16) gives
$$\#\roman{im}\ u={h_1(V_{\lambda^n}/W^0_{\lambda^n})h_2(V_{\lambda^n})\over
h_2(W^0_{\lambda^n})h_2(V_{\lambda^n}/W^0_{\lambda^n})}.\leqno(1.17)$$
Hence
$$[H^1(\bold Q_p,V_{\lambda^n}):L_{n,p}]=\#\roman{im}\delta\ge\#(\Cal
O/\lambda)^{3n}h_0(V^*_{\lambda^n})/h_0(W^0_{\lambda^n})^*.$$
The inequality in (iii) follows for $X=V_{\lambda^n}$ and the case $X=W_{\lambda^n}$ is similar. Case (ii) is similar. In
case (iv) we just need $\#\roman{im}\ u$ which is given by (1.17) with $W_{\lambda^n}$ replacing $V_{\lambda^n}.$ In case (v)
we have already observed in Section 1 that Raynaud's results imply that $\#H^0(\bold Q_p,V^*_{\lambda^n})=1$ in the flat
case. Moreover $\#H^1_\roman f(\bold Q_p,V_{\lambda^n})$ can be computed to be $\#(\Cal O/\lambda)^{2n}$ from
$$H^1_\roman f(\bold Q_p,V_{\lambda^n})\simeq H^1_\roman f(\bold Q_p,V)_{\lambda^n}\simeq\roman{Hom}_\Cal O(\frak p_R/\frak
p^2_R,K/\Cal O)_{\lambda^n}$$
where $R$ is the universal local flat deformation ring of $\rho_0$ for $\Cal O$-algebras. Using the relation $R\simeq
R^\roman{fl}\mathop{\otimes}\limits_{W(k)}\Cal O$ where $R^\roman{fl}$ is the corresponding ring for $W(k)$-algebras, and
the main theorem of [Ram] (Theorem 4.2) which computes $R^\roman{fl},$ we can deduce the result.
We now prove (vi). From the definitions
$$\#H^1_F(\bold Q_p,V_{\lambda^n})=\cases(\#\Cal O/\lambda^n)^r\#H^0(\bold Q_p,W_{\lambda^n})&\roman{if}\
\rho_{f,\lambda}|_{D_p}\ \roman{does\ not\ split}\\
(\#\Cal O/\lambda^n)^r&\roman{if}\ \rho_{f,\lambda}|_{D_p}\ \roman{splits}\endcases$$
where $r=\dim_KH^1_F(\bold Q_p,\Cal V).$ This we can compute using the calculations in [BK, Cor. 3.8.4]. We find that $r=2$
in the non-split case and $r=3$ in the split case and (vi) follows easily.\hfill$\square$
\
\
\centerline{\bf 3. Some results on subgroups of $\roman{GL}_2(k)$}
\
We now give two group-theoretic results which will not be used until\linebreak Chapter 3. Although these could be phrased in
purely group-theoretic terms it will be more convenient to continue to work in the setting of Section 1, i.e., with $\rho_0$
as in (1.1) so that $\roman{im}\ \rho_0$ is a subgroup of $\roman{GL}_2(k)$ and $\roman{det}\ \rho_0$ is assumed odd.
\
{\smc Lemma 1.10.} {\it If $\roman{im}\ \rho_0$ has order divisible by $p$ then}:
\
(i) {\it It contains an element $\gamma_0$ of order $m\ge3$ with $(m,p)=1$ and $\gamma_0$ trivial on any abelian quotient
of $\roman{im}\ \rho_0$.}
(ii) {\it It contains an element $\rho_0(\sigma)$ with any prescribed image in the Sylow $2$-subgroup of $(\roman{im}\
\rho_0)/(\roman{im}\ \rho_0)'$ and with the ratio of the eigenvalues not equal to $\omega(\sigma)$. $($Here $(\roman{im}\
\rho_0)'$ denotes the derived subgroup of $(\roman{im}\ \rho_
0)$.$)$}
\eject
{\it The same results hold if the image of the projective representation $\tilde\rho_0$ associated to $\rho_0$ is isomorphic
to $A_4,S_4$ or $A_5$.
\
Proof.} (i) Let $G=\roman{im}\ \rho_0$ and let $Z$ denote the center of $G$. Then we\linebreak have a surjection
$G'\rightarrow(G/Z)'$ where the ${}'$ denotes the derived group. By Dickson's classification of the subgroups of
$\roman{GL}_2(k)$ containing an element of order $p,(G/Z)$ is isomorphic to $\roman{PGL}_2(k')$ or $\roman{PSL}_2(k')$ for
some finite field $k'$ of\linebreak characteristic $p$ or possibly to $A_5$ when $p=3$, cf. [Di, \S260]. In each case we
can\linebreak find, and then lift to $G'$, an element of order $m$ with $(m,p)=1$ and $m\ge3,$ except possibly in the case
$p=3$ and $\roman{PSL}_2(\bold F_3)\simeq A_4$ or $\roman{PGL}_2(\bold F_3)\simeq S_4.$ However in these cases $(G/Z)'$ has
order divisible by 4 so the 2-Sylow subgroup of $G'$ has order greater than 2. Since it has at most one element of exact
order 2 (the eigenvalues would both be $-1$ since it is in the kernel of the determinant and hence the element would be
$-I$) it must also have an element of order 4.
The argument in the $A_4,S_4$ and $A_5$ cases is similar.
(ii) Since $\rho_0$ is assumed absolutely irreducible, $G=\roman{im}\ \rho_0$ has no fixed line.\linebreak We claim that the
same then holds for the derived group $G'$ For otherwise\linebreak since $G'\triangleleft G$ we could obtain a second fixed
line by taking $\langle gv\rangle$ where $\langle v\rangle$ is the\linebreak original fixed line and $g$ is a suitable
element of
$G$. Thus
$G'$ would be contained\linebreak in the group of diagonal matrices for a suitable basis and it would be\linebreak central in
which case
$G$ would be abelian or its normalizer in $\roman{GL}_2(k)$, and hence also $G$, would have order prime to $p$. Since neither
of these possibilities is allowed, $G'$ has no fixed line.
By Dickson's classification of the subgroups of $\roman{GL}_2(k)$ containing an element of order $p$ the image of
$\roman{im}\ \rho_0$ in $\roman{PGL}_2(k)$ is isomorphic to $\roman{PGL}_2(k')$\linebreak or $\roman{PSL}_2(k')$ for some
finite field $k'$ of characteristic $p$ or possibly to $A_5$ when $p=3.$ The only one of these with a quotient group of
order $p$ is $\roman{PSL}_2(\bold F_3)$ when $p=3$. It follows that $p\nmid[G:G']$ except in this one case which we treat
separately. So assuming now that $p\nmid[G:G']$ we see that $G'$ contains a non-\linebreak trivial unipotent element $u$.
Since $G'$ has no fixed line there must be another noncommuting unipotent element $v$ in $G'$. Pick a basis for
$\rho_0|_{G'}$ consisting of their fixed vectors. Then let $\tau$ be an element of $\roman{Gal}(\bold Q_\Sigma/\bold Q)$ for
which the image of $\rho_0(\tau)$ in $G/G'$ is prescribed and let $\rho_0(\tau)=(\!{a\atop c}{b\atop d}\!)$. Then
$$\delta=\pmatrix a&b\\ c&d\endpmatrix\pmatrix1&s\alpha\\ {}&1\endpmatrix\pmatrix1&{}\\
r\beta&1\endpmatrix$$
has $\det\ (\delta)=\det\rho_0(\tau)$ and $\roman{trace}\ \delta=s\alpha(ra\beta+c)+br\beta+a+d.$ Since $p\ge3$ we can
choose this trace to avoid any two given values (by varying $s$) unless $ra\beta+c=0$ for all $r$. But $ra\beta+c$ cannot be
zero for all $r$ as otherwise $a=c=0.$ So we can find a $\delta$ for which the ratio of the eigenvalues is not
$\omega(\tau),\det(\delta)$ being, of course, fixed.
\eject
Now suppose that $\roman{im}\ \rho_0$ does not have order divisible by $p$ but that the associated projective representation
$\widetilde{\hskip1pt\rho_0{}}$ has image isomorphic to $S_4$ or $A_5$, so necessarily $p\ne3.$ Pick an element $\tau$ such
that the image of $\rho_0(\tau)$ in $G/G'$ is any prescribed class. Since this fixes both $\det\rho_0(\tau)$ and
$\omega(\tau)$ we have to show that we can avoid at most two particular values of the trace for $\tau$. To achieve this we
can adapt our first choice of $\tau$ by multiplying by any element og $G'$. So pick $\sigma\in G'$ as in (i) which we can
assume in these two cases has order 3. Pick a basis for $\rho_0$, by expending scalars if necessary, so that
$\sigma\mapsto({\alpha\atop{}}{{}\atop{\alpha^{-1}}}).$ Then one\linebreak checks easily that if $\rho_0(\tau)=(\!{a\atop
c}{b\atop d}\!)$ we cannot have the traces of all of $\tau,\sigma\tau$ and $\sigma^2\tau$ lying in a set of the form
$\{\mp t\}$ unless $a=d=0$. However we can ensure that $\rho_0(\tau)$ does not satisfy this by first multiplying $\tau$ by a
suitable element of $G'$ since $G'$ is not contained in the diagonal matrices (it is not abelian).
In the $A_4$ case, and in the $\roman{PSL}_2(\bold F_3)\simeq A_4$ case when $p=3,$ we use a different argument. In both
cases we find that the 2-Sylow subgroup of $G/G'$ is generated by an element $z$ in the centre of $G$. Either a power of $z$
is a suitable candidate for $\rho_0(\sigma)$ or else we must multiply the power of $z$ by an element of $G'$, the ratio of
whose eigenvalues is not equal to 1. Such an element exists because in $G'$ the only possible elements without this property
are $\{\mp I\}$ (such elements necessary have determinant 1 and order prime to $p$) and we know that $\#G'\gt2$ as was noted
in the proof of part (i).\hfill$\square$
\
{\it Remark.} By a well-known result on the finite subgroups of $\roman{PGL}_2(\overline{\bold F}_p)$ this lemma covers all
$\rho_0$ whose images are absolutely irreducible and for which $\widetilde{\hskip1pt\rho_0{}}$ is not dihedral.
\
Let $K_1$ be the splitting field of $\rho_0$. Then we can view $W_\lambda$ and $W^*_\lambda$ as
$\roman{Gal}(K_1(\zeta_p)/\bold Q)$-modules. We need to analyze their cohomology. Recall that we are assuming that $\rho_0$
is absolutely irreducible. Let $\widetilde{\hskip1pt\rho_0{}}$ be the associated projective representation to
$\roman{PGL}_2(k).$
\
The following proposition is based on the computations in [CPS].
\
{\smc Proposition 1.11.} {\it Suppose that $\rho_0$ is absolutely irreducible. Then
$$H^1(K_1(\zeta_p)/\bold Q,W^*_\lambda)=0.$$
\
Proof.} If the image of $\rho_0$ has order prime to $p$ the lemma is trivial. The subgroups of $\roman{GL}_2(k)$ containing
an element of order $p$ which are not contained in a Borel subgroup have been classified by Dickson [Di, \S260] or [Hu,
II.8.27]. Their images inside $\roman{PGL}_2(k')$ where $k'$ is the quadratic extension of $k$ are conjugate to
$\roman{PGL}_2(F)$ or $\roman{PSL}_2(F)$ for some subfield $F$ of $k'$, or they are isomorphic to one of the exceptional
groups $A_4,S_4,A_5$.
Assume then that the cohomology group $H^1(K_1(\zeta_p)/\bold Q,W^*_\lambda)\ne0.$ Then by considering the
inflation-restriction sequence with respect to the normal\linebreak
\eject
\noindent subgroup $\roman{Gal}(K_1(\zeta_p)/K_1)$ we see that $\zeta_p\in K_1.$ Next, since the representation\linebreak is
(absolutely) irreducible, the center $Z$ of $\roman{Gal}(K_1/\bold Q)$ is contained in the\linebreak diagonal matrices and so
acts trivially on $W_\lambda$. So by considering the inflation-restriction sequence with respect to $Z$ we see that $Z$ acts
trivially on $\zeta_p$ (and on $W^*_\lambda).$ So $\roman{Gal}(\bold Q(\zeta_p)/\bold Q)$ is a quotient of
$\roman{Gal}(K_1/\bold Q)/Z.$ This rules out all cases when $p\ne3,$ and when $p=3$ we only have to consider the case where
the image of the projective representation is isomporphic as a group to $\roman{PGL}_2(F)$ for some finite field of
characteristic 3. (Note that $S_4\simeq\roman{PGL}_2(\bold F_3).)$
Extending scalars commutes with formation of duals and $H^1$, so we may assume without loss of generality $F\subseteq k$. If
$p=3$ and $\#F\gt3$ then $H^1(\roman{PSL}_2(F),W_\lambda)=0$ by results of [CPS]. Then if $\widetilde{\hskip1pt\rho_0}$ is
the projective\linebreak representation associated to $\rho_0$ suppose that $g^{-1}\roman{im}\
\widetilde{\hskip1pt\rho_0}g=\roman{PGL}_2(F)$ and let $H=g\roman{PSL}_2(F)g^{-1}.$ Then $W_\lambda\simeq W^*_\lambda$ over
$H$ and
$$H^1(H,W_\lambda)\mathop{\otimes}\limits_F\bar F\simeq
H^1(g^{-1}Hg,g^{-1}(W_\lambda\mathop{\otimes}\limits_F\bar F))=0.\leqno(1.18)$$
We deduce also that $H^1(\roman{im}\ \rho_0,W^*_\lambda)=0.$
Finally we consider the case where $F=\bold F_3$. I am grateful to Taylor for the following argument. First we consider the
action of $\roman{PSL}_2(\bold F_3)$ on $W_\lambda$ explicitly by considering the conjugation action on matrices $\{A\in
M_2(\bold F_3):\roman{trace}\ A=0\}.$ One sees that no such matrix is fixed by all the elements of order 2, whence
$$H^1(\roman{PSL}_2(\bold F_3),W_\lambda)\simeq H^1(\bold Z/3,(W_\lambda)^{C_2\times C_2})=0$$
where $C_2\times C_2$ denotes the normal subgroup of order 4 in $\roman{PSL}_2(\bold F_3)\simeq A_4.$ Next we verify that
there is a unique copy of $A_4$ in $\roman{PGL}_2(\bar\bold F_3)$ up to conjugation.\linebreak For suppose that
$A,B\in\roman{GL}_2(\bar\bold F_3)$ are such that $A^2=B^2=I$ with the images\linebreak of $A,B$ representing distinct
nontrivial commuting elements of $\roman{PGL}_2(\bar\bold F_3).$ We can choose $A=(\!{1\atop0}{\hskip6.5pt0\atop-1}\!)$ by a
suitable choice of basis, i.e., by a suitable conjugation. Then $B$ is diagonal or antidiagonal as it commutes with $A$ up
to a scalar, and as $B,A$ are distinct in $\roman{PGL}_2(\overline{\bold F}_3)$ we have $B=(\!{0\atop
a}{-a^{-1}\atop\hskip-3pt0}\!)$ for some $a$. By conjugating by a diagonal matrix (which does not change $A$) we can assume
that $a=1$. The group generated by $\{A,B\}$ in $\roman{PGL}_2(\bold F_3)$ is its own centralizer so it has index at most 6
in its normalizer $N$. Since $N/\langle A,B\rangle\simeq S_3$ there is a unique subgroup of $N$ in which $\langle
A,B\rangle$ has index 3 whence the image of the embedding of $A_4$ in $\roman{PGL}_2(\bar\bold F_3)$ is indeed unique (up to
conjugation). So arguing as in (1.18) by extending scalars we see that $H^1(\roman{im}\ \rho_0,W^*_\lambda)=0$ when $F=\bold
F_3$ also.\hfill$\square$
\
The following lemma was pointed out to me by Taylor. It permits most dihedral cases to be covered by the methods of Chapter
3 and [TW].
\
{\smc Lemma 1.12.} {\it Suppose that $\rho_0$ is absolutely irreducible and that}
\
\hskip-12pt(a) $\tilde\rho_0$ {\it is dihedral}\ ({\it the case where the image is $\bold Z/2\times\bold Z/2$ is allowed}),
\eject
\hskip-12pt(b) $\rho_0|_L$ {\it is absolutely irreducible where $L=\bold Q\Big(\sqrt{(-1)^{(p-1)/2}p}\Big)$.}
\
\noindent{\it Then for any positive integer $n$ and any irreducible Galois stable subspace $X$ of $W_\lambda\otimes\bar k$
there exists an element $\sigma\in\roman{Gal}(\bar\bold Q/\bold Q)$ such that}
\
\hskip-12pt(i) $\tilde\rho_0(\sigma)\ne1,$
\
\hskip-15pt(ii) $\sigma$ {\it fixes} $\bold Q(\zeta_
{p^n}),$
\
\hskip-18pt(iii) $\sigma$ {\it has an eigenvalue $1$ on $X$.
\
Proof.} If $\tilde\rho_0$ is dihedral then $\rho_0\otimes\bar k=\roman{Ind}^G_H\chi$ for some $H$ of index 2 in
$G$,\linebreak where
$G=\roman{Gal}(K_1/\bold Q).$ (As before, $K_1$ is the splitting field of $\rho_0$.) Here $H$ can be taken as the full
inverse image of any of the normal subgroups of index 2 defining the dihedral group. Then $W_\lambda\otimes\bar
k\simeq\delta\oplus\roman{Ind}^G_H(\chi/\chi')$ where $\delta$ is the quadratic character $G\rightarrow G/H$ and $\chi'$ is
the conjugate of $\chi$ by any element of $G-H$. Note that $\chi\ne\chi'$ since $H$ has nontrivial image in
$\roman{PGL}_2(\bar k).$
To find a $\sigma$ such that $\delta(\sigma)=1$ and conditions (i) and (ii) hold, observe that $M(\zeta_{p^n})$ is abelian
where $M$ is the quadratic field associated to $\delta$. So conditions (i) and (ii) can be satisfied if $\tilde\rho_0$ is
non-abelian. If $\tilde\rho_0$ is abelian (i.e., the image has the form $\bold Z/2\times\bold Z/2),$ then we use
hypothesis (b). If $\roman{Ind}^G_H(\chi/\chi')$ is irreducible over $\bar k$ then $W_\lambda\otimes\bar k$ is a sum of
three distinct quadratic characters, none of which is the quadratic character associated to $L$, and we can repeat the
argument by changing the choice of $H$ for the other two characters. If $X=\roman{Ind}^G_H(\chi/\chi')\otimes\bar k$ is
absolutely irreducible then pick any $\sigma\in G-H.$ This satisfies (i) and can be made to satisfy (ii) if (b) holds.
Finally, since $\sigma\in G-H$ we see that $\sigma$ has trace zero and $\sigma^2=1$ in its action on $X$. Thus it has an
eigenvalue equal to 1.\hfill$\square$
\
\
\centerline{\bf Chapter 2}
\
In this chapter we study the Hecke rings. In the first section we recall some of the well-known properties of these rings
and especially the Goren-\linebreak stein property whose proof is rather technical, depending on a characteristic\linebreak
$p$ version of the $q$-expansion principle. In the second section we compute the relations between the Hecke rings as the
level is augmented. The purpose is to find the change in the $\eta$-invariant as the level increases.
In the third section we state the conjecture relating the deformation rings of Chapter 1 and the Hecke rings. Finally we end
with the critical step of showing that if the conjecture is true at a minimal level then it is true at all levels. By the
results of the appendix the conjecture is equivalent to the\linebreak
\eject
\noindent equality of the $\eta$-invariant for the Hecke rings and the $\frak p/\frak p^2$-invariant for the deformation
rings. In Chapter 2, Section 2, we compute the change in the $\eta$-invariant and in Chapter 1, Section 1, we estimated the
change in the $\frak p/\frak p^2$-invariant.
\
\centerline{\bf 1. The Gorenstein property}
\
For any positive integer $N$ let $X_1(N)=X_1(N)_{/\bold Q}$ be the modular curve over $\bold Q$ corresponding to the group
$\Gamma_1(N)$ and let $J_1(N)$ be its Jacobian. Let $\bold T_1(N)$ be the ring of endomorphisms of $J_1(N)$ which is generated
over $\bold Z$ by the standard Hecke operators $\{T_l=T_{l*}\ \roman{for}\ l\nmid N,U_q=U_{q*}\ \roman{for}\ q|N,\langle
a\rangle=\langle a\rangle_*\mathbreak\ \roman{for}\ (a,N)=1\}.$ For precise definitions of these see [MW1, Ch. 2,\S5]. In
particular if one identifies the cotangent space of $J_1(N)(\bold C)$ with the space of cusp forms of weight 2 on
$\Gamma_1(N)$ then the action induced by $\bold T_1(N)$ is the usual one on cusp forms. We let $\Delta=\{\langle
a\rangle:(a,N)=1\}$.
The group $(\bold Z/N\bold Z)^*$ acts naturally on $X_1(N)$ via $\Delta$ and for any sub-\linebreak group $H\subseteq(\bold
Z/N\bold Z)^*$ we let $X_H(N)=X_H(N)_{/\bold Q}$ be the quotient $X_1(N)/H$. Thus for $H=(\bold Z/N\bold Z)^*$ we have
$X_H(N)=X_0(N)$ corresponding to the group $\Gamma_0(N)$. In Section 2 it will sometimes be convenient to assume that $H$
decomposes as a product $H=\prod H_q$ in $(\bold Z/N\bold Z)^*\simeq\prod(\bold Z/q^r\bold Z)^*$ where the product\linebreak is
over the distinct prime powers dividing $N$. We let $J_H(N)$ denote the Ja-\linebreak cobian of $X_H(N)$ and note that the
above Hecke operators act naturally on $J_H(N)$ also. The ring generated by these Hecke operators is denoted $\bold T_H(N)$
and sometimes, if $H$ and $N$ are clear from the context, we addreviate this\linebreak to $\bold T$.
Let $p$ be a prime $\ge3.$ Let $\frak m$ be a maximal ideal of $\bold T=\bold T_H(N)$ with $p\in\frak m$. Then associated to
$\frak m$ there is a continuous odd semisimple Galois representation $\rho_{\frak m}$,$$\rho_{\frak
m}:\roman{Gal}(\overline{\bold Q}/\bold Q)\rightarrow\roman{GL}_2(\bold T/\frak m)\leqno(2.1)$$ unramified outside $Np$ which
satisfies $$\roman{trace}\ \rho_{\frak m}(\roman{Frob}\ q)=T_q,\ \det\rho_{\frak m}(\roman{Frob}\ q)=\langle q\rangle q$$ for
each prime $q\nmid Np.$ Here Frob $q$ denotes a Frobenius at $q$ in Gal$(\overline{\bold Q}/\bold Q)$.\linebreak The
representation $\rho_{\frak m}$ is unique up to isomorphism. If $p\nmid N$ (resp. $p|N)$ we say that $\frak m$ is ordinary if
$T_p\notin\frak m$ (resp. $U_p\notin\frak m$). This implies (cf., for example, theorem 2 of [Wi1]) that for our fixed
decomposition group $D_p$ at $p$, $$\rho_{\frak m}\Big|_{D_p}\approx\pmatrix\chi_1&*\\ 0&\chi_2\endpmatrix$$ for a suitable
choic of basis, with $\chi_2$ unramified and $\chi_2(\roman{Frob}\ p)=T_p$ mod\linebreak$\frak m$ (resp. equal to $U_p$). In
particular $\rho_{\frak m}$ is ordinary in the sense of Chapter 1\linebreak
\eject
\noindent provided $\chi_1\ne\chi_2$. We will say that $\frak m$ is $D_p$-distinguished if $\frak m$ is ordinary and
$\chi_1\ne\chi_2$. (In practice $\chi_1$ is usually ramified so this imposes no extra condition.) We caution the reader that
if $\rho_{\frak m}$ is ordinary in the sense of Chapter 1 then we can only conclude that $\frak m$ is $D_p$-distinguished if
$p\nmid N$.
Let $\bold T_{\frak m}$ denote the completion of $\bold T$ at $\frak m$ so that $\bold T_{\frak m}$ is a direct factor of the
complete semi-local ring $\bold T_p=\bold T\otimes\bold Z_p.$ Let $\Cal D$ be the points of the associated $\frak m$-divisible
group $$\Cal D=J_H(N)(\overline{\bold Q})_{\frak m}\simeq J_H(N)(\overline{\bold Q})_{p^\infty}\mathop{\otimes}\limits_{\bold
T_p}\bold T_{\frak m}.$$ It is known that $\hat\Cal D=\roman{Hom}_{\bold Z_p}(\Cal D,\bold Q_p/\bold Z_p)$ is a rank 2 $\bold
T_{\frak m}$-module, i.e., that $\hat\Cal D\mathop{\otimes}\limits_{\bold Z_p}\bold Q_p\simeq(\bold T_{\frak
m}\mathop{\otimes}\limits_{\bold Z_p}\bold Q_p)^2.$ Briefly it is enough to show that $H^1(X_H(N),\bold C)$ is free of rank 2
over $\bold T\otimes\bold C$ and this reduces to showing that $S_2(\Gamma_H(N),\bold C),$\linebreak the space of cusp forms of
weight 2 on $\Gamma_H(N)$, is free of rank 1 over $\bold T\otimes\bold C$. One shows then that if $\{f_1,\dots,f_r\}$ is a
complete set of normalized newforms in $S_2(\Gamma_H(N),\bold C)$ of levels $m_1,\dots,m_r$ then if we set $d_i=N/m_i$, the
form $f=\Sigma f_i(d_iz)$ is a basis vector of $S_2(\Gamma_H(N),\bold C)$ as a $\bold T\otimes\bold C$-module.
If $\frak m$ is ordinary then Theorem 2 of [Wi1], itself a straightforward generalization of Proposition 2 and (11) of [MW2],
shows that (for our fixed decomposition group $D_p$) there is a filtration of $\Cal D$ by Pontrjagin duals of rank 1 $\bold
T_{\frak m}$-modules (in the sense explained above) $$0\rightarrow\Cal D^0\rightarrow\Cal D\rightarrow\Cal
D^E\rightarrow 0\leqno(2.2)$$ where $\Cal D^0$ is stable under $D_p$ and the induced action on $\Cal D^E$ is unramified with
Frob $p=U_p$ on it if $p|N$ and Frob $p$ equal to the unit root of $x^2-T_px+p\langle p\rangle\mathbreak=0$ in $\bold
T_{\frak m}$ if $p\nmid N$. We can describe $\Cal D^0$ and $\Cal D^E$ as follows. Pick a $\sigma\in\mathbreak I_p$ which
induces a generator of Gal$(\bold Q_p(\zeta_{Np^\infty})/\bold Q_p(\zeta_{Np})).$ Let $\varepsilon:D_p\rightarrow\bold
Z_p^\times$ be the cyclotomic character. Then $\Cal D^0=\ker(\sigma-\varepsilon(\sigma))^\roman{div}$, the kernel
being\linebreak taken inside
$\Cal D$ and `div' meaning the maximal divisible subgroup. Although\linebreak in [Wi1] this filtration is given only for a
factor
$A_f$ of
$J_1(N)$ it is easy to\linebreak deduce the result for $J_H(N)$ itself. We note that this filtration is defined without
reference to characteristic $p$ and also that if $\frak m$ is $D_p$-distinguished, $\Cal D^0$ (resp. $\Cal D^E$) can be
described as the maximal submodule on which $\sigma-\tilde\chi_1(\sigma)$ is topologically nilpotent for all
$\sigma\in\roman{Gal}(\overline{\bold Q}_p/\bold Q_p)$ (resp. quotient on which $\sigma-\tilde\chi_2(\sigma)$ is topologically
nilpotent for all $\sigma\in\roman{Gal}(\overline{\bold Q}_p/\bold Q_p))$, where $\tilde\chi_i(\sigma)$ is any lifting of
$\chi_i(\sigma)$ to $\bold T_{\frak m}$.
The Weil pairing $\langle\ ,\ \rangle$ on $J_H(N)(\overline{\bold Q})_{p^M}$ satisfies the relation $\langle
t_*x,y\rangle=\langle x,t^*y\rangle$ for any Hecke operator $t$. It is more convenient to use an adapted pairing defined as
follows. Let $w_\zeta$, for $\zeta$ a primitive $N^\roman{th}$ root of 1, be the\linebreak involution of $X_1(N)_{/\bold
Q(\zeta)}$ defined in [MW1, p. 235]. This induces an involution\linebreak of $X_H(N)_{/\bold Q(\zeta)}$ also. Then we can
define a new pairing [ , ] by setting (for a\linebreak
\eject
\noindent fixed choice of $\zeta$) $$[x,y]=\langle x,w_\zeta y\rangle.\leqno(2.3)$$ Then $[t_*x,y]=[x,t_*y]$ for all Hecke
operators $t$. In particular we obtain an induced pairing on $\Cal D_{p^M}$.
The following theorem is the crucial result of this section. It was first proved by Mazur in the case of prime level [Ma2]. It
has since been generalized in [Ti1], [Ri1] [M Ri], [Gro] and [E1], but the fundamental argument remains that of [Ma2]. For a
summary see [E1, \S9]. However some of the cases we need are not covered in these accounts and we will present these here.
\vskip6pt
{\smc Theorem} 2.1. (i) {\it If $p\nmid N$ and $\rho_{\frak m}$ is irreducible then $$J_H(N)(\overline{\bold Q})[\frak
m]\simeq(\bold T/\frak m)^2.$$}
(ii) {\it If $p\nmid N$ and $\rho_{\frak m}$ is irreducible and $\frak m$ is $D_p$-distinguished then
$$J_H(Np)(\overline{\bold Q})[\frak m]\simeq(\bold T/\frak m)^2.$$} ({\it In case} (ii) $\frak m$ {\it is a maximal ideal of
$\bold T=\bold T_H(Np).)$}
\vskip6pt
{\smc Corollary} 1. {\it In case} (i), $J_H\widehat{(N)(\overline{\bold Q})}_{\frak m}\simeq\bold T_{\frak m}^2$ {\it and}
$\roman{Ta}_{\frak m}\Big(J_H(N)(\overline{\bold Q})\Big)\simeq\bold T_{\frak m}^2$.
{\it In case} (ii), $J_H\widehat{(Np)(\overline{\bold Q})}_{\frak m}\simeq\bold T_{\frak m}^2$ {\it and} $\roman{Ta}_{\frak
m}\Big(J_H(Np)(\overline{\bold Q})\Big)\simeq\bold T_{\frak m}^2$ ({\it where} $\bold T_{\frak m}=\bold T_H(Np)_{\frak m}$).
\vskip6pt
{\smc Corollary} 2. {\it In either of cases} (i) {\it or} (ii) $\bold T_{\frak m}$ {\it is a Gorenstein ring.}
\vskip6pt
In each case the first isomorphisms of Corollary 1 follow from the theorem together with the rank 2 result alluded to
previously. Corrollary 2 and the second isomorphisms of corollory 1 then follow on applying duality (2.4). (In the proof and
in all applications we will only use the notion of a Gorenstein $\bold Z_p$-algebra as defined in the appendix. For finite
flat local $\bold Z_p$-algebras the notions of Gorenstein ring and Gorenstein $\bold Z_p$-algebra are the same.) Here
$\roman{Ta}_{\frak m}\Big(J_H(N)(\overline{\bold Q})\Big)=\roman{Ta}_p\Big(J_H(N)(\overline{\bold
Q})\Big)\mathop{\otimes}\limits_{\bold T_p}{\bold T_{\frak m}}$ is the $\frak m$-adic Tate module of $J_H(N).$
We should also point out that although Corollary 1 gives a representation from the $\frak m$-adic Tate module
$$\rho=\rho_{\bold T_{\frak m}}:\roman{Gal}(\overline{\bold Q}/\bold Q)\rightarrow\roman{GL}_2(\bold T_{\frak m})$$ this can
be constructed in a much more elementary way. (See [Ca3] for another argument.) For, the representation exists with $\bold
T_{\frak m}\otimes\bold Q$ replacing $\bold T_{\frak m}$ when we use the fact that Hom$(\bold Q_p/\bold Z_p,\Cal
D)\otimes\bold Q$ was free of rank 2. A standard argument\linebreak
\eject
\noindent using the Eichler-Shimura relations implies that this representation $\rho'$ with values in $\roman{GL}_2(\bold
T_{\frak m}\otimes\bold Q)$ has the property that $$\roman{trace}\ \rho'(\roman{Frob}\ \ell)=T_\ell,\ \ \det\
\rho'(\roman{Frob}\ \ell)=\ell\langle\ell\rangle$$ for all $\ell\nmid Np.$ We can normalize this representation by picking a
complex\linebreak conjugation $c$ and choosing a basis such that $\rho'(c)\!=\!\big({1\atop0}\ {\ \ 0\atop-1}\big)$, and then
by picking\linebreak a
$\tau$ for which $\rho'(\tau)=\big({a_\tau\atop c_\tau}\ {b_\tau\atop d_\tau}\big)$ with $b_\tau c_\tau\not\equiv0(\frak m)$
and by rescaling the basis so that $b_\tau=1.$ (Note that the explicit description of the traces shows that if $\rho_{\frak m}$
is also normalized so that $\rho_{\frak m}(c)=\big({1\atop0}\ {\ \ 0\atop-1}\big)$ then $b_\rho c_\tau\mod\frak
m=b_{\tau,\frak m}c_{\tau,\frak m}$ where $\rho_{\frak m}(\tau)=\big({a_{\tau,\frak m}\atop c_{\tau,\frak m}}\
{b_{\tau,\frak m}\atop d_{\tau,\frak m}}\big).$ The existence of a $\tau$ such that $b_\tau c_\tau\not\equiv0(\frak m)$ comes
from the irreducibility of $\rho_{\frak m}$.) With this normalization one checks that $\rho'$ actually takes values in the
(closed) subring of $\bold T_{\frak m}$ generated over $\bold Z_p$ by the traces. One can even construct the representation
directly from the representations in Theorem 0.1 using this ring which is reduced. This is the method of Carayol which requires
also the characterization of $\rho$ by the traces and determinants (Theorem 1 of [Ca3]). One can also often interpret the
$U_q$ operators in terms of $\rho$ for $q|N$ using the $\pi_q\simeq\pi(\sigma_q)$ theorem of Langlands (cf. [Ca1]) and
the\linebreak
$U_q$ operator in case (ii) using Theorem 2.1.4 of [Wi1].
\vskip6pt
{\it Proof} ({\it of theorem}). The important technique for proving such multiplicity-one results is due to Mazur and is based
on the $q$-expansion principle in characteristic $p$. Since the kernel of $J_H(N)(\overline{\bold Q})\rightarrow
J_1(N)(\overline{\bold Q})$ is an abelian group on which Gal$(\overline{\bold Q}/\bold Q)$ acts through an abelian extension
of $\bold Q$, the intersection with $\ker\frak m$ is trivial when $\rho_{\frak m}$ is irreducible. So it is enough to verify
the theorem for $J_1(N)$ in part (i) (resp. $J_1(Np)$ in part (ii)). The method for part (i) was developed by Mazur in [Ma2,
Ch. II, Prop. 14.2]. It was extended to the case of $\Gamma_0(N)$ in [Ri1, Th. 5.2] which summarizes Mazur's argument. The
case of $\Gamma_1(N)$ is similar (cf. [E1, Th. 9.2]).
Now consider case (ii). Let $\Delta_{(p)}=\{\langle a\rangle:a\equiv1(N)\}\subseteq\Delta$. Let us first\linebreak assume that
$\Delta_{(p)}$ is nontrivial mod $\frak m$, i.e., that $\delta-1\!\notin\!\frak m$ for some $\delta\!\in\!\Delta_{(p)}$.
This\linebreak case is essentially covered in [Ti1] (and also in [Gro]). We briefly review the argument for use later. Let
$K=\bold Q_p(\zeta_p),\zeta_p$ being a primitive $p^\roman{th}$ root of unity, and let $\Cal O$ be the ring of integers of the
completion of the maximal unramified extension of $K$. Using the fact that $\Delta_{(p)}$ is nontrivial mod $\frak m$ together
with Proposition 4, p. 269 of [MW1] we find that $$J_1(Np)^\roman{\acute et}_{\frak m/\Cal O}(\overline{\bold
F}_p)\simeq(\roman{Pic}^0\Sigma^\roman{\acute et}_1\times\roman{Pic}^0\Sigma^\mu_1)_{\frak m}(\overline{\bold F}_p)$$ where
the notation is taken from [MW1] loc. cit. Here $\Sigma^\roman{\acute et}_1$ and $\Sigma^\mu_1$ are the two smooth irreducible
components of the special fibre of the canonical model of $X_1(Np)_{/\Cal O}$ described in [MW1, Ch. 2]. (The smoothness in
this case was proved in [DR].) Also $J_1(Np)^\roman{\acute et}_{\frak m/\Cal O}$ denotes the canonical \'etale quotient of the
$\frak m$-divisible group over $\Cal O$. This makes sense because $J_1(Np)_{\frak m}$ does extend to\linebreak
\eject
\noindent a $p$-divisible group over $\Cal O$ (again by a theorem of Deligne and Rapoport [DR] and because $\Delta_{(p)}$ is
nontrivial mod $\frak m$). It is ordinary as follows from (2.2) when we use the main theorem of Tate ([Ta]) since $\Cal D^0$
and $\Cal D^E$ clearly correspond\linebreak to ordinary $p$-divisible groups.
Now the $q$-expansion principle implies that $\dim_{\overline{\bold F}_p}X[\frak m']\le1$ where
$$X=\{H^0(\Sigma_1^\mu,\Omega^1)\oplus H^0(\Sigma_1^{\roman{\acute et}},\Omega^1)\}$$ and $\frak m'$ is defined by embedding
$\bold T/\frak m\hookrightarrow\overline{\bold F}_p$ and setting $\frak m'=\ker:\bold T\otimes\overline{\bold
F}_p\rightarrow\overline{\bold F}_p$ under the map $t\otimes a\mapsto at\mod\frak m$. Also $\bold T$ acts on
$\roman{Pic}^0\Sigma_1^\mu\times\roman{Pic}^0\Sigma_1^{\roman{\acute et}}$, the\linebreak abelian variety part of the closed
fibre of the Neron model of $J_1(Np)_{/\Cal O}$, and hence also on its cotangent space $X$. (For a proof that $X[\frak m']$ is
at most one-dimensional, which is readily adapted to this case, see Lemma 2.2 below. For similar versions in slightly simpler
contexts see [Wi3, \S6] or [Gro, \S12]. Then the Cartier map induces an injection 9cf. Prop. 6.5 of [Wi3])
$$\delta:\{\roman{Pic}^0\Sigma_1^\mu\times\roman{Pic}^0\Sigma_1^{\roman{\acute et}}\}[p](\overline{\bold
F}_p)\mathop{\otimes}\limits_{\bold F_p}\overline{\bold F}_p\hookrightarrow X.$$
The composite $\delta\circ w_\zeta$ can be checked to be Hecke invariant (cf. Prop. 6.5 of [Wi3]. In checking the
compatibility for $U_p$ use the formulas of Theorem 5.3 of [Wi3] but note the correction in [MW1, p. 188].) It follows that
$$J_1(Np)_{\frak m/\Cal O}(\overline{\bold F}_p)[\frak m]\simeq\bold T/\frak m$$
as a $\bold T$-module. This shows that if $\hat H$ is the Pontrjagin dual of\linebreak $H=J_1(Np)_{\frak m/\Cal
O}(\overline{\bold F}_p)$ then $\hat H\simeq\bold T_{\frak m}$ since $\hat H/\frak m\simeq\bold T/\frak m$. Thus
$$J_1(Np)_{\frak m/\Cal O}(\overline{\bold F}_p)[p]\mathop{\rightarrow}\limits^\sim\roman{Hom}(\bold T_{\frak m}/p,\bold
Z/p\bold Z).$$
Now our assumption that $\frak m$ is $D_p$-distinguished enables us to identify
$$\Cal D^0=J_1(Np)^0_{\frak m/\Cal O}(\overline{\bold Q}_p)\ \ ,\ \ \Cal D^E=J_1(Np)^{\roman{\acute et}}_{\frak m/\Cal
O}(\overline{\bold Q}_p).$$
For the groups on the right are unramified and those on the left are dual to groups where inertia acts via a character of
finite order (duality with respect to $\roman{Hom}(\ \ ,\bold Q_p/\bold Z_p(1))).$ So
$$\Cal D^0[p]\mathop{\rightarrow}\limits^\sim\bold T_\frak m/p,\ \ \Cal
D^E[p]\mathop{\rightarrow}\limits^\sim\roman{Hom}(\bold T_\frak m/p,\bold Z/p\bold Z)$$
as $\bold T_\frak m$-modules, the former following from the latter when we use duality under the pairing [\ ,\ ]. In
particular as $\frak m$ is $D_p$-distinguished,
$$\Cal D[p]\simeq\bold T_{\frak m}/p\oplus\roman{Hom}(\bold T_\frak m/p,\bold Z/p\bold Z).\leqno(2.4)$$
We now use an argument of Tilouine [Ti1]. We pick a complex conjugation $\tau$. This has distinct eigenvalues $\pm1$ on
$]\rho_\frak m$ so we may decompose $\Cal D[p]$ into eigenspaces for $\tau$:
$$\Cal D[p]=\Cal D[p]^+\oplus\Cal D[p]^-.$$
\eject
\noindent Since $\bold T_{\frak m}/p$ and $\roman{Hom}(\bold T_\frak m/p,\bold Z/p\bold Z)$ are both indecomposable
Hecke-modules, by the Krull-Schmidt theorem this decomposition has factors which are isomorphic to those in (2.4) up to order.
So in the decomposition
$$\Cal D[\frak m]=\Cal D[\frak m]^+\oplus\Cal D[\frak m]^-$$
one of the eigenspaces is isomorphic to $\bold T_\frak m$ and the other to $(\bold T_\frak m/p)[\frak m]$. But since
$\rho_\frak m$ is irreducible it is easy to see by considering $\Cal D[\frak m]\oplus\roman{Hom}(\Cal D[\frak
m],\det\rho_\frak m)$ that $\tau$ has the same number of eigenvalues equal to $+1$ as equal to $-1$ in $\Cal D[\frak m]$,
whence $\#(\bold T_\frak m/p)[\frak m]=\#(\bold T/\frak m).$ This shows that $\Cal D[\frak
m]^+\mathop{\rightarrow}\limits^\sim\Cal D[\frak m]^-\simeq\bold T/\frak m$ as required.
Now we consider the case where $\Delta_{(p)}$ is trivial mod $\frak m$. This case was treated (but only for the group
$\Gamma_0(Np)$ and $\rho_{\frak m}$ `new' at $p$-----the crucial restriction being the last one) in [M Ri]. Let
$X_1(N,p)_{/\bold Q}$ be the modular curve corresponding to $\Gamma_1(N)\cap\Gamma_0(p)$ and let $J_1(N,p)$ be its Jacobian.
Then since the composite of natural maps $J_1(N,p)\rightarrow J_1(Np)\rightarrow J_1(N,p)$ is multiplication by an integer
prime to $p$ and since $\Delta_{(p)}$ is trivial mod $\frak m$ we see that $$J_1(N,p)_\frak m(\overline{\bold Q})\simeq
J_1(Np)_\frak m(\overline{\bold Q}).$$ It will be enough then to use $J_1(N,p)$, and the corresponding ring $\bold T$ and
ideal $\frak m$.
\vskip6pt
The curve $X_1(N,p)$ has a canonical model $X_1(N,p)_{/\bold Z_p}$ which over $\overline{\bold F}_p$ consists of two smooth
curves $\Sigma^\roman{\acute et}$ and $\Sigma^\mu$ intersecting transversally at the supersingular points (again this is a
theorem of Deligne and Rapoport; cf.\linebreak [DR, Ch. 6, Th. 6.9], [KM] or [MW1] for more details). We will use the
models\linebreak described in [MW1, Ch. II] and in particular the cusp $\infty$ will lie on $\Sigma^\mu$. Let $\Omega$ denote
the sheaf of regular differentials on $X_1(N,p)_{/{\bold F_p}}$ (cf. [DR, Ch. 1 \S2],\linebreak [M Ri, \S7]). Over
$\overline{\bold F}_p$, since $X_1(N,p)_{/\overline{\bold F}_p}$ has ordinary double point singularities, the differentials
may be identified with the meromorphic differentials on the normalization $X_1\widetilde{(N,p)}_{/\overline{\bold
F}_p}=\Sigma^\roman{\acute et}\cup\Sigma^\mu$ which have at most simple poles at the supersingular points (the intersection
points of the two components) and satisfy $\roman{res}_{x_1}+\roman{res}_{x_2}=0$ if $x_1$ and $x_2$ are the two points above
such a supersingular point. We need the following lemma:
\vskip6pt
{\smc Lemma} 2.2. $\dim_{\bold T/\frak m}H^0(X_1(N,p)_{/\bold F_p},\Omega)[\frak m]=1.$
\vskip6pt
{\it Proof.} First we remark that the action of the Hecke operator $U_p$ here is most conveniently defined using an extension
from characteristic zero. This is\linebreak explained below. We will first show that $\dim_{\bold T/\frak
m}H^0(X_1(N,p)_{/\bold F_p},\Omega)[\frak m]\le1,$ this being the essential step. If we embed $\bold T/\frak
m\hookrightarrow\overline{\bold F}_p$ and then set\linebreak $\frak m'=\ker:\bold T\otimes\overline{\bold
F}_p\rightarrow\overline{\bold F}_p$ (the map given by
$t\otimes a\mapsto at\mod\frak m)$ then it is enough to show that $\dim_{\overline{\bold F}_p}H^0(X_1(N,p)_{/\overline{\bold
F}_p},\Omega)[\frak m']\le1.$ First we will suppose\linebreak
\eject
\noindent that there is no nonzero holomorphic differential in $H^0(X_1(N,p)_{/\overline{\bold F}_p},\Omega)[\frak m'],$ i.e.,
no differential form which pulls back to {\it holomorphic} differentials on $\Sigma^\roman{\acute et}$ and $\Sigma^\mu$. Then
if $\omega_1$ and $\omega_2$ are two differentials in $H^0(X_1(N,p)_{/\overline{\bold F}_p},\Omega)[\frak m']$, the
$q$-expansion principle shows that $\mu\omega_1-\lambda\omega_2$ has zero $q$-expansion at $\infty$ for some pair
$(\mu,\lambda)\ne(0,0)$ in $\overline{\bold F}^2_p$ and thus is zero on $\Sigma^\mu$. As $\mu\omega_1-\lambda\omega_2=0$ on
$\Sigma^\mu$ it is holomorphic on $\Sigma^\roman{\acute et}$. By our hypothesis it would then be zero which shows that
$\omega_1$ and $\omega_2$ are linearly dependent.
This use of the $q$-expansion principle in characteristic $p$ is crucial and due to Mazur [Ma2]. The point is simply that all
the coefficients in the $q$-expansion are determined by elementary formulae from the coefficient of $q$ provided that $\omega$
is an eigenform for all the Hecke operators. The formulae for the action of these operators in characteristic $p$ follow from
the formulae in characteristic zero. To see this formally (especially for the $U_p$ operator) one checks first that
$H^0(X_1(N,p)_{/{\bold Z}_p},\Omega)$, where $\Omega$ denotes the sheaf of regular differentials on $X_1(N,p)_{/{\bold Z}_p}$,
behaves well under the base changes $\bold Z_p\rightarrow\overline{\bold Z}_p$ and $\bold Z_p\rightarrow\overline{\bold
Q}_p$;\linebreak cf. [Ma2, \S II.3] or [Wi3, Prop. 6.1]. The action of the Hecke operators on $J_1(N,p)$ induces an action on
the connected component of the Neron model of $J_1(N,p)_{/\bold Q_p}$, so also on its tangent space and cotangent space. By
Grothendieck duality the cotangent space is isomorphic to $H^0(X_1(N,p)_{/\bold Z_p},\Omega)$; see (2.5) below. (For a summary
of the duality statements used in this context, see [Ma2, \S II.3]. For explicit duality over fields see [AK, Ch. VIII].) This
then defines an action of the Hecke operators on this group. To check that over $\overline{\bold Q}_p$ this gives the standard
action one uses the commutativity of the diagram after Proposition 2.2 in [Mi1].
Now assume that there is a nonzero holomorphic differential in $$H^0(X_1(N,p)_{/\overline{\bold F}_p},\Omega)[\frak m'].$$ We
claim that the space of holomorphic differentials then has dimension 1 and that any such differential $\omega\ne0$ is actually
nonzero on $\Sigma^\mu$. The dimension claim follows from the second assertion by using the $q$-expansion principle. To prove
that $\omega\ne0$ on $\Sigma^\mu$ we use the formula $$U_{p*}(x,y)=(Fx,y')$$ for
$(x,y)\in(\roman{Pic}^0\Sigma^{\roman{\acute et}}\times\roman{Pic}^0\Sigma^\mu)(\overline{\bold F}_p)$, where $F$ denotes the
Frobenius endo-\linebreak morphism. The value of $y'$ will not be needed. This formula is a variant\linebreak on the second
part of Theorem 5.3 of [Wi3] where the corresponding re-\linebreak sult is proved for $X_1(Np).$ (A correction to the first
part of Theorem 5.3 was noted in [MW1, p. 188].) One check then that the action of $U_p$ on
$X_0=H^0(\Sigma^\mu,\Omega^1)\oplus H^0(\Sigma^{\roman{\acute et}},\Sigma^1)$ viewed as a subspace of
$H^0(X_1(N,p)_{/\overline{\bold F}_p},\Omega)$ is the same as the action on $X_0$ viewed as the cotangent space of
$\roman{Pic}^0\Sigma^\mu\times\mathbreak\roman{Pic}^0\Sigma^{\roman{\acute et}}$. From this we see that if $\omega=0$ on
$\Sigma^\mu$ then $U_{p^\omega}=0$ on $\Sigma^{\roman{\acute et}}$. But $U_p$\linebreak
\eject
\noindent acts as a nonzero scalar which gives a contradiction if $\omega\ne0$. We can thus as-\linebreak sume that the space
of $\frak m'$-torsion holomorphic differentials has dimension 1 and is generated by $\omega$. So if $\omega_2$ is now any
differential in $H^0(X_1(N,p)_{/\overline{\bold F}_p},\Omega)[\frak m']$ then $\omega_2-\lambda\omega$ has zero $q$-expansion
at $\infty$ for some choice of $\lambda$. Then $\omega_2-\lambda\omega=0$ on $\Sigma^\mu$ whence $\omega_2-\lambda\omega$ is
holomorphic and so $\omega_2=\lambda\omega.$ We have now shown\linebreak in general that
$\dim(H^0(X_1(N,p)_{/\overline{\bold F}_p},\Omega)[\frak m'])\le1.$
The singularities of $X_1(N,p)_{/\bold Z_p}$ at the supersingular points are formally isomorphic over $\widehat{\bold
Z_p^{\roman{unr}}}$ to $\widehat{\bold Z_p^{\roman{unr}}}[[X,Y]]/(XY-p^k)$ with $k=1,2$ or 3 [cf. [DR, Ch. 6, Th. 6.9]). If we
consider a minimal regular resolution $M_1(N,p)_{/\bold Z_p}$\linebreak then $H^0(M_1(N,p)_{/\bold F_p},\Omega)\simeq
H^0(X_1(N,p)_{/\bold F_p},\Omega)$ (see the argument in [Ma2, Prop. 3.4]), and a similar isomorphism holds for
$H^0(M_1(N,p)_{/\bold Z_p},\Omega).$
As $M_1(N,p)_{/\bold Z_p}$ is regular, a theorem of Raynaud [Ray2] says that the connected component of the Neron model of
$J_1(N,p)_{/\bold Q_p}$ is $J_1(N,p)^0_{/\bold Z_p}\simeq\roman{Pic}^0(M_1(N,p)_{/\bold Z_p}).$ Taking tangent spaces at the
origin, we obtain $$\roman{Tan}(J_1(N,p)^0_{/\bold Z_p})\simeq H^1(M_1(N,p)_{/\bold Z_p},\Cal O_{M_1(N,p)}).\leqno(2.5)$$
Reducing both sides mod $p$ and applying Grothendieck duality we get an isomorphism
$$\roman{Tan}(J_1(N,p)^0_{/\bold F_p})\mathop{\rightarrow}\limits^\sim\roman{Hom}(H^0(X_1(N,p)_{/\bold
F_p},\Omega),\bold F_p).\leqno(2.6)$$ (To justify the reduction in detail see the arguments in [Ma2, \S II. 3]). Since
$\roman{Tan}(J_1(N,p)^0_{/\bold Z_p})$ is a faithful $\bold T\otimes\bold Z_p$-module it follows that $$H^0(X_1(N,p)_{/\bold
F_p},\Omega)[\frak m]$$ is nonzero. This completes the proof of the lemma.\hfill$\qed$
\vskip6pt
To complete the proof of the theorem we choose an abelian subvariety\linebreak $A$ of $J_1(N,p)$ with multiplicative reduction
at $p$. Specifically let $A$ be the connected part of the kernel of $J_1(N,p)\rightarrow J_1(N)\times J_1(N)$ under the natural
map $\hat\varphi$ described in Section 2 (see (2.10)). Then we have an exact sequence $$0\rightarrow A\rightarrow
J_1(N,p)\rightarrow B\rightarrow0$$ and $J_1(N,p)$ has semistable reduction over $\bold Q_p$ and $B$ has good reduction. By
Proposition 1.3 of [Ma3] the corresponding sequence of connected group schemes $$0\rightarrow A[p]^0_{/\bold Z_p}\rightarrow
J_1(N,p)[p]^0_{/\bold Z_p}\rightarrow B[p]^0_{/\bold Z_p}\rightarrow0$$ is also exact, and by Corollary 1.1 of the same
proposition the corresponding sequence of tangent spaces of Neron models is exact. Using this we may check that the natural
map $$\roman{Tan}(J_1(N,p)[p]^t_{/\overline{\bold F}_p})\mathop{\otimes}\limits_{\bold T_p}\bold T_{\frak
m}\rightarrow\roman{Tan}(J_1(N,p)_{/\overline{\bold F}_p})\mathop{\otimes}\limits_{\bold T_p}\bold T_{\frak m}\leqno(2.7)$$
\eject
\noindent is an isomorphism, where $t$ denotes the maximal multiplicative-type subgroup scheme (cf. [Ma3, \S1]). For it is
enough to check such a relation on $A$ and $B$ separately and on $B$ it is true because the $\frak m$-divisible group is
ordinary. This follows from (2.2) by the theorem of Tate [Ta] as before.
Now (2.6) together with the lemma shows that $$\roman{Tan}(J_1(N,p))_{/\bold Z_p}\mathop{\otimes}\limits_{\bold T_p}\bold
T_{\frak m}\simeq\bold T_{\frak m}.$$ We claim that (2.7) together with this implies that as $\bold T_{\frak m}$-modules
$$V:=J_1(N,p)[p]^t(\overline{\bold Q}_p)_\frak m\simeq(\bold T_\frak m/p).$$ To see this it is sufficient to exhibit an
isomorphism of $\overline{\bold F}_p$-vector spaces $$\roman{Tan}(G_{/\overline{\bold F}_p})\simeq
G(\overline{\bold Q}_p)\mathop{\otimes}\limits_{\bold F_p}\overline{\bold F}_p\leqno(2.8)$$ for any multiplicative-type group
scheme (finite and flat) $G_{/\bold Z_p}$ which is killed by $p$ and moreover to give such an isomorphism that respects the
action of endomorphism of $G_{/\bold Z_p}$. To obtain such an isomorphism observe that we have isomorphisms
$$\leqalignno{\roman{Hom}_{\overline{\bold
Q}_p}(\boldsymbol\mu_p,G)\mathop{\otimes}\limits_{\bold F_p}{\overline{\bold
F}_p}&\simeq\roman{Hom}_{\overline{\bold F}_p}(\boldsymbol\mu_p,G)\mathop{\otimes}\limits_{\bold
F_p}{\overline{\bold
F}_p}&(2.9)\cr&\simeq\roman{Hom}\Big(\roman{Tan}({\boldsymbol\mu_p}_{/\overline{\bold
F}_p}),\roman{Tan}(G_{/\overline{\bold F}_p})\Big)}$$ where $\roman{Hom}_{\overline{\bold Q}_p}$ denotes homomorphisms of the
group schemes viewed over $\overline{\bold Q}_p$ and similarly for $\roman{Hom}_{\overline{\bold F}_p}$. The second
isomorphism can be checked by reducing to the case $G=\boldsymbol\mu_p$. Now picking a primitive $p^\roman{th}$ root of unity
we can identify the left-hand term in (2.9) with $G(\overline{\bold Q}_p)\mathop{\otimes}\limits_{{\bold F}_p}\overline{\bold
F}_p.$ Picking an isomorphism of $\roman{Tan}(\boldsymbol\mu_{p/\overline{\bold F}_p})$ with $\overline{\bold F}_p$ we can
identify the last term in (2.9) with $\roman{Tan}(G_{/\overline{\bold F}_p}).$ Thus after these choices are made we have an
isomorphism in (2.8) which respects the action of endomorphisms of $G$.
On the other hand the action of $\roman{Gal}(\overline{\bold Q}_p/\bold Q_p)$ on $V$ is ramified on every subquotient, so
$V\subseteq\Cal D^0[p]$. (Note that our assumption that $\Delta_{(p)}$ is trivial\linebreak mod $\frak m$ implies that the
action on $\Cal D^0[p]$ is ramified on every subquotient and on $\Cal D^E[p]$ is unramified on every subquotient.) By again
examining $A$ and $B$ separately we see that in fact $V=\Cal D^0[p].$ For $A$ we note that $A[p]/A[p]^t$ is unramified because
it is dual to $\hat A[p]^t$ where $\hat A$ is the dual abelian variety. We can now proceed as we did in the case where
$\Delta_{(p)}$ was nontrivial mod $\frak m$.\hfill$\qed$
\eject
\centerline{\bf2. Congruences between Hecke rings}
\
Suppose that $q$ is a prime not dividing $N$. Let $\Gamma_1(N,q)=\Gamma_1(N)\cap\Gamma_0(q)$ and let
$X_1(N,q)=X_1(N,q)_{/\bold Q}$ be the corresponding curve. The two natural maps $X_1(N,q)\rightarrow X_1(N)$ induced by the
maps $z\rightarrow z$ and $z\rightarrow qz$ on the upper half plane permit us to define a map $J_1(N)\times J_1(N)\rightarrow
J_1(N,q)$. Using a theorem of Ihara, Ribet shows that this map is injective (cf. [Ri2, Cor. 4.2]). Thus we can define
$\varphi$ by $$0\rightarrow J_1(N)\times J_1(N)\mathop{\longrightarrow}\limits^\varphi J_1(N,q).\leqno(2.10)$$ Dualizing, we
define $B$ by $$0\rightarrow B\mathop{\longrightarrow}\limits^\psi
J_1(N,q)\mathop{\longrightarrow}\limits^{\hat\varphi}J_1(N)\times J_1(N)\rightarrow0.$$ Let $\bold T_1(N,q)$ be the ring of
endomorphisms of $J_1(N,q)$ generated by the\linebreak standard Hecke operators $\{T_{l*}$ for $l\nmid Nq,U_{l*}$ for
$l|Nq,\langle a\rangle=\langle a\rangle_*$ for\linebreak $(a,Nq)=1\}$. One can check that $U_p$ preserves $B$ either by an
explicit calcu-lation or by noting that $B$ is the maximal abelian subvariety of $J_1(N,q)$ with multiplicative reduction at
$q$. We set $J_2=J_1(N)\times J_1(N).$
More generally, one can consider $J_H(N)$ and $J_H(N,q)$ in place of $J_1(N)$ and $J_1(\!N,\!q)$ (where $J_H(N,q)$ corresponds
to
$X_1(N,q)/H$) and we write $\bold T_H(N)$ and $\bold T_H(N,q)$ for the associated Hecke rings. In this case the corresponding
map $\varphi$ may have a kernel. However since the kernel of $J_H(N)\rightarrow J_1(N)$ does not meet $\ker\frak m$ for any
maximal ideal $\frak m$ whose associated $\rho_\frak m$ is irreducible,\linebreak the above sequence remain exact if we
restrict to $\frak m^{(q)}$-divisible groups, $\frak m^{(q)}$ being the maximal ideal associated to $\frak m$ of the ring
$\bold T^{(q)}_H(N,q)$ generated by the standard Hecke operators but ommitting $U_q$. With this minor modifica-tion the proofs
of the results below for $H\ne1$ follow from the cases of full level. We will use the same notation in the general case. Thus
$\varphi$ is the map $J_2=J_H(N)^2\rightarrow J_H(N,q)$ induced by $z\rightarrow z$ and $z\rightarrow qz$ on the two factors,
and $B=\ker\hat\varphi$. ($B$ will not be an abelian variety in general.)
The following lemma is a straightforward generalization of a lemma of Ribet ([Ri2]). Let $n_q$ be an integer satisfying
$n_q\equiv q(N)$ and $n_q\equiv1(q)$, and write $\langle q\rangle=\langle n_q\rangle\in\bold T_H(Nq).$
\vskip6pt
{\smc Lemma} 2.3 (Ribet). $\psi(B)\cap\varphi(J_2)_{\frak m^{(q)}}=\varphi(J_2)[U_q^2-\langle q\rangle]_{\frak m^{(q)}}$ {\it
for irre-\linebreak ducible} $\rho_\frak m$.
\vskip6pt
{\it Proof.} The left-hand side is $(\roman{im}\varphi\cap\ker\hat\varphi)$, so we compute
$\varphi^{-1}(\roman{im}\varphi\cap\mathbreak\ker\hat\varphi)=\ker(\hat\varphi\circ\varphi).$
An explicit calculation shows that
$$\hat\varphi\circ\varphi=\bmatrix q+1&T_q\\
T^*_q&q+1\endbmatrix\ \roman{on}\ J_2$$
where $T^*_q=T_q\cdot\langle q\rangle^{-1}$. The matrix action here is on the left. We also find that on $J_2$
$$U_q\circ\varphi=\varphi\circ\bmatrix0&-\langle q\rangle\\
q&T_q\endbmatrix,\leqno(2.11)$$
whence
$$(U_q^2-\langle q\rangle)\circ\varphi=\varphi\circ\bmatrix-\langle q\rangle&0\\
T_q&-\langle q\rangle\endbmatrix\circ(\hat\varphi\circ\varphi).$$
\vskip-0.35in\hfill$\qed$\vskip0.35in
Now suppose that $\frak m$ is a maximal ideal of $\bold T_H(N),p\in\frak m$ and $\rho_\frak m$ is irreducible. We will now
give a slightly stronger result than that given in the lemma in the special case $q=p$. (The case $q\ne p$ we will also
strengthen but we will do this separately.) Assume the that $p\nmid N$ and $T_p\not\in\frak m$. Let $a_p$ be the unit root of
$x^2-T_px+p\langle p\rangle=0$ in $\bold T_H(N)_\frak m$. We first define a maximal ideal $\frak m_p$ of $\bold T_H(N,p)$ with
the same associated representation as $\frak m$. To do this consider the ring $$S_1=\bold T_H(N)[U_1]/(U_1^2-T_pU_1+p\langle
p\rangle)\subseteq\roman{End}(J_H(N)^2)$$ where $U_1$ is the endomorphism of $J_H(N)^2$ given by the matrix
$$\bmatrix T_p&-\langle p\rangle\\
p&0\endbmatrix.$$
It is thus compatible with the action of $U_p$ on $J_H(N,p)$ when compared using $\hat\varphi$. Now $\frak m_1=(\frak
m,U_1-\widetilde{\ \!a_p\ \!})$ is a maximal ideal of $S_1$ where $\widetilde{\ \!a_p\ \!}$ is any element of $\bold
T_H(N)$ representing the class $\bar a_p\in\bold T_H(N)_\frak m/\frak m\simeq\bold T_H(N)/\frak m$. Moreover $S_{1,\frak
m_1}\simeq\bold T_H(N)_\frak m$ and we let $\frak m_p$ be the inverse image of $\frak m_1$ in $\bold T_H(N,p)$ under the
natural map $\bold T_H(N,p)\rightarrow S_1$. One checks that $\frak m_p$ id $D_p$-distinguished. For any standard Hecke
operator $t$ except $U_p$ (i.e., $t=T_l,U_{q'}$ for $q'\ne p$ or $\langle a\rangle$) the image of $t$ is $t$. The image of
$U_p$ is $U_1$.
We need to check that the induced map $$\alpha:\bold T_H(N,p)_{\frak m_p}\longrightarrow S_{1,\frak m_1}\simeq\bold
T_H(N)_\frak m$$ is surjective. The only problem is to show that $T_p$ is in the image. In the present\linebreak context one
can prove this using the surjectivity of $\hat\varphi$ in (2.12) and using the fact that the Tate-modules in the range and
domain of $\hat\varphi$ are free of rank 2 by\linebreak Corollary 1 to Theorem 2.1. The result then follows from Nakayama's
lemma as\linebreak one deduces easily that $\bold T_H(\!N\!)_\frak m$ is a cyclic $\bold T_H(N,p)_{\frak m_p}$-module. This
argument was suggested by Diamond. A second argument using representations can be found at the end of Proposition 2.15. We will
now give a third and more direct proof due to Ribet (cf. [Ri4, Prop. 2]) but found independently and shown to us by Diamond.
\eject
For the following lemma we let $\bold T^M$, for an integer $M$, denote the subring of\linebreak
$\roman{End}\Big(S_2(\Gamma_1(N))\Big)$ generated by the Hecke operators $T_n$ for positive integers $n$ relatively prime to
$M$. Here $S_2\Big(\Gamma_1(N)\Big)$ denotes the vector space of weight 2 cusp forms on $\Gamma_1(N)$. Write $\bold T$ for
$\bold T^1$. It will be enough to show that $T_p$ is\linebreak a redundant operator in $\bold T^1$, i.e., that $\bold
T^p=\bold T$. The result for $\bold T_H(N)_\frak m$ then follows.
\vskip6pt
{\smc Lemma} (Ribet). {\it Suppose that $(M,N)=1.$ If $M$ is odd then $\bold T^M=\bold T$.\linebreak If $M$ is even then $\bold
T^M$ has finite index in $\bold T$ equal to a power of $2$.}
\vskip6pt
As the rings are finitely generated free $\bold Z$-modules, it suffices to prove that $\bold T^M\otimes\bold
F_l\rightarrow\bold T\otimes\bold F_l$ is surjective unless $l$ and $M$ are both even. The claim follows from
\vskip6pt
1. $\bold T^M\otimes\bold F_l\rightarrow\bold T^{M/p}\otimes\bold F_l$ is surjective if $p|M$ and $p\nmid lN$.
\vskip6pt
2. $\bold T^l\otimes\bold F_l\rightarrow\bold T\otimes\bold F_l$ is surjective if $l\nmid2N.$
\vskip6pt
{\it Proof of} 1. Let $A$ denote the Tate module $\roman{Ta}_l(J_1(N)).$ Then $R=\bold T^{M/p}\otimes\bold Z_l$ acts
faithfully on $A$. Let $R'=(R\mathop{\otimes}\limits_{\bold Z_l}\bold Q_l)\cap\roman{End}_{\bold Z_l}A$ and choose $d$ so that
$l^dR'\subset lR$. Consider the $\roman{Gal}(\bar{\bold Q}/\bold Q)$-module $B=J_1(N)[l^d]\times\mu_{Nl^d}$. By \v Cebotarev
density, there is a prime $q$ not dividing $MNl$ so that $\roman{Frob}p=\roman{Frob}q$ on $B$. Using the fact that
$T_r=\roman{Frob}r+\langle r\rangle r(\roman{Frob}r)^{-1}$ on $A$ for $r=p$ and $r=q$, we see that $T_p=T_q$ on $J_1(N)[l^d]$.
It follows that $T_p-T_q$ is in $l^d\roman{End}_{\bold Z_l}A$ and therefore in $l^dR'\subset lR.$
\vskip6pt
{\it Proof of} 2. Let $S$ be the set of cusp forms in $S_2(\Gamma_1(N))$ whose $q$-expansions at $\infty$ have coefficients in
$\bold Z$. Recall that $S_2(\Gamma_1(N))=S\otimes\bold C$ and that $S$ is stable under the action of $\bold T$ (cf. [Sh1, Ch.
3] and [Hi4, \S4]). The pairing $\bold T\otimes S\rightarrow\bold Z$ defined by $T\otimes f\mapsto a_1(Tf)$ is easily checked
to induce an isomorphism of $\bold T$-modules
$$S\cong\roman{Hom}_\bold Z(\bold T,\bold Z).$$
The surjectivity of $\bold T^l/l\bold T^l\rightarrow\bold T/l\bold T$ is equivalent to the injectivity of the dual map
$$\roman{Hom}(\bold T,\bold F_l)\rightarrow\roman{Hom}(\bold T^l,\bold F_l).$$
Now use the isomorphism $S/lS\cong\roman{Hom}(\bold T,\bold F_l)$ and note that if $f$ is in the\linebreak kernel of
$S\rightarrow\roman{Hom}(\bold T^l,\bold F_l)$, then $a_n(f)=a_1(T_nf)$ is divisible by $l$ for all $n$ prime to $l$. But then
the mod $l$ form defined by $f$ is in the kernel of the operator $q{d\over dq}$, and is therefore trivial if $l$ is odd. (See
Corollary 5 of the main theorem of [Ka].) Therefore $f$ is in $lS$.
\vskip6pt
{\it Remark.} The argument does not prove that $\bold T^{Md}=\bold T^d$ if $(d,N)\ne1.$
\eject
We now return to the assumptions that $\rho_\frak m$ is irreducible, $p\nmid N$ and $T_p\not\in\frak m$. Next we define a
principal ideal $(\Delta_p)$ of $\bold T_H(N)_\frak m$ as follows. Since $\bold T_H(N,p)_{\frak m_p}$ and $\bold T_H(N)_\frak
m$ are both Gorenstein rings (by Corollary 2 of Theo-\linebreak rem 2.1) we can define an adjoint $\hat\alpha$ to
$$\alpha:\bold T_H(N,p)_{\frak m_p}\longrightarrow S_{1,\frak m_1}\simeq\bold T_H(N)_\frak m$$
in the manner described in the appendix and we set $\Delta_p=(\alpha\circ\hat\alpha)(1)$. Then $(\Delta_p)$ is independent of
the choice of (Hecke-module) pairings on $\bold T_H(N,p)_{\frak m_p}$ and $\bold T_H(N)_\frak m$. It is equal to the ideal
generated by any composite map $$\bold T_H(N)_\frak m\mathop{\longrightarrow}\limits^\beta\bold T_H(N,p)_{\frak
m_p}\mathop{\longrightarrow}\limits^\alpha\bold T_H(N)_\frak m$$
provided that $\beta$ is an injective map of $\bold T_H(N,p)_{\frak m_p}$-modules with $\bold Z_p$ torsion-free cokernel. (The
module structure on $\bold T_H(N)_\frak m$ is defined via $\alpha$.)
\vskip6pt
{\smc Proposition} 2.4. {\it Assume that $\frak m$ is $D_p$-distinguished and that $\rho_\frak m$ is irreducible of level $N$
with $p\nmid N$. Then}
$$(\Delta_p)=\Big(T_p^2-\langle p\rangle(1+p)^2\Big)=(a_p^2-\langle p\rangle).$$
{\it Proof.} Consider the maps on $p$-adic Tate-modules induced by $\varphi$ and $\hat\varphi$:
$$\roman{Ta}_p\Big(J_H(N)^2\Big)\mathop{\longrightarrow}\limits^\varphi\roman{Ta}_p\Big(J_H(N,p)\Big)
\mathop{\longrightarrow}^{\widehat\varphi}\roman{Ta}_p\Big(J_H(N)^2\Big).$$
These maps commute with the standard Hecke operators with the exception of $T_p$ or $U_p$ (which are not even defined on all
the terms). We define
$$S_2=\bold T_H(N)[U_2]/(U_2^2-T_pU_2+p\langle p\rangle)\subseteq\roman{End}\Big(J_H(N)^2\Big)$$
where $U_2$ is the endomorphism of $J_H(N)^2$ defined by $({0\atop p}{-\langle p\rangle\atop T_p})$. It satisfies $\varphi
U_2=U_p\varphi$. Again $\frak m_2=(\frak m,U_2-\widetilde{\ \!a_p\ \!})$ is a maximal ideal of $S_2$ and we have, on
restricting to the $\frak m_1,\frak m_p$ and $\frak m_2$-adic Tate-modules:
$$\matrix\roman{Ta}_{\frak
m_2}\Big(J_H(N)^2\Big)&\mathop{\longrightarrow}\limits^\varphi&\roman{Ta}_{\frak
m_p}\Big(J_H(N,p)\Big)&\mathop{\longrightarrow}\limits^{\widehat\varphi}&\roman{Ta}_{\frak m_1}\Big(J_H(N)^2\Big)\\ \\
\uparrow\!\wr\ \ v_2&&&&\uparrow\!\wr\ \ v_1\\ \\
\roman{Ta}_\frak m\Big(J_H(N)\Big)&&&&\roman{Ta}_\frak m\Big(J_H(N)\Big).\endmatrix\leqno(2.12)$$
The vertical isomorphisms are defined by $v_2:x\rightarrow(-\langle p\rangle x,a_px)$ and $v_1:x\rightarrow(a_px,px).$ (Here
$a_p\in\bold T_H(N)_\frak m$ can be viewed as an element of $\bold T_H(N)_p\simeq\prod\bold T_H(N)_\frak n$ where the product
is taken over the maximal ideals containing\linebreak $p$. So $v_1$ and $v_2$ can be viewed as maps to
$\roman{Ta}_p\Big(J_H(N)^2\Big)$ whose images are respectively $\roman{Ta}_{\frak m_1}\Big(J_H(N)^2\Big)$ and
$\roman{Ta}_{\frak m_2}\Big(J_H(N)^2\Big).)$
Now $\widehat\varphi$ is surjective and $\varphi$ is injective with torsion-free cokernel by the result of Ribet mentioned
before. Also $\roman{Ta}_\frak m\Big(J_H(N)\Big)\simeq\bold T_H(N)_\frak m^2$ and\linebreak
\eject
\noindent$\roman{Ta}_{\frak m_p}\Big(J_H(N,p)\Big)\simeq\bold T_H(N,p)^2_{\frak m_p}$ by Corollary 1 to Theorem 2.1. So as
$\varphi,\widehat\varphi$\linebreak are maps of $\bold T_H(N,p)_{\frak m_p}$-modules we can use this diagram to compute
$\Delta_p$ as remarked just prior to the statement of the proposition. (The compatibility of the $U_p$ actions requires that,
on identifying the completions $S_{1,\frak m_1}$ and $S_{2,\frak m_2}$ with $\bold T_H(N)_\frak m$, we get $U_1=U_2$ which is
indeed the case.) We find that
$$v_1^{-1}\circ\widehat\varphi\circ\varphi\otimes v_2(z)=a_p^{-1}(a_p^2-\langle p\rangle)(z).$$
\vskip-0.35in\hfill$\qed$\vskip0.225in
We now apply to $J_1(N,q^2)$ (but $q\ne p$) the same analysis that we have just\linebreak applied to $J_1(N,q^2)$. Here
$X_1(A,B)$ is the curve corresponding to $\Gamma_1(\!A\!)\cap\Gamma_0(\!B\!)$\linebreak and
$J_1(A,B)$ its Jacobian. First we need the analogue of Ihara's result. It is convenient to work in a slightly more general
setting. Let us denote the maps $X_1(Nq^{r-1},q^r)\rightarrow X_1(Nq^{r-1})$ induced by $z\rightarrow z$ and $z\rightarrow qz$
by $\pi_{1,r}$ and $\pi_{2,r}$ respectively. Similarly we denote the maps $X_1(Nq^r,q^{r+1})\rightarrow X_1(Nq^r)$ induced by
$z\rightarrow z$ and $z\rightarrow qz$ by $\pi_{3,r}$ and $\pi_{4,r}$ respectively. Also let $\pi:X_1(Nq^r)\rightarrow
X_1(Nq^{r-1},q^r)$ denote the natural map induced by $z\rightarrow z$.
In the following lemma if $\frak m$ is a maximal ideal of $\bold T_1(Nq^{r-1})$ or $\bold T_1(Nq^r)$ we use $\frak m^{(q)}$ to
denote the maximal ideal of $\bold T_1^{(q)}(Nq^r,q^{r+1})$ compatible with $\frak m$, the ring $\bold
T_1^{(q)}(Nq^r,q^{r+1})\subset\bold T_1(Nq^r,q^{r+1})$ being the subring obtained by omitting $U_q$ from the list of
generators.
\vskip6pt
{\smc Lemma} 2.5. {\it If $q\ne p$ is a prime and $r\ge1$ then the sequence of abelian varieties
$$0\rightarrow J_1(Nq^{r-1})\mathop{\longrightarrow}\limits^{\xi_1}J_1(Nq^r)\times
J_1(Nq^r)\mathop{\longrightarrow}\limits^{\xi_2}J_1(Nq^r,q^{r+1})$$
where $\xi_1=\Big((\pi_{1,r}\circ\pi)^*,-(\pi_{2,r}\circ\pi)^*\Big)$ and $\xi_2=(\pi^*_{4,r},\pi^*_{3,r})$ induces a
corresponding sequence of $p$-divisible groups which becomes exact when localized at any $\frak m^{(q)}$ for which
$\rho_\frak m$ is irreducible.}
\vskip6pt
{\it Proof.} Let $\Gamma^1(Nq^r)$ denote the group $\Big\{\big(({a\atop c}{b\atop d})\big)\in\Gamma_1(N):a\equiv
d\equiv1(q^r),\mathbreak c\equiv0(q^{r-1}),b\equiv0(q)\Big\}.$ Let $B_1$ and $B^1$ be given by
$$B_1=\Gamma_1(Nq^r)/\Gamma_1(Nq^r)\cap\Gamma(q),\ \ \ B^1=\Gamma^1(Nq^r)/\Gamma_1(Nq^r)\cap\Gamma(q)$$
and let $\Delta_q=\Gamma_1(Nq^{r-1})/\Gamma_1(Nq^r)\cap\Gamma(q)$. Thus $\Delta_q\simeq\roman{SL}_2(\bold Z/q)$ if $r=1$ and
is of order a power of $q$ if $r\gt1.$
The exact sequences of inflation-restriction give:
$$H_1(\Gamma_1(Nq^r),\bold
Q_p/\bold Z_p)\mathop{\longrightarrow}\limits^{\lambda_1\atop\sim}H^1
(\Gamma_1(Nq^r)\cap\Gamma(q),\bold Q_p/\bold Z_p)^{B_1},$$
together with a similar isomorphism with $\lambda^1$ replacing $\lambda_1$ and $B^1$ replacing $B_1$.
We also obtain
$$H^1(\Gamma_1(Nq^{r-1}),\bold Q_p/\bold Z_p)\mathop{\longrightarrow}^\sim H^1(\Gamma_1(Nq^r)\cap\Gamma(q),\bold Q_p/\bold
Z_p)^{\Delta_q}.$$
\eject
\noindent The vanishing of $H^2(\roman{SL}_2(\bold Z/q),\bold Q_p/\bold Z_p)$ can be checked by restricting to the Sylow
$p$-subgroup which is cyclic. Note that $\roman{im}\lambda_1\cap\roman{im}\lambda^1\subseteq
H^1(\Gamma_1(Nq^r)\cap\Gamma(q),\mathbreak\bold Q_p/\bold Z_p)^{\Delta_q}$ since $B_1$ and $B^1$ together generate $\Delta_q$.
Now consider the sequence
$$\leqalignno{0&{}^{\textstyle\underarrow{\hskip0.55in}} H^1(\Gamma_1(Nq^{r-1}),\bold Q_p/\bold Z_p)&(2.13)\cr
&{}^{\textstyle\underarrow{\scriptstyle\roman{res}_1\oplus-\roman{res}^1}}H^1(\Gamma_1(Nq^r),\bold Q_p/\bold Z_p)\oplus
H^1(\Gamma^1(Nq^r),\bold Q_p/\bold Z_p)\cr
&{}^{\textstyle\underarrow{\scriptstyle\hskip0.11in\lambda_1\oplus\lambda^1\hskip0.11in}}H^1(\Gamma_1(Nq^r)\cap\Gamma(q),\bold
Q_p/\bold Z_p).}$$ We claim it is exact. To check this, suppose that $\lambda_1(x)=-\lambda^1(y).$ Then\linebreak
$\lambda_1(x)\in H^1(\Gamma_1(Nq^r)\cap\Gamma(q),\bold Q_p/\bold Z_p)^{\Delta_q}.$ So $\lambda_1(x)$ is the restriction of an\linebreak $x'\in
H^1\Big(\Gamma_1(Nq^{r-1}),\bold Q_p/\bold Z_p\Big)$ whence $x-\roman{res}_1(x')\in\ker\lambda_1=0.$ It follows also that
$y=-\roman{res}^1(x').$
Now conjugation by the matrix $({q\atop0}{0\atop1})$ induces isomorphisms
$$\Gamma^1(Nq^r)\simeq\Gamma_1(Nq^r),\ \ \ \Gamma_1(Nq^r)\cap\Gamma(q)\simeq\Gamma_1(Nq^r,q^{r+1}).$$
So our sequence (2.13) yields the exact sequence of the lemma, except that we have to change from group cohomology to the
cohomology of the associated complete curves. If the groups are torsion-free then the difference between these cohomologies is
Eisenstein (more precisely $T_l-1-l$ for $l\equiv1\roman{mod}Nq^{r+1}$\linebreak is nilpotent) so will vanish when we localize
at the preimage of $\frak m^{(q)}$ in the abstract Hecke ring generated as a polynomial ring by all the standard Hecke
operators excluding $T_q$. If $M\le3$ then the group $\Gamma_1(M)$ has torsion. For\linebreak $M=1,2,3$ we can restrict to
$\Gamma(3),\Gamma(4),\Gamma(3)$, respectively, where the co-homology is Eisenstein as the corresponding curves have genus
zero and the groups are torsion-free. Thus one only needs to check the action of the Hecke operators on the kernels of the
restriction maps in these three exceptional cases. This can be done explicitly and again they are Eisenstein. This completes
the proof of the lemma.\hfill$\qed$
\vskip6pt
Let us denote the maps $X_1(N,q)\rightarrow X_1(N)$ induced by $z\rightarrow z$ and $z\rightarrow qz$ by $\pi_1$ and $\pi_2$
respectively. Similarly we denote the maps $X_1(N,q^2)\rightarrow X_1(N,q)$ induced by $z\rightarrow z$ and $z\rightarrow qz$
by $\pi_3$ and $\pi_4$ respectively.
>From the lemma (with $r=1$) and Ihara's result (2.10) we deduce that there is a sequence
$$0\rightarrow J_1(N)\times J_1(N)\times J_1(N)\mathop{\longrightarrow}\limits^\xi J_1(N,q^2)\leqno(2.14)$$
where $\xi=(\pi_1\circ\pi_3)^*\times(\pi_2\circ\pi_3)^*\times(\pi_2\circ\pi_4)^*$ and that the induced map of $p$-divisible
groups becomes injective after localization at $\frak m^{(q)}$'s which correspond to irreducible $\rho_\frak m$'s. By duality
we obtain a sequence
$$J_1(N,q^2)\mathop{\longrightarrow}\limits^{\hat\xi}J_1(N)^3\rightarrow0$$
which is `surjective' on Tate modules in the same sense. More generally we can prove analogous results for $J_H(N)$ and
$J_H(N,q^2)$ although there may be\linebreak
\eject
\noindent a kernel of order divisible by $p$ in $J_H(N)\rightarrow J_1(N).$ However this kernel will not meet the $\frak
m^{(q)}$-divisible group for any maximal ideal $\frak m^{(q)}$ whose associated $\rho_\frak m$ is irreducible and hence, as in
the earlier cases, will not affect the results if after passing to $p$-divisible groups we localize at such an $\frak
m^{(q)}$. We use the same notation in the general case when $H\ne1$ so $\xi$ is the map $J_H(N)^3\rightarrow J_H(N,q^2).$
We suppose now that $\frak m$ is a maximal ideal of $\bold T_H(N)$ (as always with $p\in\mathbreak\frak m$) associated to an
irreducible representation and that $q$ is a prime, $p\nmid Np$. We now define a maximal ideal $\frak m_q$ of $\bold
T_H(N,q^2)$ with the same associated representation as $\frak m$. To do this consider the ring
$$S_1=\bold T_H(N)[U_1]/U_1(U_1^2-T_qU_1+q\langle q\rangle)\subseteq\roman{End}\Big(J_H(N)^3\Big)$$
where the action of $U_1$ on $J_H(N)^3$ is given by the matrix
$$\bmatrix T_q&-\langle q\rangle&0\\
q&0&0\\
0&q&0\endbmatrix.$$
Then $U_1$ satisfies the compatibility
$$\widehat\xi\circ U_q=U_1\circ\widehat\xi.$$
One checks this using the actions on cotangent spaces. For we may identify the cotangent spaces with spaces of cusp forms and
with this identification any Hecke operator $t_*$ induces the usual action on cusp forms. There is a maximal ideal $\frak
m_1=(U_1,\frak m)$ in $S_1$ and $S_{1,\frak m_1}\simeq\bold T_H(N)_\frak m.$ We let $\frak m_q$ denote the reciprocal image of
$\frak m_1$ in $\bold T_H(N,q^2)$ under the natural map $\bold T_H(N,q^2)\rightarrow S_1$.
Next we define a principal ideal $(\Delta_q')$ of $\bold T_H(N)_\frak m$ using the fact that $\bold T_H(N,q^2)_{\frak m_q}$
and $\bold T_H(N)_\frak m$ are both Gorenstein rings (cf. Corollary 2 to Theorem 2.1). Thus we set
$(\Delta_q')=(\widehat\alpha\circ\alpha')$ where
$$\alpha':\bold T_H(N,q^2)_{\frak m_q}\rightarrow S_{1,\frak m_1}\simeq\bold T_H(N)_\frak m$$
is the natural map and $\widehat\alpha'$ is the adjoint with respect to selected Hecke-module pairings on $\bold
T_H(N,q^2)_{\frak m_q}$ and $\bold T_H(N)_\frak m$. Note that $\alpha'$ is surjective. To show that the $T_q$ operator is in
the image one can use the existence of the associated 2-dimensional representation (cf. \S1) in which
$T_q=\roman{trace}(\roman{Frob}\ \!q)$ and apply the \v Cebotarev density theorem.
\vskip6pt
{\smc Proposition} 2.6. {\it Suppose that $frak m$ is a maximal ideal of $\bold T_H(N)$ associated to an irreducible
$\rho_\frak m$. Suppose also that $q\nmid Np$. Then
$$(\Delta'_p)=(q-1)(T_q^2-\langle q\rangle(1+q)^2).$$}
\vskip6pt
{\it Proof.} We prove this in the same manner as we proved Proposition 2.4. Consider the maps on $p$-adic
Tate-modules induced by $\xi$ and $\widehat\xi$:
$$\roman{Ta}_p\Big(J_H(N)^3\Big)\mathop{\longrightarrow}^\xi\roman{Ta}_p\Big(J_H(N,q^2)\Big)\mathop{\longrightarrow}
\limits^{\widehat\xi}\roman{Ta}_p\Big(J_H(N)^3\Big).\leqno(2.15)$$
These maps commute with the standard Hecke operators with the exception of $T_q$ and $U_q$ (which are not even defined on all
the terms). We define
$$S_2=\bold T_H(N)[U_2]/U_2(U_2^2-T_qU_2+q\langle q\rangle)\subseteq\roman{End}\Big(J_H(N)^3\Big)$$
where $U_2$ is the endomorphism of $J_H(N)^3$ given by the matrix
$$\bmatrix0&0&0\\
q&0&-\langle q\rangle\\
0&q&T_q\endbmatrix.$$
Then $U_q\xi=\xi U_2$ as one can verify by checking the equality $(\widehat\xi\circ\xi)U^2=U_1(\widehat\xi\circ\xi)$ because
$\widehat\xi\circ\xi$ is an isogeny. The formula for $\widehat\xi\circ\xi$ is given below. Again $\frak m_2=(\frak m,U_2)$ is
a maximal ideal of $S_2$ and $S_{2,\frak m_2}\simeq\bold T_H(N)_\frak m.$ On restricting (2.15) to the $\frak m_2,\frak m_q$
and $\frak m_1$-adic Tate modules we get
$$\matrix\roman{Ta}_{\frak m_2}(J_H(N)^3)&\mathop{\longrightarrow}\limits^\xi&\roman{Ta}_{\frak
m_q}(J_H(N,q^2))&\mathop{\longrightarrow}\limits^{\widehat\xi}&\roman{Ta}_{\frak m_1}(J_H(N)^3)\\
\\
\bigg\uparrow\wr u_2&&&&\bigg\uparrow\wr u_1\\
\\
\roman{Ta}_\frak m(J_H(N))&&&&\roman{Ta}_\frak m(J_H(N)).\endmatrix\leqno(2.16)$$
The vertical isomorphisms are induced by $u_2:z\rightarrow(\langle q\rangle z,-T_qz,qz)$ and $u_1:z\rightarrow(0,0,z)$. Now a
calculation shows that on $J_H(N)^3$
$$\hat\xi\circ\xi=\bmatrix q(q+1)&T_q\cdot q&T_q^2-\langle q\rangle(1+q)\\
T_q^*\cdot q&q(q+1)&T_q\cdot q\\
T_q^{*2}-\langle q\rangle^{-1}(1+q)&T_q^*\cdot q&q(q+1)\endbmatrix$$
where $T_q^*=\langle q\rangle^{-1}T_q$.
We compute then that
$$(u_1^{-1}\circ\widehat\xi\circ\xi\circ u_2)=-\langle q^{-1}\rangle(q-1)\Big(T_q^2-\langle q\rangle(1+q)^2\Big).$$
Now using the surjectivity of $\widehat\xi$ and that $\xi$ has torsion-free cokernel in (2.16) (by Lemma 2.5) and that
$\roman{Ta}_\frak m\Big(J_H(N)\Big)$ and $\roman{Ta}_{\frak m_q}\Big(J_H(N,q^2)\Big)$ are each free of rank 2 over the
respective Hecke rings (Corollary 1 of Theorem 2.1), we deduce the result as in Proposition 2.4.\hfill$\qed$
\vskip6pt
There is one further (and completely elementary) generalization of this result. We let $\pi:X_H(Nq,q^2)\rightarrow X_H(N,q^2)$
be the map given by $z\rightarrow z$. Then $\pi^*:J_H(N,q^2)\rightarrow J_H(Nq,q^2)$ has kernel a cyclic group and as before
this will vanish when we localize at $\frak m^{(q)}$ if $\frak m$ is associated to an irreducible representation. (As before
the superscript $q$ denotes the omission of $U_q$ from the list of generators of $\bold T_H(Nq,q^2)$ and $\frak m^{(q)}$
denotes the maximal ideal of $\bold T_H^{(q)}(Nq,q^2)$ compatible with $\frak m.$)
\eject
We thus have a sequence (not necessarily exact)
$$0\rightarrow J_H(N)^3\mathop{\longrightarrow}\limits^\kappa J_H(Nq,q^2)\rightarrow Z\rightarrow0$$
where $\kappa=\pi^*\circ\xi$ which induces a corresponding sequence of $p$-divisible groups which becomes exact when localized
at an $\frak m^{(q)}$ corresponding to an irreducible $\rho_\frak m$. Here $Z$ is the quotient abelian variety
$J_H(Nq,q^2)/\roman{im}\kappa$. As before there is a natural surjective homomorphism
$$\alpha:\bold T_H(Nq,q^2)_{\frak m_q}\rightarrow S_{1,\frak m_1}\simeq\bold T_H(N)_\frak m$$
where $\frak m_q$ is the inverse image of $\frak m_1$ in $\bold T_H(Nq,q^2).$ (We note that one can replace $\bold
T_H(Nq,q^2)$ by $\bold T_H(Nq^2)$ in the definition of $\alpha$ and Proposition 2.7 below would still hold unchanged.) Since
both rings are again Gorenstein we can define an adjoint $\widehat\alpha$ and a principal ideal
$$(\Delta_q)=(\alpha\circ\widehat\alpha).$$
\vskip6pt
{\smc Proposition} 2.7. {\it Suppose that $\frak m$ is a maximal ideal of $\bold T=\bold T_H(N)$ associated to an irreducible
representation. Suppose that $q\nmid Np$. Then
$$(\Delta_q)=(q-1)^2\Big(T_q^2-\langle q\rangle(1+q)^2\Big)).$$}
The proof is a trivial generalization of that of Proposition 2.6.
\vskip6pt
{\it Remark} 2.8. We have included the operator $U_q$ in the definition of $\bold T_{\frak m_q}=\bold T_H(Nq,q^2)_{\frak m_q}$
as in the application of the $q$-expansion principle it is important to have all the Hecke operators. However $U_q=0$ in
$\bold T_{\frak m_q}$. To see this we recall that the absolute values of the eigenvalues $c(q,f)$ of $U_q$ on newforms of
level $Nq$ with $q\nmid N$ are known (cf. [Li]). They satisfy $c(q,f)^2=\langle q\rangle$ in $\Cal O_f$ (the ring of integers
generated by the Fourier coefficients of $f$) if $f$ is on $\Gamma_1(N,q)$, and $|c(q,f)|=q^{1/2}$ if $f$ is on $\Gamma_1(Nq)$
but not on $\Gamma_1(N,q)$. Also when $f$ is a newform of level dividing $N$ the roots of $x^2-c(q,f)x+q\chi_f(q)=0$ have
absolute value $q^{1/2}$ where $c(q,f)$ is the eigenvalue of $T_q$ and $\chi_f(q)$ of $\langle q\rangle$. Since for $f$ on
$\Gamma_1(Nq,q^2),U_qf$ is a form on $\Gamma_1(Nq)$ we see that
$$U_q(U_q^2-\langle q\rangle)\prod_{f\in\Cal S_1}(U_q-c(q,f))\prod_{f\in\Cal S_2}\Big(U_q^2-c(q,f)U_q+q\langle
q\rangle\Big)=0$$
in $\bold T_H(Nq,q^2)\otimes\bold C$ where $\Cal S_1$ is the set of newforms on $\Gamma_1(Nq)$ which are not on
$\Gamma_1(n,q)$ and $\Cal S_2$ is the set of newforms of level dividing $N$. In particular as $U_q$ is in $\frak m_q$ it must
be zero in $\bold T_{\frak m_q}.$
\vskip6pt
A slightly different situation arises if $\frak m$ is a maximal ideal of $\bold T=\bold T_H(N,q)\mathbreak (q\ne p)$ which is
not associated to any maximal ideal of level $N$ (in the sense of having the same associated $\rho_\frak m$). In this case we
may use the map $\xi_3=(\pi_4^*,\pi_3^*)$ to give
$$J_H(N,q)\times
J_H(N,q)\mathop{\longrightarrow}\limits^{\xi_3}J_H(N,q^2)\mathop{\longrightarrow}\limits^{\hat\xi_3}J_H(N,q)\times
J_H(N,q).\leqno(2.17)$$
\eject
\noindent Then $\hat\xi_3\circ\xi_3$ is given by the matrix
$$\hat\xi_3\circ\xi_3=\bmatrix q&U_q^*\\
U_q&q\endbmatrix$$
on $J_H(N,q)^2$, where $U_q^*=U_q\langle q\rangle^{-1}$ and $U_q^2=\langle q\rangle$ on the $\frak m$-divisible group. The
second of these formulae is standard as mentioned above; cf. for example [Li, Th. 3], since $\rho_m$ is not associated to any
maximal ideal of level $N$. For the first consider any newform $f$ of level divisible by $q$ and observe that the
Petersson\linebreak inner product $\Big\langle(U_q^*U_q-1)f(rz),f(mz)\Big\rangle$ is zero for any $r,m|(Nq/\roman{level}\
f)$\linebreak by [Li, Th. 3]. This shows that $U_q^*U_qf(rz)$, {\it a priori} a linear combination of\linebreak $f(m_iz)$, is
equal to $f(rz)$. So $U_q^*U_q=1$ on the space of forms on $\Gamma_H(N,q)$ which are new at $q$, i.e. the space spanned by
forms $\{f(sz)\}$ where $f$ runs through newforms with $q|\roman{level}\ f.$ In particular $U_q^*$ preserves the $\frak
m$-divisible group and satisfies the same relation on it, again because $\rho_\frak m$ is not associated to any maximal ideal
of level $N$.
\vskip6pt
{\it Remark} 2.9. Assume that $\rho_\frak m$ is of type (A) at $q$ in the terminology of Chapter 1, \S1 (which ensures that
$\rho_\frak m$ does not occur at level $N$). In this case $\bold T_\frak m=\bold T_H(N,q)_\frak m$ is already generated by
the standard Hecke operators with the omission of $U_q$. To see this, consider the $\roman{GL}_2(\bold T_\frak m)$
representation of $\roman{Gal}(\overline{\bold Q}/\bold Q)$ associated to the $\frak m$-adic Tate module of $J_H(N,q)$ (cf.
the discussion following Corollary 2 of Theorem 2.1). Then this representation is already defined over the $\bold
Z_p$-subalgebra $\bold T_\frak m^\roman{tr}$ of $\bold T_\frak m$ generated by the traces of Frobenius elements, i.e. by the
$T_\ell$ for $\ell\nmid Nqp$. In particular $\langle q\rangle\in\bold T_\frak m^\roman{tr}$. Furthermore, as\linebreak $\bold
T_m^\roman{tr}$ is local and complete, and as $U_q^2=\langle q\rangle$, it is enough to solve $X^2=\langle q\rangle$ in the
residue field of $\bold T_\frak m^\roman{tr}$. But we can even do this in $k_0$ (the minimal field of definition of
$\rho_\frak m$) by letting $X$ be the eigenvalue of $\roman{Frob}\ \!q$ on the unique unramified rank-one free quotient of
$k_0^2$ and invoking the $\pi_q\simeq\pi(\sigma_q)$ theorem of Langlands (cf. [Ca1]). (It is to ensure that the unramified
quotient is free of rank one that we assume $\rho_\frak m$ to be of type (A).)
\vskip6pt
We assume now that $\rho_\frak m$ is of type (A) at $q$. Define $S_1$ this time by setting
$$S_1=\bold T_H(N,q)[U_1]/U_1(U_1-U_q)\subseteq\roman{End}\Big(J_H(N,q)^2\Big)$$
where $U_1$ is given by the matrix
$$U_1=\bmatrix0&q\\
0&U_q\endbmatrix\leqno(2.18)$$
on $J_H(N,q)^2$. The map $\widehat\xi_3$ is not necessarily surjective and to remedy this we introduce $\frak m^{(q)}=\frak
m\cap\bold T_H^{(q)}(N,q)$ where $\bold T_H^{(q)}(N,q)$ is the subring of $\bold T_H(N,q)$ generated by the standard Hecke
operators but omitting $U_q$. We also write $\frak m^{(q)}$\linebreak
\eject
\noindent for the corresponding maximal ideal of $\bold T_H^{(q)}(Nq,q^2)$. Then on $\frak m^{(q)}$-divisible groups,
$\widehat\xi_3$ and $\widehat\xi_3\circ\pi_*$ are surjective and we get a natural restriction map of localization $\bold
T_H(Nq,q^2)_{(\frak m^{(q)})}\rightarrow S_{1(\frak m^{(q)})}.$ (Note that the image of $U_q$ under this map is $U_1$ and not
$U_q$.) The ideal $\frak m_1=(\frak m,U_1)$ is maximal in $S_1$ and so also in $S_{1,(\frak m^{(q)})}$ and we let $\frak m_q$
denote the inverse image of $\frak m_1$ under this restriction map. The inverse image of $\frak m_q$ in $\bold T_H(Nq,q^2)$ is
also a maximal ideal which we agin write $\frak m_q$. Since the completions $\bold T_H(Nq,q^2)_{\frak m_q}$ and $S_{1,\frak
m_1}\simeq\bold T_H(N,q)_\frak m$ are both Gorenstein rings (by Corollary 2 of Theorem 2.1) we can define a principal ideal
$(\Delta_q)$ of $\bold T_H(N,q)_\frak m$ by
$$(\Delta_q)=(\alpha\circ\widehat\alpha)$$
where $\alpha:\bold T_H(Nq,q^2)_{\frak m_q}\twoheadrightarrow S_{1,\frak m_1}\simeq\bold T(N,q)_\frak m$ is the restriction
map induced by the restriction map on $\frak m^{(q)}$-localizations described above.
\vskip6pt
{\smc Proposition} 2.10. {\it Suppose that $\frak m$ is a maximal ideal of $\bold T_H(N,q)$\linebreak associated to an
irreducible
$\frak m$ of type} (A){\it . Then
$$(\Delta_q)=(q-1)^2(q+1).$$}
{\it Proof.} The method is a straightforward adaptation of that used for Propositions 2.4 and 2.6. We let $S_2=\bold
T_H(N,q)[U_2]/U_2(U_2-U_q)$ be the ring of endomorphisms of $J_H(N,q)^2$ where $U_2$ is given by the matrix
$$\bmatrix U_q&q\\
0&0\endbmatrix.$$
This satisfies the compatability $\xi_3U_2=U_q\xi_3$. We define $\frak m_2=(\frak m,U_2)$ in $S_2$ and observe that $S_2,\frak
m_2\simeq\bold T_H(N,q)_\frak m.$
Then we have maps
$$\matrix\roman{Ta}_{\frak
m_2}\Big(J_H(N,q)^2\Big)&\mathop{\hookrightarrow}\limits^{\pi^*\circ\xi_3}&\roman{Ta}_{\frak
m_q}\Big(J_H(Nq,q^2)\Big)&\mathop{\twoheadrightarrow}\limits^{\hat\xi_3\circ\pi_*}&\roman{Ta}_{\frak
m_1}\Big(J_H(N,q)^2\Big)\\
\\
\uparrow\wr\ v_2&&&&\uparrow\wr\ v_1\\
\\
\roman{Ta}_\frak m\Big(J_H(N,q)\Big)&&&&\roman{Ta}_\frak m\Big(J_H(N,q)\Big).\endmatrix$$
The maps $v_1$ and $v_2$ are given by $v_2:z\rightarrow(-qz,a_qz)$ and $v_1:z\rightarrow(z,0)$ where $U_q=a_q$ in $\bold
T_H(N,q)_\frak m$. One checks then that $v_1^{-1}\circ(\hat\xi_3\circ\pi_*)\circ(\pi^*\circ\xi_3)\circ v_2$ is equal to
$-(q-1)(q^2-1)$ or $-{1\over2}(q-1)(q^2-1).$
The surjectivity of $\widehat\xi_3\circ\pi_*$ on the completions is equivalent to the statement that
$$J_H(Nq,q^2)[p]_{\frak m_q}\rightarrow J_H(N,q)^2[p]_{\frak m_1}$$
is surjective. We can replace this condition by a similar one with $\frak m^{(q)}$ substituted for $\frak m_q$ and for $\frak
m_1$, i.e., the surjectivity of
$$J_H(Nq,q^2)[p]_{\frak m^{(q)}}\rightarrow J_H(N,q)^2[p]_{\frak m^{(q)}}.$$
\eject
\noindent By our hypothesis that $\rho_\frak m$ be of type (A) at $q$ it is even sufficient to show that the cokernel of
$J_H(Nq,q^2)[p]\otimes\overline{\bold F}_p\rightarrow J_H(N,q)^2[p]\otimes\overline{\bold F}_p$ has no subquotient as a
Galois-module which is irreducible, two-dimensional and ramified at $q$. This statement, or rather its dual, follows from
Lemma 2.5. The injectivity of $\pi^*\circ\xi_3$ on the completions and the fact that it has torsion-free cokernel also follows
from Lemma 2.5 and our hypothesis that $\rho_\frak m$ be of type (A) at $q$.\hfill$\qed$
\vskip6pt
The case that corresponds to type (B) is similar. We assume in the anal-ysis of type (B) (and also of type (C) below) that $H$
decomposes as $\Pi H_q$ as described at the beginning of Section 1. We assume that $\frak m$ is a maximal ideal of $\bold
T_H(Nq^r)$ where $H$ contains the Sylow $p$-subgroup $S_p$ of $(\bold Z/q^r\bold Z)^*$ and that
$$\rho_\frak m\Big|_{I_q}\approx\pmatrix\chi_q&{}\\
{}&1\endpmatrix\leqno(2.19)$$
for a suitable choice of basis with $\chi_q\ne1$ and $\roman{cond}\chi_q=q^r.$ Here $q\nmid Np$ and we assume also that
$\rho_\frak m$ is irreducible. We use the sequence
$$J_H(Nq^r)\times J_H(Nq^r)\mathop{\hbox to
0.5in{\rightarrowfill}}\limits^{(\pi')^*\circ\xi_2}J_{H'}(Nq^r,q^{r+1})\mathop{\hbox to
0.5in{\rightarrowfill}}\limits^{\hat\xi_2\circ\pi'_*}J_H(Nq^r)\times J_H(Nq^r)$$
defined analogously to (2.17) where $\xi_2$ was as defined in Lemma 2.5 and where $H'$ is defined as follows. Using the
notation $H=\Pi H_l$ as at the beginning of Section 1 set $H'_l=H_l$ for $l\ne q$ and $H'_q\times S_p=H_q$. Then define
$H'=\Pi H'_l$ and let $\pi':X_{H'}(Nq^r,q^{r+1})\rightarrow X_H(Nq^r,q^{r+1})$ be the natural map $z\rightarrow z$. Using
Lemma 2.5 we check that $\xi_2$ is injective on the $\frak m^{(q)}$-divisible group. Again we set $S_1=\bold
T_H(Nq^r)[U_1]/U_1(U_1-U_q)\subseteq\roman{End}(J_H\Big(Nq^r)^2\Big)$ where $U_1$ is given by the matrix in (2.18). We define
$\frak m_1=(\frak m,U_1)$ and let $\frak m_q$ be the inverse image of $\frak m_1$ in $\bold T_{H'}(Nq^r,q^{r+1}).$ The
natural map (in which $U_q\rightarrow U_1$) $$\alpha:\bold T_{H'}(Nq^r,q^{r+1})_{\frak m_q}\rightarrow S_{1,\frak
m_1}\simeq\bold T_H(Nq^r)_\frak m$$
is surjective by the following remark.
\vskip6pt
{\it Remark} 2.11. When we assume that $\rho_\frak m$ is of type (B) then the $U_q$ operator is redundant in $\bold T_\frak
m=\bold T_H(Nq^r)_\frak m$. To see this, first assume that $\bold T_\frak m$ is reduced and consider the $\roman{GL}_2(\bold
T_\frak m)$ representation of $\roman{Gal}(\overline{\bold Q}/\bold Q)$ associated to the $\frak m$-adic Tate module. Pick a
$\sigma_q\in I_q$, the inertia group in $D_q$ in $\roman{Gal}(\overline{\bold Q}/\bold Q)$, such that $\chi_q(\sigma_q)\ne1.$
Then because the eigenvalues of $\sigma_q$ are distinct mod $\frak m$ we can\linebreak diagonalize the representation with
respect to
$\sigma_q$. If $\roman{Frob}\ \!q$ is a Frobenius in $D_q$,\linebreak then in the $\roman{GL}_2(\bold T_\frak m)$
representation the image of Frob $q$ normalizes $I_q$ and we can recover $U_q$ as the entry of the matrix giving the value of
Frob $q$ on the unit\linebreak eigenvector for $\sigma_q$. This is by the $\pi_q\simeq\pi(\sigma_q)$ theorem of Langlands as
before (cf. [Ca1]) applied to each of the representations obtained from maps $\bold T_\frak m\rightarrow\Cal O_{f,\lambda}.$
Since the representation is defined over the $\bold Z_p$-algebra $\bold T_\frak m^{\roman tr}$ generated by the traces, the
same reasoning applied to
$\bold T_\frak m^\roman{tr}$ shows that $U_q\in\bold T_\frak m^\roman{tr}$.
\eject
If $\bold T_\frak m$ is not reduced the above argument shows only that there is an operator $v_q\in\bold T_\frak m^\roman{tr}$
such that $(U_q-v_q)$ is nilpotent. Now $\bold T_H(Nq^r)$ can be viewed as a ring of endomorphisms of $S_2(\Gamma_H(Nq^r))$,
the space of cusp forms of weight 2 on $\Gamma_H(Nq^r)$. There is a restriction map $\bold T_H(Nq^r)\rightarrow\bold
T_H(Nq^r)^\roman{new}$ where $\bold T_H(Nq^r)^\roman{new}$ is the image of $\bold T_H(Nq^r)$ in the ring of endomorphisms of
$S_2(\Gamma_H(Nq^r))/S_2(\Gamma_H(Nq^r))^\roman{odd}$, the old part being defined as the sum of two copies of
$S_2(\Gamma_H(Nq^{r-1}))$ mapped via $z\rightarrow z$ and $z\rightarrow qz$. One sees that on $\frak m$-completions $\bold
T_\frak m\simeq(\bold T_H(Nq^r)^\roman{new})_\frak m$ since the conductor of $\rho_\frak m$ is divisible by $q^r$. It follows
that $U_q\in\bold T_\frak m$ satisfies an equation of the form $P(U_q)=0$ where $P(x)$ is a polynomial with coefficients in
$W(k_\frak m)$ and with distinct roots. By extending scalars to $\Cal O$ (the integers of a local field containing $W(k_\frak
m)$) we can assume that the roots lie in $T\simeq\bold T_\frak m\mathop{\otimes}\limits_{W(k_\frak m)}\Cal O.$
Since $(U_q-v_q)$ is nilpotent it follows that $P(v_q)^r=0$ for some $r$. Then since $v_q\in\bold T_\frak m^\roman{tr}$ which
is reduced, $P(v_q)=0.$ Now consider the map $T\rightarrow\Pi T_{(\frak p)}$ where the product is taken over the localizations
of $T$ at the minimal primes $\frak p$ of $T$. The map is injective since the associated primes of the kernel are all maximal,
whence the kernel is of finite cardinality and hence zero. Now in each $T_{(\frak p)},U_q=\alpha_i$ and $v_q=\alpha_j$ for
roots $\alpha_i,\alpha_j$ of $P(x)=0$ because the roots are distinct. Since $U_q-v_q\in\frak p$ for each $\frak p$ it follows
that $\alpha_i=\alpha_j$ for each $\frak p$ whence $U_q=v_q$ in each $T_{(\frak p)}$. Hence $U_q=v_q$ in $T$ also and this
finally shows that $U_q\in\bold T_\frak m^\roman{tr}$ in general.
\vskip6pt
We can therefore define a principal ideal
$$(\Delta_q)=(\alpha\circ\widehat\alpha)$$
using,as previously, that the rings $\bold T_{H'}(Nq^r\!,q^{r+1})_{\frak m_q}$ and $\bold T_H(Nq^r)_\frak m$ are
Gorenstein. We compute $(\Delta_q)$ in a similar manner to the type (A) case, but using this time that $U_q^*U_q=q$ on the
space of forms on $\Gamma_H(Nq^r)$ which are new at\linebreak $q$, i.e., the space spanned by forms $\{f(sz)\}$ where $f$ runs
through newforms with $q^r|\roman{level}\ \!f$. To see this let $f$ be any newform of level divisible by $q^r$ and\linebreak
observe that the Petersson inner product $\Big\langle(U_q^*U_q-q)f(rz),f(mz)\Big\rangle=0$ for any $m|(Nq^r/\roman{level}\
\!f)$ by [Li, Th. 3(ii)]. This shows that $(U_q^*U_q-q)f(rz)$,\linebreak {\it a priori} a linear combination of $\{f(m_iz)\}$,
is zero. We obtain the following result.
\vskip6pt
{\smc Proposition} 2.12. {\it Suppose that $\frak m$ is a maximal ideal of $\bold T_H(Nq^r)$\linebreak associated to an
irreducible
$\rho_\frak m$ of type} (B) {\it at $q$, i.e., satisfying} (2.19) {\it including the hypothesis that $H$ cantains $S_p$.}
({\it Again $q\nmid Np.$}) {\it Then}
$$(\Delta_q)=\Big((q-1)^2\Big).$$
Finally we have the case where $\rho_\frak m$ is of type (C) at $q$. We assume then that $\frak m$ is a maximal ideal of
$\bold T_H(Nq^r)$ where $H$ contains the Sylow $p$-subgroup
\eject
\noindent$S_p$ of $(\bold Z/q^r\bold Z)^*$ and that
$$H^1(\bold Q_q,W_\lambda)=0\leqno(2.20)$$
where $W_\lambda$ is defined as in (1.6) but with $\rho_\frak m$ replacing $\rho_0$, i.e., $W_\lambda=\roman{ad}^0\rho_\frak
m.$
This time we let $\frak m_q$ be the inverse image of $\frak m$ in $\bold T_{H'}(Nq^r)$ under the natural restriction map
$\bold T_{H'}(Nq^r)\longrightarrow\bold T_H(Nq^r)$ with $H'$ defined as in the case of type $B$. We set
$$(\Delta_q)=(\alpha\circ\hat\alpha)$$
where $\alpha:\bold T_{H'}(Nq^r)_{\frak m_q}\twoheadrightarrow\bold T_H(Nq^r)_\frak m$ is the induced map on the completions,
which as before are Gorenstein rings. The proof of the following proposition is analogous (but simpler) to the proof of
Proposition 2.10. (Notice that the proposition does not require the condition that $\rho_\frak m$ satisfy (2.20) but this is
the case in which we will use it.)
\vskip6pt
{\smc Proposition} 2.13. {\it Suppose that $\frak m$ is a maximal of $\bold T_H(Nq^r)$ asso-\linebreak ciated to an irreducible
$\rho_\frak m$ with $H$ containing the Sylow $p$-subgroup of $(\bold Z/q^r\bold Z)^*.$ Then $$(\Delta_q)=(q-1).$$}
Finally, in this section we state Proposition 2.4 in the case $q\ne p$ as this will be used in Chapter 3. Let $q$ be a prime,
$q\nmid Np$ and let $S_1$ denote the ring $$T_H(N)[U_1]/\{U_1^2-T_qU_1+\langle q\rangle
q\}\subseteq\roman{End}(J_H(N)^2)\leqno(2.21)$$
where $\hat\varphi:J_H(N,q)\rightarrow J_H(N)^2$ is the map defined after (2.10) and $U_1$ is the matrix
$$\bmatrix T_q&-\langle q\rangle\\
q&0\endbmatrix.$$
Thus, $\hat\varphi U_q=U_1\hat\varphi$. Also $\langle q\rangle$ is defined as $\langle n_q\rangle$ where $n_q\equiv
q(N),n_q\equiv1(q).$ Let $\frak m_1$ be a maximal ideal of $S_1$ containing the image of $\frak m$, where $\frak m$ is a
maximal ideal of $\bold T_H(N)$ with associated irreducible $\rho_\frak m$. We will also assume\linebreak that $\rho_\frak
m(\roman{Frob}\ \!q)$ has distinct eigenvalues. (We will only need this case and\linebreak it simplifies the exposition.) Let
$\frak m_q$ denote the corresponding maximal ide-als of $\bold T_H(N,q)$ and $\bold T_H(Nq)$ under the natural restriction maps
$\bold T_H(Nq)\rightarrow\bold T_H(N,q)\rightarrow S_1.$ The corresponding maps on completions are
$$\leqalignno{\bold T_H(Nq)_{\frak m_q}&\mathop{\longrightarrow}\limits^\beta\bold T_H(N,q)_{\frak m_q}&(2.22)\cr
&\mathop{\longrightarrow}\limits^\alpha S_{1,\frak m_1}\simeq\bold T_H(N)_\frak m\mathop{\otimes}\limits_{W(k_\frak
m)}W(k^+)}$$
where $k^+$ is the extension of $k_\frak m$ generated by the eigenvalues of $\{\rho_\frak m(\roman{Frob}\ \!q)\}.$ That $k^+$
is either equal to $k_\frak m$ or its quadratic extension. The maps $\beta,\alpha$ are surjective, the latter because $T_q$ is
a trace in the 2-dimensional representation
\eject
\noindent to $\roman{GL}_2(\bold T_H(N)_\frak m)$ given after Theorem 2.1 and hence is `redundant' by the \v Cebotarev density
theorem. The completions are Gorenstein by Corollary 2 to Theorem 2.1 and so we define invariant ideals of $S_{1,\frak m_1}$
$$(\Delta)=(\alpha\circ\hat\alpha),\ \ \ (\Delta')=(\alpha\circ\beta)\circ(\widehat{\alpha\circ\beta}).\leqno(2.23)$$
Let $\alpha_q$ be the image of $U_1$ in $\bold T_H(N)_\frak m\mathop{\otimes}\limits_{W(k_\frak m)}W(k^+)$ under the last
isomorphism in (2.22). The proof of Proposition 2.4 yields
\vskip6pt
{\smc Proposition} $2.4'$. {\it Suppose that $\rho_m$ is irreducible where $\frak m$ is a maximal ideal of $\bold T_H(N)$
and that $\rho_\frak m(\roman{Frob}\ \!q)$ has distinct eigenvalues. Then
$$\aligned(\Delta)&=(\alpha_q^2-\langle q\rangle),\\
(\Delta')&=(\alpha_q^2-\langle q\rangle)(q-1).\endaligned$$}
\vskip6pt
{\it Remark.} Note that if we suppose also that $q\equiv1(p)$ then $(\Delta)$ is the unit ideal and $\alpha$ is an isomorphism
in (2.22).
\
\
\centerline{\bf3. The main conjectures}
\
As we suggested in Chapter 1, in order to study the deformation theory of $\rho_0$ in detail we need to assume that it is
modular. That this should always be so for $\det\rho_0$ odd was conjectured by Serre. Serre also made a conjecture (the
`$\varepsilon$'-conjecture) making precise where one could find a lifting of $\rho_0$ once one assumed it to be modular (cf.
[Se]). This has now been proved by the combined efforts of a number of authors including Ribet, Mazur, Carayol, Edixhoven and
others. The most difficult step was to show that if $\rho_0$ was unramified at a prime $l$ then one could find a lifting in
which $l$ did not divide the level. This was proved (in slightly less generality) by Ribet. For a precise statement and
complete references we refer to Diamond's paper [Dia] which removed the last restrictions referred to in Ribet's survey
article [Ri3]. The following is a minor adaptation of the epsilon conjecture to our situation which can be found in [Dia, Th.
6.4]. (We wish to use weight 2 only.) Let $N(\rho_0)$ be the prime to $p$ part of the conductor of $\rho_0$ as defined for
example in [Se].
\vskip6pt
{\smc Theorem} 2.14. {\it Suppose that $\rho_0$ is modular and satisfies} (1.1) ({\it so in\linebreak particular is
irreducible}) {\it and is of type $\Cal D=(\cdot,\Sigma,\Cal O,\Cal M)$ with $\cdot=\roman{Se,\ str}$ or $\roman{fl}$. Suppose
that at least one of the following conditions holds} (i) $p\gt3$ {\it or} (ii) $\rho_0$ {\it is not induced from a character
of $\bold Q(\sqrt{-3})$. Then there exists a newform $f$ of weight $2$ and a prime $\lambda$ of $\Cal O_f$ such that
$\rho_{f,\lambda}$ is of type $\Cal D'=(\cdot,\Sigma,\Cal O',\Cal M)$ for some $\Cal O',$ and such that
$(\rho_{f,\lambda}\roman{\ \!mod\ \!}\lambda)\simeq\rho_0$ over $\overline{\bold F}_p$. Moreover we can\linebreak assume that
$f$ has character $\chi_f$ of order prime to $p$ and has level $N(\rho_0)p^{\delta(\rho_0)}$}\linebreak
\eject
\noindent{\it where} $\delta(\rho_0)=0$ {\it if} $\rho_0|_{D_p}$ {\it is associated to a finite flat group scheme over $\bold
Z_p$\linebreak and $\det\rho_0\Big|_{I_p}=\omega,$ and $\delta(\rho_0)=1$ otherwise. Furthermore in the Selmer case we can
assume that
$a_p(f)\equiv\chi_2(\roman{Frob}\ \!p)\roman{\ \!mod\ \!}\lambda$ in the notation of} (1.2) {\it where $a_p(f)$ is the
eigenvalue of $U_p$.}
\vskip6pt
For the rest of this chapter we will assume that $\rho_0$ is modular and that if $p=3$ then $\rho_0$ is not induced from a
character of $\bold Q(\sqrt{-3})$. Here and in the rest of the paper we use the term `induced' to signify that the
representation is induced after an extension of scalars to the algebraic closure.
For each $\Cal D=\{\cdot,\Sigma,\Cal O,\Cal M\}$ we will now define a Hecke ring $\bold T_\Cal D$ except where $\cdot$ is
unrestricted. Suppose first that we are in the flat, Slemer or strict cases. Recall that when referring to the flat case we
assume that $\rho_0$ is not ordinary and that $\det\rho_0|_{I_p}=\omega$. Suppose that $\Sigma=\{q_i\}$ and that
$N(\rho_0)=\mathbreak\Pi q_i^{s_i}$ with $s_i\ge0.$ If $U_\lambda\simeq k^2$ is the representation space of $\rho_0$ we set
$n_q=\dim_k(U_\lambda)^{I_q}$ where $I_q$ in the inertia group at $q$. Define $M_0$ and $M$ by
$$M_0=N(\rho_0)\prod_{n_{q_i}=1\atop q_i\not\in\Cal M\cup\{p\}}q_i\cdot\prod_{n_{q_i}=2}q_i^2,\ \ \
M=M_0p^{\tau(\rho_0)}\leqno(2.24)$$
where $\tau(\rho_0)=1$ if $\rho_0$ is ordinary and $\tau(\rho_0)=0$ otherwise. Let $H$ be the subgroup of $(\bold Z/M\bold
Z)^*$ generated by the Sylow $p$-subgroup of $(\bold Z/q_i\bold Z)^*$ for each $q_i\in\Cal M$ as well as by all of $(\bold
Z/q_i\bold Z)^*$ for each $q_i\in\Cal M$ of type (A). Let $\bold T_H'(M)$ denote the ring generated by the standard Hecke
operators $\{T_l$ for $l\nmid Mp,\langle a\rangle$ for $(a,Mp)=1\}$. Let $\frak m'$ denote the maximal ideal of $\bold
T'_H(M)$ associated to the\linebreak $f$ and $\lambda$ given in the theorem and let $k_{\frak m'}$ be the residue field $\bold
T'_H(M)/\frak m$. Note that $\frak m'$ does not depend on the particular choice of pair $(f,\lambda)$ in theorem 2.14. Then
$k_{\frak m'}\simeq k_0$ where $k_0$ is the smallest possible field of definition for $\rho_0$ because $k_{\frak m'}$ is
generated by the traces. Henceforth we will identify $k_0$ with $k_{\frak m'}$. There is one exceptional case where $\rho_0$
is ordinary and $\rho_0|_{D_p}$ is isomorphic to a sum of two distinct unramified characters ($\chi_1$ and $\chi_2$ in the
notation of Chapter 1, \S1). If $\rho_0$ is not exceptional we define
$$\bold T_{\Cal D}=\bold T_H'(M)_{\frak m'}\mathop{\otimes}\limits_{W(k_0)}\Cal O.\leqno(2.25(\roman a))$$
If $\rho_0$ is exceptional we let $\bold T''_H(M)$ denote the ring generated by the operators $\{T_l$ for $l\nmid Mp,\langle
a\rangle$ for $(a,Mp)=1,U_p\}$. We choose $\frak m''$ to be a maximal\linebreak ideal of $\bold T''_H(M)$ lying above $\frak
m'$ for which there is an embedding $k_{\frak m''}\hookrightarrow k$ (over\linebreak $k_0=k_{\frak m'}$) satisfying
$U_p\rightarrow\chi_2(\roman{Frob}\ \!p$). (Note that $\chi_2$ is specified by $\Cal D$.) Then in the exceptional case
$k_{\frak m''}$ is either $k_0$ or its quadratic extension and we define
$$\bold T_\Cal D=\bold T_H''(M)_{\frak m''}\mathop{\otimes}\limits_{W(k_{\frak m''})}\Cal O.\leqno(2.25(\roman b))$$
The omission of the Hecke operators $U_q$ for $q|M_0$ ensures that $\bold T_\Cal D$ is reduced.\linebreak
\eject
We need to relate $\bold T_\Cal D$ to a Hecke ring with no missing operators in order to apply the results of Section 1.
\vskip6pt
{\smc Proposition} 2.15. {\it In the nonexceptional case there is a maximal ideal\linebreak$\frak m$ for $\bold T_H(M)$ with
$\frak m\cap_H'(M)=\frak m'$ and $k_0=k_\frak m$, and such that the natural map $\bold T'_H(M)_{\frak m'}\rightarrow\bold
T_H(M)_\frak m$ is an isomorphism}, {\it thus given}
$$\bold T_\Cal D\simeq\bold T_H(M)_\frak m\mathop{\otimes}\limits_{W(k_0)}\Cal O.$$
{\it In the exceptional case the same statements hold with $\frak m''$ replacing $\frak m',\bold T_H''(M)$ replacing $\bold
T'_H(M)$ and $k_{\frak m''}$ replacing $k_0$.}
\vskip6pt
{\it Proof.} For simplicity we describe the nonexceptional case indicating where appropriate the slight modifications needed
in the exceptional case. To construct $\frak m$ we take the eigenform $f_0$ obtain from the newform $f$ of Theorem 2.14 by
removing the Euler factors at all primes $q\in\Sigma-\{\Cal M\cup p\}.$ If $\rho_0$ is ordinary and $f$ has level prime to $p$
we also remove the Euler factor $(1-\beta_p\cdot p^{-s})$ where $\beta_p$ is the non-unit eigenvalue in $\Cal O_{f\lambda}$.
(By `removing Euler factors' we mean take the eigenform whose $L$-series is that of $f$ with these Euler factors removed.)
Then $f_0$ is an eigenform of weight 2 on $\Gamma_H(M)$ (this is ensured by the choice of $f$) with $\Cal O_{f,\lambda}$
coefficients. We have a corresponding homomorphism $\pi_{f_0}:\bold T_H(M)\rightarrow\Cal O_{f,\lambda}$ and we let $\frak
m=\pi_{f_0}^{-1}(\lambda).$
Since the Hecke operators we have used to generate $\bold T_H'(M)$ are prime to the level these is an inclusion with finite
index
$$\bold T'_H(M)\hookrightarrow\prod\Cal O_g$$
where $g$ runs over representatives of the Galois conjugacy classes of newforms associated to $\Gamma_H(M)$ and where we note
that by multiplicity one $\Cal O_g$ can also be described as the ring of integers generated by the eigenvalues of the
operators in $\bold T_H'(M)$ acting on $g$. If we consider $\bold T_H(M)$ in place of $\bold T'_H(M)$ we get a similar map but
we have to replace the ring $\Cal O_g$ by the ring
$$S_g=\Cal O_g[X_{q_1},\dots,X_{q_r},X_p]/\{Y_i,Z_p\}^r_{i=1}$$
where $\{p,p_1,\dots,q_r\}$ are the distinct primes dividing $Mp$. Here
$$Y_i=\cases X^{r_i-1}_{q_i}\Big(X_{q_i}-\alpha_{q_i}(g)\Big)\Big(X_{q_i}-\beta_{q_i}(g)\Big)&\roman{if}\
q_i\nmid\roman{level}(g)\\
X^{r_i}_{q_i}\Big(X_{q_i}-a_{q_i}(g)\Big)&\roman{if}\ q_i|\ \roman{level}(g),\endcases\leqno(2.26)$$
where the Euler factor of $g$ at $q_i$ (i.e., of its associated $L$-series) is\linebreak
$(1-\alpha_{q_i}(g)q_i^{-s})(1-\beta_{q_i}(g)q_i^{-s})$ in the first cases and $(1-a_{q_i}(g)q_i^{-s})$ in the second\linebreak
case, and
$q_i^{r_i}||\Big(M/\roman{level}(g)\Big).$ (We allow $a_{q_i}(g)$ to be zero here.) Similarly $Z_p$ is
\eject
\noindent defined by
$$Z_p=\cases X^2_p-a_p(g)X_p+p\chi_g(p)&\roman{if}\ p|M,p\nmid\roman{level}(g)\\
X_p-a_p(g)&\roman{if}\ p\nmid M\\
X_p-a_p(g)&\roman{if}\ p|\roman{level}(g),\endcases$$
where the Euler factor of $g$ at $p$ is $(1-a_p(g)p^{-s}+\chi_g(p)p^{1-2s})$ in the first two cases and $(1-a_p(g)p^{-s})$ in
the third case. We then have a commutative diagram
$$\matrix \bold T_H'(M)&\subset\!\!\!{\lower2.69pt\hbox{$\longrightarrow$}}&\prod\limits_g\Cal O_g\\
\cap\hskip-9pt{\lower10pt\hbox{$\Big\downarrow$}}&&\cap\hskip-9pt{\lower10pt\hbox{$\Big\downarrow$}}\\
\bold T_H(M)&\subset\!\!\!{\lower2.69pt\hbox{$\longrightarrow$}}&\prod\limits_gS_g=\prod\limits_g\Cal
O_g[X_{q_1},\dots,X_{q_r},X_p]/\{Y_i,Z_p\}^r_{i=1}\endmatrix\leqno(2.27)$$
where the lower map is given on $\{U_q,U_p$ or $T_p\}$ by $U_{q_i}\longrightarrow X_{q_i},U_p$ or\linebreak $T_p\longrightarrow
X_p$ (according as $p|M$ or $p\nmid M$). To verify the existence of such a homomorphism one considers the action of $\bold
T_H(M)$ on the space of forms of weight 2 invariant under $\Gamma_H(M)$ and uses that $\sum_{j=1}^rg_j(m_jz)$ is a free
generator as a $\bold T_H(M)\otimes\bold C$-module where $\{g_j\}$ runs over the set of newforms and
$m_j=M/\roman{level}(g_j).$
Now we tensor all the rings in (2.27) with $\bold Z_p$. Then completing the top row of (2.27) with respect to $\frak m'$ and
the bottom row with respect to $\frak m$ we get a commutative diagram
$$\matrix\bold T'_H(M)_{\frak
m'}&\subset\!\!\!{\lower2.69pt\hbox{$\longrightarrow$}}&\Big(\prod\limits_g\Cal O_g\Big)_{\frak
m'}&\simeq&\prod\limits_{g\atop\frak m'\rightarrow\mu}\Cal O_{g,\mu}\\
\Bigg\downarrow&&\Bigg\downarrow&&\Bigg\downarrow\\
\bold T_H(M)_\frak
m&\subset\!\!\!{\lower2.69pt\hbox{$\longrightarrow$}}&\Big(\prod\limits_gS_g\Big)_\frak
m&\simeq&\prod(S_g)_\frak m.\endmatrix\leqno(2.28)$$
Here $\mu$ runs through the primes above $p$ in each $\Cal O_g$ for which $\frak m'\rightarrow\mu$ under $\bold
T_{H'}(M)\rightarrow\Cal O_g.$ Now $(S_g)_\frak m$ is given by
$$\leqalignno{(S_g\otimes\bold Z_p)_\frak m&\simeq\Big((\Cal O_g\otimes\bold
Z_p)[X_{q_1},\dots,X_{q_r},X_p]/\{Y_i,Z_p\}^r_{i=1}\Big)_\frak m&(2.29)\cr
&\simeq\Bigg(\textstyle\prod\limits_{\mu|p}\Cal O_{g,\mu}[X_{q_1},\dots,X_{q_r},X_p]/\{Y_i,Z_p\}^r_{i=1}\Bigg)_\frak m\cr
&\simeq\Bigg(\textstyle\prod\limits_{\mu|p}A_{g,\mu}\Bigg)_m}$$
where $A_{g,\mu}$ denotes the product of the factors of the complete semi-local ring $\Cal
O_{g,\mu}[X_{q_1},\dots,X_{q_r},X_p]/\{Y_i,Z_p\}^r_{i=1}$ in which $X_{q_i}$ is topologically nilpotent for\linebreak
\eject
\noindent$q_i\not\in\Cal M$ and in which $X_p$ is a unit if we are in the ordinary case (i.e., when $p|M$). This is because
$U_{q_i}\in\frak m$ if $q_i\not\in\Cal M$ and $U_p$ is a unit at $\frak m$ in the ordinary case.
Now if $\frak m'\rightarrow\mu$ then in $(A_{g,\mu})_\frak\mu$ we claim that $Y_i$ is given up to a unit by $X_{q_i}-b_i$ for
some $b_i\in\Cal O_{g,\mu}$ with $b_i=0$ if $q_i\not\in\Cal M.$ Similarly $Z_p$ is given up to a\linebreak unit by
$X_p-\alpha_p(g)$ where $\alpha_p(g)$ is the unit root of $x^2-a_p(g)x+p\chi_g(p)=0$ in\linebreak $\Cal O_{g,\mu}$ if
$p\nmid\roman{level}\ \!g$ and $p|M$ and by $X_p-a_p(g)$ if $p|\roman{level}\ \!g$ or $p\nmid M$. This\linebreak will show that
$(A_{g,\mu})_\frak m\simeq\Cal O_{g,\mu}$ when $\frak m'\rightarrow\mu$ and $(A_{g,\mu})_\frak m=0$ otherwise.
For $q_i\in\Cal M$ and for $p$, the claim is straightforward. For $q_i\not\in\Cal M$, it amounts to the following. Let
$U_{g,\mu}$ denote the 2-dimensional $K_{g,\mu}$-vector space with Galois action via $\rho_{g,\mu}$ and let
$n_{q_i}(g,\mu)=\dim(U_{g,\mu})^{I_{q_i}}$. We wish to check that $Y_i=\roman{unit}.X_{q_i}$ in $(A_{g,\mu})_\frak m$ and
from the definition of $Y_i$ in (2.26) this reduces\linebreak to checking that $r_i=n_{q_i}(g,\mu)$ by the
$\pi_q\simeq\pi(\sigma_q)$ of theorem (cf. [Ca1]). We use here that $\alpha_{q_i}(g),\beta_{q_i}(g)$ and $a_{q_i}(g)$ are
$p$-adic units when they are nonzero since they are eigenvalues of $\roman{Frob}(q_i)$. Now by definition the power of $q_i$
dividing $M$\linebreak is the same as that dividing $N(\rho_0)q_i^{n_{q_i}}$ (cf. (2.21)). By an observation of Livn\'e (cf.
[Liv], [Ca2,\S1]),
$$\roman{ord}_{q_i}(\roman{level}\ \!g)=\roman{ord}_{q_i}\Big(N(\rho_0)q_i^{n_{q_i}-n_{q_i}(g,\mu)}\Big).\leqno(2.30)$$
As by definition $q_i^{r_i}||(M/\roman{level}\ \!g)$ we deduce that $r_i=n_{q_i}(g,\mu)$ as reqired.
We have now shown that each $A_{g,\mu}\simeq\Cal O_{g,\mu}$ (when $\frak m'\rightarrow\mu$) and it follows from (2.28) and
(2.29) that we have homomorphisms
$$\bold T_H'(M)_{\frak m'}\subset\!\!\!{\lower2.69pt\hbox{$\longrightarrow$}}\bold T_H(M)_\frak
m\subset\!\!\!{\lower2.69pt\hbox{$\longrightarrow$}}\prod\limits_{g\atop\frak m'\rightarrow\mu}\Cal O_{g,\mu}$$
where the inclusions are of finite index. Moreover we have seen that $U_{q_i}=0$ in $\bold T_H(M)_\frak m$ for $q_i\not\in\Cal
M.$ We now consider the primes $q_i\in\Cal M.$ We have to show that the operators $U_q$ for $q\in\Cal M$ are redundant in the
sense that they lie in $\bold T'_H(M)_{\frak m'}$, i.e., in the $\bold Z_p$-subalgebra of $\bold T_H(M)_\frak m$ generated by
the $\{T_l:l\nmid Mp,\langle a\rangle:a\in(\bold Z/M\bold Z)^*\}$. For $q\in\Cal M$ of type (A), $U_q\in\bold T'_H(M)_{\frak
m'}$ as explained in Remark 2.9 are for $q\in\Cal M$ of type (B), $U_q\in\bold T'_H(M)_{\frak m'}$ as explained in Remark
2.11. For $q\in\Cal M$ of type (C) but not of type (A), $U_q=0$ by the $\pi_q\simeq\pi(\sigma_q)$ theorem (cf. [Ca1]). For in
this case $n_q=0$ whence also $n_q(g,\mu)=0$ for each pair $(g,\mu)$ with $\frak m'\rightarrow\mu.$ If $\rho_0$ is strict or
Selmer at $p$ then $U_p$ can be recovered from the two-dimensional representation $\rho$ (described after the corollaries to
Theorem 2.1) as the eigenvalue of $\roman{Frob}\ \!p$ on the (free, of rank one) unramified quotient (cf. Theorem 2.1.4 of
[Wi4]). As this representation is defined over the $\bold Z_p$-subalgebra generated by the traces, it follows that $U_p$ is
contained in this subring. In the exceptional case $U_p$ is in $\bold T_H''(M)_{\frak m''}$ by definition.
Finally we have to show that $T_p$ is also redundant in the sense explained above when $p\nmid M$. A proof of this has already
been given in Section 2 (Ribet's\linebreak
\eject
\noindent lemma). Here we give an alternative argument using the Galois representa-tions. We know that $T_p\in\frak m$ and it
will be enough to show that $T_p\in(\frak m^2,p).$ Writing $k_\frak m$ for the residue field $\bold T_H(M)_\frak m/\frak m$
we reduce to the following situation. If $T_p\not\in(\frak m^2,p)$ then there is a quotient
$$\bold T_H(M)_\frak m/(\frak m^2,p)\twoheadrightarrow k_\frak m[\varepsilon]=\bold T_H(M)_\frak m/\frak a$$
where $k_\frak m[\varepsilon]$ is the ring of dual numbers (so $\varepsilon^2=0$) with the property that
$T_p\mapsto\lambda\varepsilon$ with $\lambda\ne0$ and such that the image of $\bold T'_H(M)_{\frak m'}$ lies in $k_\frak m$.
Let $G_{/\bold Q}$ denote the four-dimensional $k_\frak m$-vector space associated to the representation
$$\rho_\varepsilon:\roman{Gal}(\overline{\bold Q}/\bold Q)\longrightarrow\roman{GL}_2(k_\frak m[\varepsilon])$$
induced from the representation in Theorem 2.1. It has the form
$$G_{/\bold Q}\simeq{G_0}_{/\bold Q}\oplus{G_0}_{/\bold Q}$$
where $G_0$ is the corresponding space associated to $\rho_0$ by our hypothesis that the traces lie in $k_\frak m$. The
semisimplicity of $G_{/\bold Q}$ here is obtained from the main\linebreak theorem of [BLR]. Now $G_{/\bold Q_p}$ extends to a
finite flat group scheme $G_{/\bold Z_p}$.\linebreak Explicitly it is a quotient of the group scheme $J_H(M)_\frak
m[p]_{/\bold Z_p}$. Since extensions\linebreak to $\bold Z_p$ are unique (cf. [Ray1]) we know
$$G_{/\bold Z_p}\simeq {G_0}_{/\bold Z_p}\oplus {G_0}_{/\bold Z_p}.$$
Now by the Eichler-Shimura relation we know that in $J_H(M)_{/\bold F_p}$
$$T_p=F+\langle p\rangle F^T.$$
Since $T_p\in\frak m$ it follows that $F+\langle p\rangle F^T=0$ on ${G_0}_{/\bold F_p}$ and hence the same holds on
$G_{/\bold F_p}$. But $T_p$ is an endomorphism of $G_{/\bold Z_p}$ which is zero on the special fibre, so by [Ray1, Cor.
3.3.6], $T_p=0$ on $G_{/\bold Z_p}$. It follows that $T_p=0$ in $k_\frak m[\varepsilon]$ which contradicts our earlier
hypothesis. So $T_p\in(\frak m^2,p)$ as required. This completes the proof of the proposition.\hfill$\qed$
\vskip6pt
>From the proof of the proposition it is also clear that $\frak m$ is the unique max-imal ideal of $\bold T_H(M)$ extending
$\frak m'$ and satisfying the conditions that $U_q\in\frak m$ for $q\in\Sigma-\{\Cal M\cup p\}$ and $U_p\not\in\frak m$ if
$\rho_0$ is ordinary. For the rest of this chapter we will always make this choice of $\frak m$ (given $\rho_0$).
Next we define $\bold T_\Cal D$ in the case when $\Cal D=(\roman{ord},\Sigma,\Cal O,\Cal M)$. If $\frak n$ is any ordinary
maximal ideal (i.e. $U_p\not\in\frak n$) of $\bold T_H(Np)$ with $N$ prime to $p$ then Hida has constructed a 2-dimensional
Noetherian local Hecke ring
$$\bold T_\infty=e\bold T_H(Np^\infty)_\frak n:=\lim\limits_{\longleftarrow} e\bold T_H(Np^r)_{\frak n_r}$$
which is a $\Lambda=\bold Z_p[\![T]\!]$-algebra satisfying $\bold T_\infty/T\simeq\bold T_H(Np)_\frak n$. Here $\frak n_r$ is
the inverse image of $\frak n$ under the natural restriction map. Also $T=\lim\limits_{\longleftarrow}\langle1+Np\rangle-1$
and $e=\lim\limits_{\scriptstyle\longrightarrow\atop\textstyle r}U_p^{r!}.$ For an irreducible $\rho_0$ of type $\Cal D$ we
have defined
$\bold T_{\Cal D'}$ in (2.25(a)), where $\Cal D'=(\roman{Se},\Sigma,\Cal O,\Cal M)$ by
$$\bold T_{\Cal D'}\simeq\bold T_H(M_0p)_\frak m\mathop{\otimes}\limits_{W(k_\frak m)}\Cal O,$$
the isomorphism coming from Proposition 2.15. We will define $\bold T_\Cal D$ by
$$\bold T_\Cal D=e\bold T_H(M_0p^\infty)_\frak m\mathop{\otimes}_{W(k_\frak m)}\Cal O.\leqno(2.31)$$
In particular we see that
$$\bold T_\Cal D/T\simeq\bold T_{\Cal D'},\leqno(2.32)$$
i.e., where $\Cal D'$ is the same as $\Cal D$ but with `Selmer' replacing `ord'. Moreover if $\frak q$ is a height one prime
ideal of $\bold T_\Cal D$ containing $\Big((1+T)^{p^n}-(1+Np)^{p^n(k-2)}\Big)$ for any integers $n\ge0,k\ge2,$ then $\bold
T_\Cal D/\frak q$ is associated to an eigenform in a natural way (generalizing the case $n=0,k=2$). For more details about
these rings as well as about $\Lambda$-adic modular forms see for example [Wi1] or [Hi1].
For each $n\ge1$ let $\bold T_n=\bold T_H(M_0p^n)_{\frak m_n}$. Then by the argument given after the statement ofTheorem 2.1
we can construct a Galois representation $\rho_n$ unramified outside $Mp$ with values in $\roman{GL}_2(\bold T_n)$ satisfying
$\roman{trace}\ \!\rho_n(\roman{Frob}\ \!l)=T_l,\mathbreak\det\rho_n(\roman{Frob}\ \!l)=l\langle l\rangle$ for $(l,Mp)=1$.
These representations can be patched together to give a continuous representation
$$\rho=\lim\limits_{\longleftarrow}\rho_n:\roman{Gal}(\bold Q_\Sigma/\bold Q)\longrightarrow\roman{GL}_2(\bold
T_\Cal D)\leqno(2.33)$$
where $\Sigma$ is the set of primes dividing $Mp$. To see this we need to check the commutativity of the maps
\vskip6pt
$$\matrix R_\Sigma\longrightarrow\bold T_n\\
\hskip12pt\searrow\hskip12pt\downarrow\\
\hskip30pt\bold T_{n-1}\endmatrix$$
where the horizontal maps are induced by $\rho_n$ and $\rho_{n-1}$ and the vertical map is the natural one. Now the
commutativity is valid on elements of $R_\Sigma$, which are traces or determinants in the universal representation, since
$\roman{trace}(\roman{Frob}\ \!l)\mapsto T_l$ under both horizontal maps and similarly for determinants. Here $R_\Sigma$ is
the universal deformation ring described in Chapter 1 with respect to $\rho_0$ viewed with residue field $k=k_\frak m$. It
suffices then to show that $R_\Sigma$ is generated (topologically) by traces and this reduces to checking that there are no
nonconstant deformations of $\rho_0$ to $k[\varepsilon]$ with traces lying in $k$ (cf. [Ma1, \S1.8]). For then if
$R_\Sigma^\roman{tr}$\linebreak denotes the closed $W(k)$-subalgebra of $R_\Sigma$ generated by the traces we see
that\linebreak
$R_\Sigma^\roman{tr}\rightarrow(R_\Sigma/m^2)$ is surjective, $m$ being the maximal ideal of $R_\Sigma$, from which\linebreak
we easily conclude that $R_\Sigma^\roman{tr}=R_\Sigma$. To see that the condition holds, assume\linebreak
\eject
\noindent that a basis is chosen so that $\rho_0(c)=({1\atop0}{\ \ 0\atop-1})$ for a chosen complex cunjugation $c$ and
$\rho_0(\sigma)=({a_\sigma\atop c_\sigma}{b_\sigma\atop d_\sigma})$ with $b_\sigma=1$ and $c_\sigma\ne0$ for some $\sigma$.
(This is possible because $\rho_0$ is irreducible.) Then any deformation $[p]$ to $k[\varepsilon]$ can be represented by a
representation $\rho$ such that $\rho(c)$ and $\rho(\sigma)$ have the same properties. It follows easily that if the traces
of $\rho$ lie in $k$ then $\rho$ takes values in $k$ whence it is equal to $\rho_0$. (Alternatively one sees that the
universal representation can be defined over $R_\Sigma^\roman{tr}$ by diagonalizing complex conjugation as before. Since the
two maps $R_\Sigma^\roman{tr}\rightarrow\bold T_{n-1}$ induced by the triangle are the same, so the associ-\linebreak ated
representations are equivalent, and the universal property then implies the commutativity of the triangle.)
The representations (2.33) were first exhibited by Hida and were the original inspiration for Mazur's deformation theory.
For each $\Cal D=\{\cdot,\Sigma,\Cal O,\Cal M\}$ where $\cdot$ is not unrestricted there is then a canonical surjective map
$$\varphi_\Cal D:R_\Cal D\rightarrow\bold T_\Cal D$$
which induces the representations described after the corollaries to Theorem 2.1\linebreak and in (2.33). It is enough to check
this when $\Cal O=W(k_0)$ (or $W(k_{\frak m''}$) in the exceptional case). Then one just has to check that for every pair
$(g,\mu)$ which appears in (2.28) the resulting representation is of type $\Cal D$. For then we claim that the image of the
canonical map $R_\Cal D\rightarrow\widetilde{\bold T_\Cal D}=\Pi\Cal O_{g,\mu}$ is $\bold T_\Cal D$ where here $\sim$
denotes the normalization. (In the case where $\cdot$ is ord this needs to be checked instead for $\bold
T_n\mathop{\otimes}\limits_{W(k_0)}\Cal O$ for each $n$.) For this we just need to see that $R_\Cal D$ is generated by traces.
(In the exceptional case we have to show also that $U_p$ is in the image. This holds because it can be identified, using
Theorem 2.1.4 of [Wi1], with the image of $u\in R_\Cal D$ where $u$ is the eigenvalue of $\roman{Frob}\ \!p$ on the unique
rank one unramified quotient of $R^2_\Cal D$ with eigenvalue $\equiv\chi_2(\roman{Frob}\ \!p)$ which is specified in the
definition of $\Cal D$.) But we saw above that this was true for $R_\Sigma$. The same then holds for $R_\Cal D$ as
$R_\Sigma\rightarrow R_\Cal D$ is surjective because the map on reduced cotangent spaces is surjective (cf. (1.5)). To check
the condition on the pairs $(g,\mu)$ observe first that for $q\in\Cal M$ we have imposed the following conditions on the level
and character of such $g$'s by our choice of $M$ and $H$:
\vskip6pt
\hskip4pt$q$ of type (A): $q||$ level $g,\det\rho_{g,\mu}\Big|_{I_q}=1,$
\vskip6pt
\hskip4pt$q$ of type (B): cond $\chi_q||$ level $g,\det\rho_{g,\mu}\Big|_{I_q}=\chi_q,$
\vskip6pt
\hskip4pt$q$ of type (C): $\det\rho_{g,\mu}\Big|_{I_q}$ is the Teichm\"uller lifting of $\det\rho_0\Big|_{I_q}$.
\vskip6pt
In the first two cases the desired form of $\rho_{q,\mu}\Big|_{D_q}$ then follows from the $\pi_q\simeq\pi(\sigma_q)$ theorem
of Langlands (cf. [Ca1]). The third case is already of\linebreak
\eject
\noindent type (C). For $q=p$ one can use Theorem 2.1.4 of [Wi1] in the ordinary case, the flat case being well-known.
The following conjecture generalized a fundamantal conjecture of Mazur and Tilouine for $\Cal
D=(\roman{ord},\Sigma,W(k_0),\phi)$; cf. [MT].
\vskip6pt
{\smc Conjecture} 2.16. $\varphi_\Cal D$ {\it is an isomorphism.}
\vskip6pt
Equivalently this conjecture says that the representation described after the corollaries to Theorem 2.1 (or in (2.33) in the
ordinary case) is the universal one for a suitable choice of $H,N$ and $\frak m$. We remind the reader that throughout this
section we are assuming that if $p=3$ then $\rho_0$ is not induced from a character of $\bold Q(\sqrt{-3}\ \!)$.
\vskip6pt
{\it Remark.} The case of most interest to us is when $p=3$ and $\rho_0$ is a representation with values in
$\roman{GL}_2(\bold F_3)$. In this case it is a theorem of Tunnell, extending results of Langlands, that $\rho_0$ is always
modular. For $\roman{GL}_2(\bold F_3)$ is a double cover of $S_4$ and can be embedded in $\roman{GL}_2(\bold Z[\sqrt{-2}\
\!])$ whence in $\roman{GL}_2(\bold C);$ cf. [Se] and [Tu]. The conjecture will be proved with a mild restriction on $\rho_0$
at the end of Chapter 3.
\vskip6pt
{\it Remark.} Our original restriction to the types (A), (B), (C) for $\rho_0$ was motivated by the wish that the deformation
type (a) be of minimal conductor among its twists, (b) retain property (a) under unramified base changes. Without this kind of
stability it can happen that after a base change of $\bold Q$ to an\linebreak extension unramified at
$\Sigma,\rho_0\otimes\psi$ has smaller `conductor' for some character $\psi$. The typical example of this is where
$\rho_0\Big|_{D_q}=\roman{Ind}_K^{\bold Q_q}(\chi)$ with $q\equiv-1(p)$ and $\chi$ is a ramified character over $K$, the
unramified quadratic extension of $\bold Q_q$. What makes this difficult for us is that there are then nontrivial ramified
local deformations $(\roman{Ind}_K^{\bold Q_p}\chi\xi$ for $\xi$ a ramified character of order $p$ of $K$) which we cannot
detect by a change of level.
\vskip6pt
For the purposes of Chapter 3 it is convenient to digress now in order to introduce a slight varient of the deformation rings
we have been considering so far. Suppose that $\Cal D=(\cdot,\Sigma,\Cal O,\Cal M)$ is a standard deformation problem
(associated to $\rho_0$) with $\cdot=$ Se, str or fl and suppose that $H,M_0,M$ and $\frak m$ are defined as in (2.24) and
Proposition 2.15. We choose a finite set of primes $Q=\{q_1,\dots,q_r\}$ with $q_i\nmid Mp$. Furthermore we assume that each
$q_i\equiv1(p)$\linebreak and that the eigenvalues $\{\alpha_i,\beta_i\}$ of $\rho_0(\roman{Frob}\ \!q_i)$ are distinct for
each $q_i\in Q$. This last condition ensures that $\rho_0$ does not occur as the residual representation of the $\lambda$-adic
representation associated to any newform on $\Gamma_H(M,q_1\dots q_r)$\linebreak where any $q_i$ divides the level of the form.
This can be seen directly by looking at $(\roman{Frob}\ \!q_i)$ in such a representation or by using Proposition 2.4' at the
end of Section 2. It will be convenient to assume that the residue field of $\Cal O$ contains $\alpha_i,\beta_i$ for each
$q_i$.
\eject
Pick $\alpha_i$ for each $i$. We let $\Cal D_Q$ be the deformation problem associated to representations $\rho$ of
$\roman{Gal}(\bold Q_{\Sigma\cup Q}/\bold Q)$ which are of type $\Cal D$ and which in addition satisfy the property that at
each $q_i\in Q$
$$\rho\Big|_{D_{q_i}}\sim\pmatrix\chi_{1,q_i}&\\
&\chi_{2,q_i}\endpmatrix\leqno(2.34)$$
with $\chi_{2,q_i}$ unramified and $\chi_{2,q_i}(\roman{Frob}\ \!q_i)\equiv\alpha_i\mod\frak m$ for a suitable choice of
basis. One checks as in Chapter 1 that associated to $\Cal D_Q$ there is a universal deformation ring $R_Q$. (These new
contions are really variants on type (B).)
We will only need a corresponding Hecke ring in a very special case and it is convenient in this case to define it using all
the Hecke operators. Let us now set $N=N(\rho_0)p^{\delta(\rho_0)}$ where $\delta(\rho_0)$ in as defined in Theorem 2.14. Let
$\frak m_0$ denote a maximal ideal of $\bold T_H(N)$ given by Theorem 2.14 with the property that $\rho_{\frak
m_0}\simeq\rho_0$ over $\overline{\bold F}_p$ relative to a suitable embedding of $k_{\frak m_0}\rightarrow k$ over $k_0$. (In
the exceptional case we also impose the same condition on $\frak m_0$ about the reduction of $U_p$ as in the definition of
$\bold T_\Cal D$ in the exceptional case before (2.25)(b).) Thus $\rho_{\frak m_0}\simeq\rho_{f,\lambda}\mod\lambda$ over the
residue field of $\Cal O_{f,\lambda}$ for some choice of $f$ and $\lambda$ with $f$ of level $N$. By dropping one of the Euler
factors at each $q_i$ as in the proof of Proposition 2.15, we obtain a form and hence a maximal ideal $\frak m_Q$ of $\bold
T_H(Nq_1\dots q_r)$ with the property that $\rho_{\frak m_Q}\simeq\rho_0$ over $\overline{\bold F}_p$ relative to a suitable
embedding $k_{\frak m_Q}\rightarrow k$ over $k_{\frak m_0}$. The field $k_{\frak m_Q}$ is the extension of $k_0$ (or
$k_{\frak m''}$ in the exceptional case) generated by the $\alpha_i,\beta_i$. We set
$$\bold T_Q=\bold T_H(Nq_1\dots q_r)_{\frak m_Q}\mathop{\otimes}\limits_{W(k_{\frak m_Q})}\Cal O.\leqno(2.35)$$
It is easy to see directly (or by the arguments of Proposition 2.15) that $\bold T_Q$ is reduced and that there is an
inclusion with finite index
$$\bold Q_Q\hookrightarrow\tilde{\bold T}_Q=\prod\Cal O_{g,\mu}\leqno(2.36)$$
where the product is taken over representatives of the Galois conjugacy classes of eigenforms $g$ of level $Nq_1\dots q_r$
with $\frak m_Q\rightarrow\mu$. Now define $\Cal D_Q$ using the choices $\alpha_i$ for which $U_{q_i}\rightarrow\alpha_i$
under the chosen embedding $k_{\frak m_Q}\rightarrow k.$ Then each of the 2-dimensional representations associated to each
factor $\Cal O_{g,\mu}$ is of type $\Cal D_Q$. We can check this for each $q\in Q$ using either the $\pi_q\simeq\pi(\sigma_q)$
theorem (cf. [Ca1]) as in the case of type (B) or using the Eichler-Shimura relation if $q$ does not divide the level of the
newform associated to $g$. So we get a homomorphism of $\Cal O$-algebras $R_Q\rightarrow\tilde{\bold T}_Q$ and hence also an
$\Cal O$-algebra map $$\varphi_Q:R_Q\rightarrow\bold T_Q\leqno(2.37)$$ as $R_Q$ is generated by traces. This is not an
isomorphism in general as we have used $N$ in place of $M$. However it is surjective by the arguments of Proposition 2.15.
Indeed, for $q|N(\rho_0)p$, we check that $U_q$ is in the image of\linebreak
\eject
\noindent$\varphi_Q$ using the arguments in the second half of the proof of Proposition 2.15. For $q\in Q$ we use the fact
that $U_q$ is the image of the value of $\chi_{2,q}(\roman{Frob\ \!q})$\linebreak in the universal representation;cf. (2.34).
For
$q|M$, but not of the previous types, $T_q$ is a trace in $\rho_{\bold T_Q}$ and we can apply the \v Cebotarev density theorem
to show that it is in the image of $\varphi_Q$.
Finally, if there is a section $\pi:\bold T_Q\rightarrow \Cal O$, then set $\frak p_Q=\ker\pi$ and let $\rho_\frak p$ denote
the 2-dimensional representation to $\roman{GL}_2(\Cal O)$ obtained from $\rho_{\bold T_Q}\mod\frak p_Q$. Let
$V=\roman{Ad}\rho_{\frak p}\mathop{\otimes}\limits_{\Cal O}K/\Cal O$ where $K$ is the field of fractions of $\Cal O$. We pick a
basis for $\rho_{\frak p}$ satisfying (2.34) and then let
$$\aligned V^{(q_i)}&=\bigg\lbrace\pmatrix a&0\\
0&0\endpmatrix\bigg\rbrace\\
\subseteq\roman{Ad}\rho_\frak m\mathop{\otimes}\limits_\Cal OK/\Cal O&=\bigg\lbrace\pmatrix
a&b\\
c&d\endpmatrix:a,b,c,d\in\Cal O\bigg\rbrace\mathop{\otimes}\limits_\Cal OK/\Cal O\endaligned\leqno(2.38)$$
and let $V_{(q_i)}=V/V^{(q_i)}$. Then as in Proposition 1.2 we have an isomorphism
$$\roman{Hom}_\Cal O(\frak p_{R_Q}/\frak p_{R_Q}^2,K/\Cal O)\simeq H_{\Cal D_Q}^1(\bold Q_{\Sigma\cup Q}/\bold
Q,V)\leqno(2.39)$$
where $\frak p_{R_Q}=\ker(\pi\circ\varphi_Q)$ and the second term is defined by
$$H_{\Cal D_Q}^1(\bold Q_{\Sigma\cup Q}/\bold Q,V)=\ker:H_\Cal D^1(\bold Q_{\Sigma\cup Q}/\bold
Q,V)\rightarrow\prod_{i=1}^rH^1(\bold Q_{q_i}^\roman{unr},V_{(q_i)}).\leqno(2.40)$$
We return now to our discussion of Conjecture 2.16. We will call a de-formation theory $\Cal D$ {\it minimal} if $\Sigma=\Cal
M\cup\{p\}$ and $\cdot$ is Selmer, strict or flat.\linebreak This notion will be critical in Chapter 3. (A slightly stronger
notion of minimality is described in Chapter 3 where the Selmer condition is replaced, when possible, by the condition that the
representations arise from finite flat group schemes - see the remark after the proof of Theorem 3.1.) Unfortunately even up
to twist, not every $\rho_0$ has an associated minimal $\Cal D$ even when $\rho_0$ is flat or ordinary at $p$ as explained in
the remarks after Conjecture 2.16. However this could be achieved if one replaced $\bold Q$ by a suitable finite extension
depending on $\rho_0$.
Suppose now that $f$ is a (normalized) newform, $\lambda$ is a prime of $\Cal O_f$ above $p$\linebreak and $\rho{f,\lambda}$ a
deformation of $\rho_0$ of type $\Cal D$ where $\Cal D=(\cdot,\Sigma,\Cal O_{f,\lambda},\Cal M)$ with $\cdot=$ Se,\linebreak
str or fl. (Strictly speaking we may be changing $\rho_0$ as we wish to choose its field of definition to be $k=\Cal
O_{f,\lambda}/\lambda$.) Suppose further that level$(f)|M$ where $M$ is defined by (2.24).
Now let us set $\Cal O=\Cal O_{f,\lambda}$ for the rest of this section. There is a homomorphism
$$\pi=\pi_{\Cal D,f}:\bold T_\Cal D\rightarrow\Cal O\leqno(2.41)$$
\eject
\noindent whose kernel is the prime ideal $\frak p_{\bold T,f}$ associated to $f$ and $\lambda$. Similary there is\linebreak a
homomorphism
$$R_\Cal D\rightarrow\Cal O$$
whose kernel is the prime ideal $\frak p_{R,f}$ associated to $f$ and $\lambda$ and which factors through $\pi_f$. Pick
perfect pairings of $\Cal O$-modules, the second one $\bold T_\Cal D$-bilinear,
$$\Cal O\times\Cal O\rightarrow\Cal O,\ \ \ \ \langle\ ,\rangle:\bold T_\Cal D\times\bold T_\Cal D\rightarrow\Cal
O.\leqno(2.42)$$
In each case we use the term perfect pairing to signify that the pairs of induced maps $\Cal O\rightarrow\roman{Hom}_\Cal
O(\Cal O,\Cal O)$ and $\bold T_\Cal D\rightarrow\roman{Hom}_\Cal O(\bold T_\Cal D,\Cal O)$ are isomorphisms. In addition the
second one is required to be $\bold T_\Cal D$-linear. The existence of the second\linebreak pairing is equivalent to the
Gorenstein property, Corollary 2 of Theorem 2.1, as we explain below. Explicitly if $h$ is a generator of the free $\bold
T_\Cal D$-module $\roman{Hom}_\Cal O(\bold T_\Cal D,\Cal O)$ we set $\langle t_1,t_2\rangle=h(t_1t_2).$
{\it A priori} $\bold T_H(M)_\frak m$ (occurring in the description of $\bold T_\Cal D$ in Proposition 2.15) is only
Gorenstein as a $\bold Z_p$-algebra but it follows immediately that it is also a Gorenstein $W(k_\frak m)$-algebra. (The
notion of Gorenstein $\Cal O$-algebra is explained in the appendix.) Indeed the map
$$\roman{Hom}_{W(k_\frak m)}\Big(\bold T_H(M)_\frak m,W(k_\frak m)\Big)\rightarrow\roman{Hom}_{\bold Z_p}\Big(\bold
T_H(M)_\frak,\bold Z_p\Big)$$
given by $\varphi\mapsto\roman{trace}\circ\varphi$ is easily seen to be an isomorphism, as the reduction mod $p$ is injective
and the ranks are equal. Thus $\bold T_\Cal D$ is a Gorenstein $\Cal O$-algebra.
Now let $\hat\pi:\Cal O\rightarrow\bold T_\Cal D$ be the adjoint of $\pi$ with respect to these pairings. Then define a
principal ideal $(\eta)$ of $\bold T_\Cal D$ by
$$(\eta)=(\eta_{\Cal D,f})=(\hat\pi(1)).$$
This is well-defined independently of the pairings and moreover one sees that $\bold T_\Cal D/\eta$ is torsion-free (see the
appendix). From its description $(\eta)$ is invariant under extensions of $\Cal O$ to $\Cal O'$ in an obvious way. Since
$\bold T_\Cal D$ is reduced $\pi(\eta)\ne0.$
One can also verify that
$$\pi(\eta)=\langle\eta,\eta\rangle\leqno(2.43)$$
up to a unit in $\Cal O$.
We will say that $\Cal D_1\supset\Cal D$ if we obtain $\Cal D_1$ by relaxing certain of the\linebreak hypotheses on $\Cal D$,
i.e., if
$\Cal D=(\cdot,\Sigma,\Cal O,\Cal M)$ and $\Cal D_1=(\cdot,\Sigma_1,\Cal O_1,\Cal M_1)$ we allow\linebreak that
$\Sigma_1\supset\Sigma$, any $\Cal O_1,\Cal M\supset\Cal M_1$ (but of the same type) and if $\cdot$ is Se or str\linebreak in
$\Cal D$ it can be Se, str, ord or unrestricted in $\Cal D_1$, if $\cdot$ is fl in $\Cal D_1$ it can be fl or unrestricted in
$\Cal D_1$. We use the term {\it restricted} to signify that
$\cdot$ is Se, str, fl or ord. The following theorem reduces conjecture 2.16 to a `class number'\linebreak criterion. For an
interpretation of the right-hand side of the inequality in\linebreak the theorem as the order of a cohomology group, see
Propostion 1.2. For an interpretation of the left-hand side in terms of the value of an inner product, see Proposition 4.4.
\eject
{\smc Theorem} 2.17. {\it Assume}, {\it as above}, {\it that} $\rho_{f,\lambda}$ {\it is a deformation of $\rho_0$ of\linebreak
type $\Cal D=(\cdot,\Sigma,\Cal O=\Cal O_{f,\lambda},\Cal M)$ with $\cdot=\roman{Se,\ str}$ or $\roman{fl.}$ Suppose that
$$\#\Cal O/\pi(\eta_{\Cal D,f})\ge\#\frak p_{R,f}/\frak p^2_{R,f}.$$
Then}
\vskip6pt
\hskip-10pt(i) $\varphi_{\Cal D_1}:R_{\Cal D_1}\simeq\bold T_{\Cal D_1}$ {\it is an isomorphism for all} ({\it restricted})
$\Cal D_1\supset\Cal D${\it.}
\vskip6pt
\hskip-12pt(ii) $\bold T_{\Cal D_1}$ {\it is a complete intersection} ({\it over $\Cal O_1$ if $\cdot$ is} Se, str {\it or}
fl) {\it for all re-\linebreak${}$\hskip26pt stricted $\Cal D_1\supset\Cal D$.}
\vskip6pt
{\it Proof.} Let us write $\bold T$ for $\bold T_\Cal D$, $\frak p_\bold T$ for $\frak p_{\bold T,f}$, $\frak p_R$ for $\frak
p_{R,f}$ and $\eta$ for $\eta_{\eta,f}$. Then we always have
$$\#\Cal O/\eta\le\#\frak p_\bold T/\frak p^2_\bold T.\leqno(2.44)$$
(Here and in what follows we sometimes write $\eta$ for $\pi(\eta)$ if the context makes this reasonable.) This is proved as
follows. $\bold T/\eta$ acts faithfully on $\frak p_\bold T$. Hence the Fitting ideal of $\frak p_\bold T$ as a $\bold
T/\eta$-module is zero. The same is then true of $\frak p_\bold T/\frak p^2_\bold T$ as an $\Cal O/\eta=(\bold T/\eta)/\frak
p_\bold T$-module. So the Fitting ideal of $\frak p_\bold T/\frak p^2_\bold T$ as an $\Cal O$-module is contained in $(\eta)$
and the conclusion is then easy. So together with the hypothesis of the theorem we get inequality (and hence equalities)
$$\#\Cal O/\pi(\eta)\ge\#\frak p_R/\frak p^2_R\ge\#\frak p_\bold T/\frak p^2_\bold T\ge\#\Cal O/\pi(\eta).$$
By Proposition 2 of the appendix $\bold T$ is a complete intersection over $\Cal O$. Part (ii)\linebreak of the theorem then
follows for
$\Cal D$. Part (i) follows for $\Cal D$ from Proposition 1 of the appendox.
We now prove inductively that we can deduce the same inequality
$$\#\Cal O_1/\eta_{\Cal D_1,f}\ge\#\frak p_{R_1,f}/\frak p^2_{R_1,f}\leqno(2.45)$$
for $\Cal D_1\supset\Cal D$ and $R_1=R_{\Cal D_1}$. The above argument will then prove the theorem for $\Cal D_1$. We explain
this first in the case $\Cal D_1=\Cal D_q$ where $\Cal D_q$ differs from $\Cal D$ only in replacing $\Sigma$ by
$\Sigma\cup\{q\}$. Let us write $\bold T_q$ for $\bold T_{\Cal D_q},\frak p_{R,q}$ for $\frak p_{R,f}$ with $R=R_{\Cal D_q}$
and $\eta_q$ for $\eta_{\Cal D_q,f}$. We recall that $U_q=0$ in $\bold T_q$.
We choose isomorphisms
$$\bold T\simeq\roman{Hom}_\Cal O(\bold T,\Cal O),\ \ \ \ \ \bold T_q\simeq\roman{Hom}_\Cal O(\bold T_q,\Cal O)\leqno(2.46)$$
coming from the fact that each of the rings is a Gorenstein $\Cal O$-algebra. If $\alpha_q:\bold T_q\rightarrow\bold T$ is the
natural map we may consider the element $\Delta_q=\alpha_q\circ\hat\alpha_q\in\bold T$ where the adjoint is with respect to
the above isomorphisms. Then it is clear that $$\Big(\alpha_q(\eta_q)\Big)=(\eta\Delta_q)\leqno(2.47)$$
as principal ideals of $\bold T$. In particular $\pi(\eta_q)=\pi(\eta\Delta_q)$ in $\Cal O$.
Now it follows from Proposition 2.7 that the principal ideal $(\Delta_q)$ is given by\linebreak
$$(\Delta_q)=\Big((q-1)^2(T_q^2-\langle q\rangle(1+q)^2)\Big).\leqno(2.48)$$
\eject
\noindent In the statement of Proposition 2.7 we used $\bold Z_p$-pairings
$$\bold T\simeq\roman{Hom}_{\bold Z_p}(\bold T,\bold Z_p),\ \ \ \ \bold T_q\simeq\roman{Hom}_{\bold Z_p}(\bold T_q,\bold
Z_p)$$
to define $(\Delta_q)=(\alpha_q\circ\hat\alpha_q).$ However, using the description of the pairings as $W(k_\frak
m)$-algebras derived from these $\bold Z_p$-pairings in the paragraph following (2.42) we see that the ideal $(\Delta_q)$
is unchanged when we use $W(k_\frak m)$-algebra pairings, and hence also when we extend scalars to $\Cal O$ as in (2.42).
On the other hand
$$\#\frak p_{R,q}/\frak p^2_{R,q}\le\#\frak p_R/\frak p^2_R\cdot\#\Big\{\Cal O/(q-1)^2\Big(T^2_q-\langle
q\rangle(1+q)^2\Big)\Big\}$$
by Propositions 1.2 and 1.7. Combining this with (2.47) and (2.48) gives (2.45).
If $\Cal M\ne\phi$ we use a similar argument to pass from $\Cal D$ to $\Cal D_q$ where this time $\Cal D_q$ signifies that
$\Cal D$ is unchanged except for dropping $q$ from $\Cal M$. In each of types (A), (B), and (C) one checks from
Propositions 1.2 and 1.8 that
$$\#\frak p_{R,q}/\frak p^2_{R,q}\le\#\frak p_R/\frak p^2_R\cdot\#H^0(\bold Q_q,V^*).$$
This is in agreement with Propositions 2.10, 2.12 and 2.13 which give the corresponding change in $\eta$ by the method
described above.
To change from an $\Cal O$-algebra to an $\Cal O_1$-algebra is straightforward (the\linebreak complete intersection property
can be checked using [Ku1, Cor. 2.8 on p. 209]),\linebreak and to change from Se to ord we use (1.4) and (2.32). The change
from str to ord reduces to this since by Proposition 1.1 strict deformations and Selmer deformations are the same. Note
that for the ord case if $R$ is a local Noetherian ring and $f\in R$ is not a unit and not a zero divisor, then $R$ is a
complete intersection if and only if $R/f$ is (cf. [BH, Th. 2.3.4]). This completes the proof of the theorem.\hfill$\qed$
{\it Remark} 2.18. If we suppose in the Selmer case that $f$ has level $N$ with $p\nmid N$ we can also consider the ring
$\bold T_H(M_0)_{\frak m_0}$ (with $M_0$ as in (2.24) and $\frak m_0$ defined in the same way as for $\bold T_H(M)).$ This
time set
$$T_0=\bold T_H(M_0)_{\frak m_0}\mathop{\otimes}\limits_{W(k_{\frak m_0})}\Cal O,\ \ \ \ T=\bold T_H(M)_\frak
m\mathop{\otimes}\limits_{W(k_\frak m)}\Cal O.$$
Define $\eta_0,\eta,\frak p_0$ and $\frak p$ with respect to these rings, and let $(\Delta_p)=\alpha_p\circ\hat\alpha_p$
where $\alpha_p:T\rightarrow T_0$ and the adjoint is taken with respect to $\Cal O$-pairings on $T$ and $T_0$. We then have
by Proposition 2.4
$$(\eta_p)=(\eta\cdot\Delta_p)=\Big(\eta\cdot\Big(T^2_p-\langle
p\rangle(1+p)^2\Big)\Big)=\Big(\eta\cdot(a_p^2-\langle p\rangle)\Big)\leqno(2.49)$$
as principal ideals of $T$, where $a_p$ is the unit root of $x^2-T_px+p\langle p\rangle=0.$
\vskip6pt
{\it Remark.} For some earlier work on how deformation rings change with $\Sigma$ see [Bo].
\centerline{\bf Chapter 3}
\
In this chapter we prove the main results about Conjecture 2.16. We begin by showing that bound for the Selmer group to
which it was reduced in Theorem 2.17 can be checked if one knows that the minimal Hecke ring is a complete intersection.
Combining this with the main result of [TW] we complete the proof of Conjecture 2.16 under a hypothesis that ensures that a
minimal Hecke ring exists.
\
\
\centerline{\bf Estimates for the Selmer group}
\
Let $\rho_0:\roman{Gal}(\bold Q_\Sigma/\bold Q)\rightarrow\roman{GL}_2(k)$ be an odd irreducible representation which we
will assume is modular. Let $\Cal D$ be a deformation theory of type $(\cdot,\Sigma,\Cal O,\Cal M)$ such that $\rho_0$ is
type $\Cal D$, where $\cdot$ is Selmer, strict or flat. We remind the reader that $k$ is assumed to be the residue field of
$\Cal O$. Then as explained in Theorem 2.14, we can pick a modular lifting $\rho_{f,\lambda}$ of $\rho_0$ of type $\Cal D$
(altering $k$ if necessary and replacing $\Cal O$ by a ring containing $\Cal O_{f,\lambda}$) provided that $\rho_0$ is not
induced from a character of $\bold Q(\sqrt{-3}\ \!)$ if $p=3$. For the rest of this chapter, we will make the assumption
that $\rho_0$ is not of this exceptional type. Theorem 2.14 also specifies a certain minimum level and character for $f$
and in particular ensures that we can pick $f$ to have level prime to $p$ when $\rho_0|_{D_p}$ is associated to a finite
flat group scheme over $\bold Z_p$ and $\det\rho_0|_{I_p}=\omega$.
In Chapter 2, Section 3, we defined a ring $\bold T_\Cal D$ associated to $\Cal D$. Here we make a slight modification of
this ring. In the case where $\cdot$ is Selmer and $\rho_0|_{D_p}$ is associated to a finite flat group scheme and
$\det\rho_0|_{I_p}=\omega$ we set
$$\bold T_{\Cal D_0}=\bold T'_H(M_0)_{\frak m'_0}\mathop{\otimes}\limits_{W(k_0)}\Cal O\leqno(3.1)$$
with $M_0$ as in (2.24), $H$ defined following (2.24) (it is actually a subgroup of $(\bold Z/M_0\bold Z)^*)$ and $\frak
m_0'$ the maximal ideal of $\bold T'_H(M_0)$ associated to $\rho_0$. The same proof as in Proposition 2.15 ensures that
there is a maximal ideal $\frak m_0$ of $\bold T_H(M_0)$ with $\frak m_0\cap\bold T'_H(M_0)=\frak m'_0$ and such that the
natural map
$$\bold T_{\Cal D_0}=\bold T'_H(M_0)_{\frak m_0'}\mathop{\otimes}\limits_{W(k_0)}\Cal O\rightarrow\bold
T_H(M_0)_{\frak m_0}\mathop{\otimes}\limits_{W(k_0)}\Cal O\leqno(3.2)$$
is an isomorphism. The maximal ideal $\frak m_0$ which we choose is characterized by the properties that $\rho_{\frak
m_0}=\rho_0$ and $U_q\in\frak m_0$ for $q\in\Sigma-\Cal M\cup\{p\}$. (The value of $T_p$ or of $U_q$ for $q\in\Cal M$ is
determined by the other operators; see the proof of Proposition 2.15.) We now define $\bold T_{\Cal D_0}$ in general by the
following:
\eject
\hskip16pt$\bold T_{\Cal D_0}$ is given by (3.1)\hskip6pt if $\cdot$ is Se and $\rho_0|_{D_p}$ is associated
\hskip1.585in to a finite flat group scheme over $\bold Z_p$ and
\hskip1.585in $\det\rho_0|_{I_p}=\omega;$
\
\noindent(3.3)
\
\hskip62pt$\bold T_{\Cal D_0}=\bold T_\Cal D$\hskip6pt if $\cdot$ is str or fl, or $\rho_0|{D_p}$ is not associated
\hskip1.585in to a finite flat group scheme over $\bold Z_p$, or
\hskip1.585in$\det\rho_0|_{I_p}\ne\omega.$
\
\noindent We choose a pair $(f,\lambda)$ of minimum level and character as given by Theo-\linebreak rem 2.14 and this gives
a homomorphism of $\Cal O$-algebras
$$\pi_f:\bold T_{Cal D_0}\rightarrow\Cal O\supseteq\Cal O_{f,\lambda}.$$
We set $\frak p_{\bold T,f}=\ker\pi_f$ and similarly we let $\frak p_{R,f}$ denote the inverse image of $\frak p_{\bold
T,f}$ in $R_\Cal D$. We define a principal ideal $(\eta_{\bold T,f})$ of $\bold T_{\Cal D_0}$ by taking an adjoint
$\hat\pi_f$ of $\pi_f$ with respect to parings as in (2.42) and write
$$\eta_{\bold T,f}=(\hat\pi_f(1)).$$
Note that $\frak p_{\bold T,f}/\frak p^2_{\bold T,f}$ is finite and $\pi_f(\eta_{\bold T,f})\ne0$ because $\bold T_{\Cal
D_0}$ is reduced. We also write $\eta_{\bold T,f}$ for $\pi_f(\eta_{\frak T,f})$ if the context makes this usage
reasonable. We let $V_f=\roman{Ad}\ \!\rho_\frak p\mathop{\otimes}\limits_\Cal OK/\Cal O$ where $\rho_\frak p$ is the
extension of scalars of $\rho_{f,\lambda}$ to $\Cal O$.
\vskip6pt
{\smc Theorem} 3.1. {\it Assume that $\Cal D$ is minimal}, {\it i.e.}, {\it $\sum=\Cal M\cup\{p\}$}, {\it and that $\rho_0$
is absolutely irreducible when restricted to $\bold Q\bigg(\sqrt{(-1)^{p-1\over2}p}\ \!\bigg).$ Then}
\vskip6pt
(i) $\#H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_f)\le\#(\frak p_{\bold T,f}/\frak p^2_{\bold T,f})^2\cdot c_p/\#(\Cal
O/\eta_{\bold T,f})$
\vskip6pt
\noindent{\it where $c_p=\#(\Cal O/U_p^2-\langle p\rangle)\lt\infty$ when $\rho_0$ is Selmer and $\rho_0|_{D_p}$ is
associated to a finite flat group scheme over $\bold Z_p$ and $\det\rho_0|_{I_p}=\omega,$ and $c_p=1$ otherwise};
(ii) {\it if} $\bold T_{\Cal D_0}$ {\it is a complete intersection over $\Cal O$ then} (i) {\it is an equality}, $R_\Cal
D\simeq\bold T_\Cal D$ {\it and} $\bold T_\Cal D$ {\it is a complete intersection.}
\vskip6pt
{\it In general}, {\it for any} ({\it not necessarily minimal}) $\Cal D$ {\it of Selmer}, {\it strict or flat type}, {\it
and any} $\rho_{f,\lambda}$ {\it of type} $\Cal D,\#H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_f)\lt\infty$ {\it if} $\rho_0$ {\it
is as above.}
\vskip6pt
{\it Remarks.} The finiteness was proved by Flach in [Fl] under some restrictions on $f,p$ and $\Cal D$ by a different
method. In particular, he did not consider the strict case. The bound we obtain in (i) is in fact the actual order of
$H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_f)$ as follows from the main result of [TW] which proves the hypothesis of part (ii).
Then applying Theorem 2.17 we obtain the order of this group for more general $\Cal D$'s associated to $\rho_0$ under the
condition that a minimal $\Cal D$ exists associated to $\rho_0$. This is stated in Theorem 3.3.
\eject
The case where the projective representation associated to $\rho_0$ is dihedral does not always have the property that a
twist of it has an associated minimal $\Cal D$. In the case where the associated quadratic field is imaginary we will give
a different argument in Chapter 4.
\vskip6pt
{\it Proof.} We will assume throughout the proof that $\Cal D$ is minimal, indicating only at the end the slight changes
needed fot the final assertion of the theorem. Let $Q$ be a finite set of primes disjoint from $\Sigma$ satisfying
$q\equiv1(p)$ and $\rho_0(\roman{Frob}\ \!q)$ having distinct eigenvalues for each $q\in Q$. For the minimal deformation
problem $\Cal D=(\cdot,\Sigma,\Cal O,\Cal M)$, let $\Cal D_Q$ be the deformation problem described before (2.34); i.e., it
is the refinement of $(\cdot,\Sigma\cup Q,\Cal O,\Cal M)$ obtained by imposing the additional restriction (2.34) at each
$q\in Q$. (We will assume for the proof that $\Cal O$ is chosen so $\Cal O/\lambda=k$ contains the eigenvalues of
$\rho_0(\roman{Frob\ \!}q)$ for each $q\in Q$.) We set
$$\bold T=\bold T_{\Cal D_0},\ \ \ \ R=R_\Cal D$$
and recall the definition of $\bold T_Q$ and $R_Q$ from Chapter 2, \S3 (cf. (2.35)). We write $V$ for $V_f$ and recall the
definition of $V_{(q)}$ following (2.38). Also remember that $\frak m_Q$ is a maximal ideal of $\bold T_H(Nq_1\dots q_r)$
as in (2.35) for which $\rho_{\frak m_Q}\simeq\rho_0$ over $\bar\bold F_p$ (recall that this uses the same choice of
embedding $k_{\frak m_Q}\longrightarrow k$ as in the definition of $\bold T_Q$). We use $\frak m_Q$ also to denote the
maximal ideal of $\bold T_Q$ if the context makes this reasonable.
Consider the exact and commutative diagram
$$\eightpoint{\matrix0&\rightarrow&H^1_\Cal D(\bold Q_\Sigma/\bold Q,V)&\rightarrow&H^1_{\Cal D_Q}(\bold
Q_{\Sigma\cup Q}/\bold Q,V)&\mathop{\rightarrow}\limits^{\delta_Q}&\prod\limits_{q\in Q}H^1(\bold
Q^\roman{unr}_q,V^{(q)})^{\roman{Gal}(\bold Q^\roman{unr}_q/\bold Q_q)}\\
\\
&&|\wr&&|\wr\\
\\
0&\rightarrow&(\frak p_R/\frak p^2_R)^*&\rightarrow&(\frak p_{R_Q}/\frak p^2_{R_Q})^*&&\hskip-1.35in\Bigg\uparrow\iota_Q\\
\\
&&\uparrow&&\uparrow\\
\\
0&\rightarrow&(\frak p_\bold T/\frak p^2_\bold T)^*&\rightarrow&(\frak p_{\bold T_Q}/\frak p^2_{\bold
T_Q})^*&\mathop{\rightarrow}\limits^{u_Q}&\hskip-1.25in K_Q\rightarrow0\endmatrix}$$
where $K_Q$ is by definition the cokernel in the horizontal sequence and $*$ denotes $\roman{Hom}_\Cal O(\ ,K/\Cal O)$ for
$K$ the field of fractions of $\Cal O$. The key result is:
\vskip6pt
{\smc Lemma} 3.2. {\it The map $\iota_Q$ is injective for any finite set of primes $Q$\linebreak satisfying
$$q\equiv1(p),T^2_q\not\equiv\langle q\rangle(1+q)^2\roman{mod}\ \!\frak m\ {\italic for\ all}\ q\in Q.$$}
\vskip6pt
{\it Proof.} Note that the hypotheses of the lemma ensure that $\rho_0(\roman{Frob}\ \!q)$ has distinct eigenvaluesw for
each $q\in Q.$ First, consider the ideal $\frak a_Q$ of $R_Q$ defined\linebreak
\eject
\noindent by
$$\frak a_Q=\bigg\{a_i-1,b_i,c_i,d_i-1:\pmatrix a_i&b_i\\
c_i&d_i\endpmatrix=\rho_{\Cal D_Q}(\sigma_i)\ \roman{with}\ \sigma_i\in I_{q_i},q_i\in Q\bigg\}.\leqno(3.4)$$
Then the universal property of $R_Q$ shows that $R_Q/\frak a_Q\simeq R.$ This permits us to identify $(\frak p_R/\frak
p^2_R)^*$ as
$$(\frak p_R/\frak p^2_R)^*=\{f\in(\frak p_{R_Q}/\frak p^2_{R_Q})^*:f(\frak a_Q)=0\}.$$
If we prove the same relation for the Hecke rings, i.e., with $\bold T$ and $\bold T_Q$ replacing $R$ and $R_Q$ then we
will have the injectivity of $\iota_Q$. We will write $\bar\frak a_Q$ for the image of $\frak a_Q$ in $\bold T_Q$ under the
map $\varphi_Q$ of (2.37).
It will be enough to check that for any $q\in Q',Q'$ a subset of $Q,\bold T_{Q'}/\bar\frak a_q\simeq\bold T_{Q'-\{q\}}$
where $\frak a_q$ is defined as in (3.4) but with $Q$ replaced by $q$. Let\linebreak
$N'=N(\rho_0)p^{\delta(\rho_0)}\cdot\prod_{q_i\in Q'-\{q\}}q_i$ where $\delta(\rho_0)$ is as defined in Theorem 2.14. Then
take an element $\sigma\in I_q\subseteq\roman{Gal}(\bar\bold Q_q/\bold Q_q)$ which restricts to a generator of
$\roman{Gal}(\bold Q(\zeta_{N'q}/\bold Q(\zeta_{N'})).$ Then $\det(\sigma)=\langle t_q\rangle\in\bold T_{Q'}$ in the
representation to $\roman{GL}_2(\bold T_{Q'})$ defined after Theorem 2.1. (Thus $t_q\equiv1(N')$ and $t_q$ is a primitive
root mod $q$.) It is easily checked that
$$J_H(N'.q)_{\frak m_{Q'}}(\bar\bold Q)\simeq J_H(N'q)_{\frak m_{Q'}}(\bar\bold Q)[\langle t
_q\rangle-1].\leqno(3.5)$$
Here $H$ is still a subgroup of $(\bold Z/M_0\bold Z)^*$. (We use here that $\rho_0$ is not reducible for the injectivity
and also that $\rho_0$ is not induced from a character of $\bold Q(\sqrt{-3}\ \!)$ for the surjectivity when $p=3$. The
latter is to avoid the ramification points of the covering $X_H(N'q)\rightarrow X_H(N',q)$ of order 3 which can give rise
to invariant divisors of $X_H(N'q)$ which are not the images of divisors on $X_H(N',q).$)
Now by Corollary 1 to Theorem 2.1 the Pontrjagin duals of the modules in (3.5) are free of rank two. It follows that
$$(\bold T_H(N'q)_{\frak m_{Q'}})^2/(\langle t_q\rangle-1)\simeq(\bold T_H(N',q)_{\frak m_{Q'}})^2.\leqno(3.6)$$
The hypotheses of the lemma imply the condition that $\rho_0(\roman{Frob}\ \!q)$ has distinct eigenvalues. So applying
Proposition 2.4' (at the end of \S2) and the remark following it (or using the fact remarked in Chapter 2, \S3 that this
condition implies that $\rho_0$ does not occur as the residual representation associated to any form which has the special
representation at $q$) we see that after tensoring over $W(k_{\frak m_{Q'}})$ with $\Cal O$, the right-hand side of (3.6)
can be replaced by $\bold T^2_{Q'-\{q\}}$ thus giving
$$\bold T^2_{Q'/\bar\frak a_q}\simeq\bold T^2_{Q'-\{q\}},$$
since $\langle t_q\rangle-1\in\bar\frak a_q$. Repeated inductively this gives the desired relation $\bold T_Q/\bar\frak
a_Q\simeq\bold T$, and completes the proof of the lemma.\hfill$\qed$
\eject
Suppose now that $Q$ is a finite set of primes chosen as in the lemma. Recall that from the theory of congruences (Prop.
2.4' at the end of \S2)
$$\eta_{\bold T_{Q,f}}/\eta_{\bold T,f}=\prod\limits_{q\in Q}(q-1),$$
the factors $(\alpha^2_q-\langle q\rangle)$ being units by our hypotheses on $q\in Q$. (We only need that the right-hand
side divides the left which is somewhat easier.) Also, from the theory of Fitting ideals (see the proof of (2.44))
$$\aligned\#(\frak p_\bold T/\frak p^2_\bold T)&\ge\#(\Cal O/\eta_{\bold T_f})\\
\#(\frak p_{\bold T_Q}/\frak p^2_{\bold T_Q}&\ge\#(\Cal O/\eta_{\bold T_{Q,f}}).\endaligned$$
We dedeuce that
$$\#K_Q\ge\#\Bigg(\Cal O\Big/\prod\limits_{q\in Q}(q-1)\Bigg)\cdot t^{-1}$$
where $t=\#(\frak p_\bold T/\frak p^2_\bold T)/\#(\Cal O/\eta_{\bold T,f}).$ Since the range of $\iota_Q$ has order given by
$$\#\Bigg\{\Cal O\Big/\prod\limits_{q\in Q}(q-1)\Bigg\},$$
we compute that the index of the image of $\iota_Q$ is $\le t$ as $\iota_Q$ is injective.
Keeping our assumption on $Q$ from Lemma 3.2, consider the kernel of $\lambda^M$ applied to the diagram at the beginning of
the proof of the theorem. Then\linebreak with $M$ chosen large enough so that $\lambda^M$ annihilates $\frak p_\bold T/\frak
p^2_\bold T$ (which is finite because $\bold T$ is reduced) we get:
\
\noindent$\eightpoint{\hskip-4pt0\rightarrow H^1_\Cal D\!(\bold Q_\Sigma/\bold
Q,V[\lambda^M])\rightarrow H^1_{\Cal D_Q}\!(\bold Q_{\Sigma\cup Q}/\bold
Q,V[\lambda^M])\mathop{\rightarrow}\limits^{\delta_Q}\prod\limits_{q\in Q}\!H^1(\bold
Q_q^\roman{unr},V^{(q)}[\lambda^M])^{\roman{Gal}(\bold Q_q^\roman{unr}/\bold Q_q)}}$
\
\noindent$\eightpoint{\hskip26pt\uparrow\hskip1.1in\uparrow\psi_Q\hskip1.35in\uparrow\iota_Q}$
\
\noindent$\eightpoint{\hskip-4pt0\rightarrow(\frak p_\bold T/\frak p^2_\bold T)^*\hskip36pt\rightarrow(\frak p_{\bold
T_Q}/\frak p^2_{\bold T_Q})^*[\lambda^M]\hskip21pt\rightarrow\hskip15pt K_Q[\lambda^M]\rightarrow(\frak p_\bold T/\frak
p^2_\bold T)^*.}$
\
\noindent See (1.7) for the justification that $\lambda^M$ can be taken inside the parentheses in the first two terms. Let
$X_Q=\psi_Q((\frak p_{\bold T_Q}/\frak p^2_{\bold T_Q})^*[\lambda^M]).$ Then we can estimate the order of $\delta_Q(X_Q)$
using the fact that the image if $\iota_Q$ has index at most $t$. We get
$$\#\delta_Q(X_Q)\ge\Bigg(\prod\limits_{q\in Q}\#\Cal O/(\lambda^M,q-1)\Bigg)\cdot(1/t)\cdot(1/\#(\frak p_\bold
T/\frak p^2_\bold T)).\leqno(3.7)$$
Now we choose $Q$ to be a set of primes with the property that
$$\varepsilon_Q:H^1_{\Cal D^*}(\bold Q_\Sigma/\bold Q,V^*_{\lambda^M})\rightarrow\prod\limits_{q\in Q}H^1(\bold
Q_q,V^*_{\lambda^M})\leqno(3.8)$$
\eject
\noindent is injective. We also keep the condition that $\iota_Q$ is injective by only allowing $Q$ to contain primes of
the form given in the lemma. In addition, we require these $q$'s to satisfy $q\equiv1(p^M).$
To see that this can be done, suppose that $x\in\ker\varepsilon_Q$ and $\lambda x=0$ but $x\ne0$. We have a commutative
diagram
$$\matrix H^1(\bold Q_\Sigma/\bold
Q,V^*_{\lambda^M}[\lambda]&\mathop{\rightarrow}\limits^{\varepsilon_Q}&\prod\limits_{q\in Q}H^1(\bold
Q_q,V^*_{\lambda^M})[\lambda]\\
\\
|\wr&&|\wr\\
\\
H^1(\bold Q_\Sigma/\bold Q,V^*_\lambda)&\mathop{\rightarrow}\limits^{\bar\varepsilon_Q}&\prod\limits_{q\in
Q}H^1(\bold Q_q,V^*_\lambda)\endmatrix$$
the left-hand isomorphisms coming from our particular choices of $q$'s and the left-hand isomorphism from our hypothesis on
$\rho_0$. The same diagram will hold if we replace $Q$ by $Q_0=Q\cup\{q_0\}$ and we now need to show that we can choose
$q_0$ so that $\bar\varepsilon_{Q_0}(x)\ne0.$
The restriction map
$$H^1(\bold Q_\Sigma/\bold Q,V^*_\lambda)\rightarrow\roman{Hom}(\roman{Gal}(\bar\bold
Q/K_0(\zeta_p)),V^*_\lambda)^{\roman{Gal}(K_0(\zeta_p)/\bold Q)}$$
has kernel $H^1(K_0(\zeta_p)/\bold Q,k(1))$ by Proposition 1.11 where here $K_0$ is the splitting field of $\rho_0$. Now if
$x\in H^1(K_0(\zeta_p)/\bold Q,k(1))$ and $x\ne0$ then $p=3$ and $x$ factors through an abelian extension $L$ of $\bold
Q(\zeta_3)$ of exponent 3 which is non-abelian over $\bold Q$. In this exeptional case, $L$ must ramify at some prime
$\frak q$ of $\bold Q(\zeta_3)$, and if $\frak q$ lies over the rational prime $q\ne3$ then the composite map
$$H^1(K_0(\zeta_3)/\bold Q,k(1))\rightarrow H^1(\bold Q^\roman{unr}_q,k(1))\rightarrow H^1(\bold Q^\roman{unr}_q,(\Cal
O/\lambda^M)(1))$$
is nonzero on $x$. But then $x$ is not of type $\Cal D^*$ which gives a contradiction. This only leaves the possibility
that $L=\bold Q(\zeta_3,\root{3}\of3)$ but again this means that $x$ is not of type $\Cal D^*$ as locally at the prime
above 3, $L$ is not generated by the cube root of a unit over $\bold Q_3(\zeta_3)$. This argument holds whether or not
$\Cal D$ is minimal.
So $x$, which we view in $\ker\bar\varepsilon_Q$, gives a nontrivial Galois-equivalent homomorphism
$f_x\in\roman{Hom}(\roman{Gal}(\bar\bold Q/K_0(\zeta_p)),V^*_\lambda)$ which factors through an abelian extension $M_x$ of
$K_0(\zeta_p)$ of exponent $p$. Specifically we choose $M_x$ to be the minimal such extension. Assume first that the
projective representation $\tilde\rho_0$ associated to $\rho_0$ is not dihedral so that $\roman{Sym}^2\rho_0$ is absolutely
irreducible. Pick a $\sigma\in\roman{Gal}(M_x(\zeta_{p^M})/\bold Q)$ satisfying
\vskip6pt
\noindent(3.9)\hskip0.7in(i) $\rho_0(\sigma)$ has order $m\ge3$ with $(m,p)=1,$
\vskip6pt
\hskip0.67in(ii) $\sigma$ fixes $\bold Q(\det\rho_0)(\zeta_{p^M}),$
\vskip6pt
\hskip0.63in(iii) $f_x(\sigma^m)\ne0.$
\vskip6pt
\noindent To show that this is possible, observe first that the first two conditions can\linebreak be achieved by Lemma
1.10(i) and the subsequent remark. Let $\sigma_1$ be an el-\linebreak
\eject
\noindent ement satisfying (i) and (ii) and let $\bar\sigma_1$ denote its image in $\roman{Gal}(K_0(\zeta_p)/\bold Q).$
Then $\langle\bar\sigma_1\rangle$ acts on $G=\roman{Gal}(M_x/K_0(\zeta_p))$ and under this action $G$ decom-\linebreak poses
as
$G\simeq G_1\oplus G'_1$ where $\sigma_1$ acts trivially on $G_1$ and without fixed points on $G'_1$. If $X$ is any
irreducible Galois stable $\bar k$-subspace of $f_x(G)\otimes_{\bold F_p}\bar k$ then $\ker(\sigma_1-1)|_X\ne0$ since
$\roman{Sym}^2\rho_0$ is assumed absolutely irreducible. So also $\ker(\sigma_1-1)|_{f_x(G)}\ne0$ and thus we can find
$\tau\in G_1$ such that $f_x(\tau)\ne0.$ Viewing $\tau$ as an element of $G$ we then take
$$\tau_1=\tau\times1\in\roman{Gal}(M_x(\zeta_{p^M})/K_0(\zeta_p))\simeq G\times\roman{Gal}(K_0(\zeta_{p^M})/K_0(\zeta_p))$$
(This decomposition holds because $M_x$ is minimal and because $\roman{Sym}^2\rho_0$ and\linebreak $\mu_p$ are distinct from
the trivial representation.) Now $\tau_1$ commutes with $\sigma_1$ and\linebreak either $f_x((\tau_1\sigma_1)^m)\ne0$ or
$f_x(\sigma_1^m)\ne0.$ Since $\rho_0(\tau_1\sigma_1)=\rho_0(\sigma_1)$ this gives\linebreak (3.9) with at least one of
$\sigma=\tau_1\sigma_1$ or $\sigma=\sigma_1$. We may then choose $q_0$ so that $\roman{Frob}q_0=\sigma$ and we will then
have $\bar\varepsilon_{Q_0}(x)\ne0$. Note that conditions (i) and (ii) imply that $q_0\equiv1(p)$ and also that
$\rho_0(\sigma)$ has distinct eigenvalues, thus giving both the hypothses of Lemma 3.2.
If on the other hand $\tilde\rho_0$ is dihedral then we pick $\sigma$'s satisfying
\vskip6pt
\hskip-12pt(i) $\tilde\rho_0(\sigma)\ne1$,
\vskip6pt
\hskip-15pt(ii) $\sigma$ fixes $\bold Q(\zeta_{p^M}),$
\vskip6pt
\hskip-18pt(iii) $f_x(\sigma^m)\ne0$,
\vskip6pt
\noindent with $m$ the order of $\rho_0(\sigma)$ (and $p\nmid m$ since $\tilde\rho_0$ is dihedral). The first two
condi-\linebreak tions can be achieved using Lemma 1.12 and, in addition, we can assume that $\sigma$ takes the eigenvalue 1
on any given irreducible Galois stable subspace $X$ of $W_\lambda\otimes\bar k$. Arguing as above, we find a $\tau\in G_1$
such that $f_x(\tau)\ne0$ and we proceed as before. Again, conditions (i) and (ii) imply the hypotheses of Lemma 3.2. So by
successively adjoining $q$'s we can assume that $Q$ is chosen so that $\varepsilon_Q$ is injective.
\vskip6pt
We have thus shown that we can choose $Q=\{q_1,\dots,q_r\}$ to be a finite\linebreak set of primes $q_i\equiv1(p^M)$
satisfying the hypotheses of Lemma 3.2 as well as the injectivity of $\varepsilon_Q$ in (3.8). By Proposition 1.6, the
injectivity of $\varepsilon_Q$ implies that
$$\#H^1_\Cal D(\bold Q_{\Sigma\cup Q}/\bold Q,V_g[\lambda^M])=h_\infty\cdot\prod\limits_{q\in\Sigma\cup Q}h_q.\leqno(3.10)$$
Here we are using the convention explained after Proposition 1.6 to define $H^1_\Cal D.$ Now, as $\Cal D$ was chosen to be
minimal, $h_q=1$ for $q\in\sum-\{p\}$ by Proposi-\linebreak tion 1.8. Also, $h_q=\#(\Cal O/\lambda^M)^2$ for $q\in Q$. If
$\cdot$ is str or fl then $h_\infty h_p=1$\linebreak by Proposition 1.9 (iv) and (v). If $\cdot$ is Se, $h_\infty h_p\le
c_p$ by Proposition 1.9 (iii). (To compute this we can assume that $I_p$ acts on $W^0_\lambda$ via $\omega$, as otherwise
we\linebreak
\eject
\noindent get $h_\infty h_p\le1$. Then with this hypothesis, $(W^0_{\lambda^n})^*$ is easily verified to be un-ramified with
$\roman{Frob}\ \!p$ acting as $U^2_p\langle p\rangle^{-1}$ by the description of $\rho_{f,\lambda}|_{D_p}$ in [Wi1, Th.
2.1.4].) On the other hand, we have constructed classes which are ramified at primes in $Q$ in (3.7). These are of type
$\Cal D_Q$. We also have classes in
$$\roman{Hom}(\roman{Gal}(\bold Q_{\Sigma\cup Q}/\bold Q),\Cal O/\lambda^M)=H^1(\bold Q_{\Sigma\cup Q}/\bold Q,\Cal
O/\lambda^M)\hookrightarrow H^1(\bold Q_{\Sigma\cup Q}/\bold Q,V_{\lambda^M})$$
coming from the cyclotomic extension $\bold Q(\zeta_{q_1}\dots\zeta_{q_r}).$ These are of type $\Cal D$ and disjoint from
the classes obtained from (3.7). Combining these with (3.10) gives
$$\#H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_f[\lambda^M])\le t\cdot\#(\frak p_\bold T/\frak p^2_\bold T)\cdot c_p$$
as required. This proves part (i) of Theorem 3.1.
Now if we assume that $\bold T$ is a complete intersection we have that $t=1$ by Proposition 2 of the appendix. In the
strict or flat cases (and indeed in all cases where $c_p=1$) this implies that $R_\Cal D\simeq\bold T_\Cal D$ by
Proposition 1 of the appendix together with Proposition 1.2. In the Selmer case we get
$$\#(\frak p_\bold T/\frak p^2_\bold T)\cdot c_p=\#(\Cal O/\eta_{\bold T,f})c_p=\#(\Cal O/\eta_{\bold
T_{\Cal D,f}})\le\#(\frak p_{\bold T_\Cal D}/\frak p^2_{\bold T_\Cal D})\leqno(3.11)$$
where the central equality is by Remark 2.18 and the right-hand inequality is from the theory of Fitting ideals. Now
applying part (i) we see that the inequality in (3.11) is an equality. By Proposition 2 of the appendix, $\bold T_\Cal D$
is also a complete intersection.
The final assertion of the theorem is proved in exactly the same way on noting that we only used the minimality to ensure
that the $h_q$'s were 1. In general, they are bounded independently of $M$ and easily computed. (The only point to note is
that if $\rho_{f,\lambda}$ is multiplicative type at $q$ then $\rho_{f,\lambda|_{D_q}}$ does not split.)\hfill$\qed$
\vskip6pt
{\it Remark.} The ring $\bold T_{\Cal D_0}$ defined in (3.1) and used in this chapter should be the deformation ring
associated to the following deformation prblem $\Cal D_0$. One alters $\Cal D$ only by replacing the Selmer condition by
the condition that the deformations be flat in the sense of Chapter 1, i.e., that each deformation $\rho$ of $\rho_0$ to
$\roman{GL}_2(A)$ has the property that for any quotient $A/\frak a$ of finite order, $\rho|_{D_p}\roman{mod}\ \!\frak a$
is the Galois representation associated to the $\bar\bold Q_p$-points of a finite flat group scheme over $\bold Z_p$. (Of
course, $\rho_0$ is ordinary here in contrast to our usual assumption for flat deformations.)
\vskip6pt
>From Theorem 3.1 we deduce our main results about representations by using the main result of [TW], which proves the
hypothesis of Theorem 3.1 (ii), and then applying Theorem 2.17. More precisely, the main result of [TW] shows that $\bold
T$ is a complete intersection and hence that $t=1$ as explained above. The hypothesis of Theorem 2.17 is then given by
Theorem 3.1 (i), together with the equality $t=1$ (and the central equality of (3.11) in the\linebreak
\eject
\noindent Selmer case) and
Proposition 1.2. Strictly speaking, Theorem 1 of [TW] refers to a slightly smaller class of $\Cal D$'s than those covered
by Theorem 3.1 but up to a twist every such $\Cal D$ is covered. It is straightforward to see that it is enough to check
Theorem 3.3 for $\rho_0$ up to a suitable twist.
\vskip6pt
{\smc Theorem} 3.3. {\it Assume that $\rho_0$ is modular and absolutely irreducible\linebreak when restricted to $\bold
Q\Big(\sqrt{(-1)^{p-1\over2}p}\ \!\Big)$. Assume also that $\rho_0$ is of type} (A), (B)\linebreak {\it or} (C) {\it at each
$q\ne p$ in $\Sigma$. Then the map $\varphi_\Cal D:R_\Cal D\rightarrow\bold T_\Cal D$ of Conjecture} 2.16 {\it is an
isomorphism for all $\Cal D$ associated to $\rho_0$}, {\it i.e.}, {\it where $\Cal D=(\cdot,\Sigma,\Cal O,\Cal M)$ with
$\cdot=\roman{Se,str,fl}$ or} ord{\it. In particular if $\cdot=\roman{Se,str}$ or} fl {\it and $f$ is any newform\linebreak
for which
$\rho_{f,\lambda}$ is a deformation of $\rho_0$ of type $\Cal D$ then
$$\#H^1_\Cal D(\bold Q_\Sigma/\bold Q,V_f)=\#(\Cal O/\eta_{\Cal D,f})\lt\infty$$
where $\eta_{\Cal D,f}$ is the invariant defined in Chapter} 2 {\it prior to} (2.43){\it.}
\vskip6pt
The condition at $q\ne p$ in $\Sigma$ ensures that there is a minimal $\Cal D$ associated to $\rho_0$. The computation of
the Selmer group follows from Theorem 2.17 and Proposition 1.2. Theorem 0.2 of the introduction follows from Theorem 3.3,
after it is checked that a twist of a $\rho_0$ as in Theorem 0.2 satisfies the hypotheses of Theorem 3.3.
\
\
\centerline{\bf Chapter 4}
\
In this chapter we give a different (and slightly more general) derivation of the bound for the Selmer group in the CM case.
In the first section we estimate the Selmer group using the main theorem of [Ru 4] which is based on Kolyvagin's method. In
the second section we use a calculation of Hida to relate\linebreak the $\eta$-invariant to special values of an
$L$-function. Some of these computations are valid in the non-CM case also. They are needed if one wishes to give the order
of the Selmer group in terms of the special value of an $L$-function.
\
\centerline{\bf 1. The ordinary CM case}
\
\
In this section we estimate the order of the Selmer group in the ordinary CM case. In Section 1 we use the proof of the
main conjecture by Rubin to bound the Selmer group in terms of an $L$-function. The methods are standard (cf. [de Sh])
and some special cases have been described elsewhere (cf. [Guo]). In Section 2 we use a calculation of Hida to relate
this to the $\eta$-invariant.
We assume that
$$\rho=\roman{Ind}^\bold Q_L\kappa:\roman{Gal}(\bar\bold Q/\bold Q)\rightarrow\roman{GL}_2(\Cal O)\leqno(4.1)$$
\eject
\noindent is the $p$-adic representation associated to a character $\kappa:\roman{Gal}(\overline L/L)\rightarrow\Cal
O^\times$ of an imaginary quadratic field $L$. We assume that $p$ is unramified in $L$ and that $\kappa$ factors through
an extension of $L$ whose Galois group has the form $A\simeq\bold Z_p\oplus T$ where $T$ is a finite group of order prime
to $p$. The ring $\Cal O$ is assumed to be the ring of integers of a local field with maximal ideal $\lambda$ and we also
assume that $\rho$ is a Selmer deformation of $\rho_0=\rho\ \!\roman{mod}\ \!\lambda$ which is supposed irreducible with
$\det\rho_0|_{I_p}=\omega.$ In particular it follows that $p$ splits in $L,p=\frak p\bar\frak p$ say, and that precisely
one of $\kappa,\kappa^*$ is ramified at $\frak p$ ($\kappa^*$ being the character
$\tau\rightarrow\kappa(\sigma\tau\sigma^{-1})$ for any $\sigma$ representing the nontrivial coset in
$\roman{Gal}(\bar\bold Q/\bold Q)/\roman{Gal}(\bar\bold Q/L)).$ We can suppose without loss of generality that $\kappa$
is ramified at $\frak p$.
We consider the representation module $V\simeq(K/\Cal O)^4$ (where $K$ is the field of fractions of $\Cal O$) and the
representation is $\roman{Ad}\ \!\rho.$ In this case $V$ splits as
$$V\simeq Y\oplus(K/\Cal O)(\psi)\oplus K/\Cal O$$
where $\psi$ is the quadratic character of $\roman{Gal}(\bar\bold Q/\bold Q)$ associated to $L$. We let $\Sigma$ denote a
finite set of primes including all those which ramify in $\rho$ (and in particular $p$). Our aim is to compute
$H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,V).$ The decomposition of $V$ gives a corresponding decomposition of $H^1(\bold
Q_\Sigma/\bold Q,V)$ and we can use it to define $H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,Y).$ Since $W^0\subset Y$ (see
Chapter 1 for the definition of $W^0$) we can define $H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,Y)$ by
$$H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,Y)=\ker\{H^1(\bold Q_\Sigma/\bold Q,Y)\rightarrow H^1(\bold
Q^\roman{unr}_p,Y/W^0)\}.$$
Let $Y^*$ be the arithmetic dual of $Y$, i.e., $\roman{Hom}(Y,\boldsymbol\mu_{p^\infty})\otimes\bold Q_p/\bold Z_p$.
Where $\nu$ for $\kappa\varepsilon/\kappa^*$ and let $L(\nu)$ be the splitting field of $\nu$. Then we claim that
$\roman{Gal}(L(\nu)/L)\simeq\bold Z_p\oplus T'$ with $T'$ a finite group of order prime to $p$. For this it is enough to
show that $\chi=\kappa\kappa^*/\varepsilon$ factors through a group of order prime to $p$ since $\nu=\kappa^2\chi^{-1}.$
Suppose that $\chi$ has order $m=m_0p^r$ with $(m_0,p)=1$. Then $\chi^{m_0}$ extends to a character of $\bold Q$ which is
then unramified at $p$ since the same is true of $\chi$. Also it factors through an abelian extension of $L$ with Galois
group isomorphic to $\bold Z^2_p$ since $\chi$ factors through such an extension with Galois group isomorphic to $\bold
Z^2_p\oplus T_1$ with $T_1$ of order prime to $p$ (the composite of the\linebreak splitting fields of $\kappa$ and
$\kappa^*$). It follows that $\chi^{m_0}$ is also unramified outside $p$, whence it is trivial. This proves the claim.
Over $L$ there is an isomorphism of Galois modules
$$Y^*\simeq(K/\Cal O)(\nu)\oplus(K/\Cal O)(\nu^{-1}\varepsilon^2).$$
In analogy to the above we define $H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,Y^*)$ by
$$H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,Y^*)=\ker\{H^1(\bold Q_\Sigma/\bold Q,Y^*)\rightarrow H^1(\bold
Q^\roman{unr}_p,(W^0)^*)\}.$$
Analogous definitions apply if $Y^*$ is replaced by $Y^*_{\lambda^n}$. Also we say informally that a cohomology class is
Selmer at $p$ if it vanishes in $H^1(\bold Q^\roman{unr}_p,(W^0)^*)$ (resp.\linebreak
\eject
\noindent $H^1(\bold Q^\roman{unr}_p,(W^0_{\lambda^n})^*)).$ Let $M_\infty$ be the maximal abelian $p$-extension of
$L(\nu)$ unramified outside $\frak p$. The following proposition generalizes [CS, Prop. 5.9].
\vskip6pt
{\smc Proposition} 4.1. {\it There is an isomorphism
$$H^1_\roman{unr}(\bold Q_\Sigma/\bold
Q,Y^*)\mathop{\rightarrow}\limits^\sim\roman{Hom}(\roman{Gal}(M_\infty/L(\nu)),(K/\Cal
O)(\nu))^{\roman{Gal}(L(\nu)/L)}$$where $H^1_\roman{unr}$ denotes the subgroup of classes which are Selmer at $p$ and
unramified everywhere else.}
\vskip6pt
{\it Proof.} The sequence is obtained from the inflation-restriction sequence as follows. First we can replace $H^1(\bold
Q_\Sigma/\bold Q,Y^*)$ by
$$\Big\{H^1(\bold Q_\Sigma/L,(K/\Cal O)(\nu))\oplus H^1\Big(\bold Q_\Sigma/L,(K/\Cal
O)(\nu^{-1}\varepsilon^2)\Big)\Big\}^\Delta$$
where $\Delta=\roman{Gal}(L/\bold Q).$ The unramified condition then translates into the\linebreak requirement that the
cohomology class should lie in
$$\Big\{H^1_{\roman{unr\ in\ }\Sigma-\frak p}(\bold Q_\Sigma/L,(K/\Cal O)(\nu))\oplus H^1_{\roman{unr\ in\
}\Sigma-\frak p^*}\Big(\bold Q_\Sigma/L,(K/\Cal O)(\nu^{-1}\varepsilon^2)\Big)\Big\}^\Delta.$$
Since $\Delta$ interchanges the two groups inside the parentheses it is enough to compute the first of them, i.e.,
$$H^1_{\roman{unr\ in\ }\Sigma-\frak p}(\bold Q_\Sigma/L,K/\Cal O(\nu)).\leqno(4.2)$$
The inflation-restriction sequence applied to this gives an exact sequence
$$\leqalignno{0&\rightarrow H^1_{\roman{unr\ in\ }\Sigma-\frak p}(L(\nu)/L,(K/\Cal O)(\nu))&(4.3)\cr
&\rightarrow H^1_{\roman{unr\ in\ }\Sigma-\frak p}(\bold Q_\Sigma/L,(K/\Cal O)(\nu))\cr
&\rightarrow\roman{Hom}(\roman{Gal}(M_\infty/L(\nu)),(K/\Cal O)(\nu))^{\roman{Gal}(L(\nu)/L)}.}$$
The first term is zero as one easily check using the divisibility of $(K/\Cal O)(\nu).$\linebreak Next note that
$H^2(L(\nu)/L,(K/\Cal O)(\nu))$ is trivial. If $\nu\not\equiv1(\lambda)$ this is straightforward (cf. Lemma 2.2 of [Ru1]).
If $\nu\equiv1(\lambda)$ then $\roman{Gal}(L(\nu)/L)\simeq\bold Z_p$ and so it is trivial in this case also. It follows
that any class in the final term of (4.3) lifts to a class $c$ in $H^1(\bold Q_\Sigma/L,(K/\Cal O)(\nu)).$ Let $L_0$ be
the splitting field of $Y^*_\lambda$. Then $M_\infty L_0/L_0$ is unramified outside $\frak p$ and $L_0/L$ has degree
prime to $p$. It follows that $c$ is unramified outside $\frak p$.\hfill$\qed$
\vskip6pt
Now write $H^1_\roman{str}(\bold Q_\Sigma/\bold Q,Y^*_n)$ (where $Y^*_n=Y^*_{\lambda^n}$ and similarly for $Y_n$)
for\linebreak the supgroup of $H^1_{\roman{unr}}(\bold Q_\Sigma/\bold Q,Y^*_n)$ given by
$$H^1_\roman{str}(\bold Q_\Sigma/\bold Q,Y^*_n)=\Big\{\alpha\in H^1_\roman{unr}(\bold Q_\Sigma/\bold Q,Y^*_n):\alpha_p=0\
\roman{in}\ H^1(\bold Q_p,Y^*_n/(Y^*_n)^0)\Big\}$$
where $(Y^*_n)^0$ is the first step in the filtration under $D_p$, thus equal to $(Y_n/Y^0_n)^*$ or equivalently to
$(Y^*)^0_{\lambda^n}$ where $(Y^*)^0$ is the divisible submodule of $Y^*$ on which the action of $I_p$ is via
$\varepsilon^2$. (If $p\ne3$ one can characterize $(Y^*_n)^0$ as the\linebreak
\eject
\noindent maximal submodule on which $I_p$ acts via $\varepsilon^2.$) A similar definition applies with $Y_n$ replacing
$Y^*_n$. It follows from an examination of the action of $I_p$ on $Y_\lambda$ that
$$H^1_\roman{str}(\bold Q_\Sigma/\bold Q,Y_n)=H^1_\roman{unr}(\bold Q_\Sigma/\bold Q,Y_n).\leqno(4.4)$$
In the case of $Y^*$ we will use the inequality
$$\#H^1_\roman{str}(\bold Q_\Sigma/\bold Q,Y^*)\le\#H^1_\roman{unr}(\bold Q_\Sigma/\bold Q,Y^*).\leqno(4.5)$$
We also need the fact that for $n$ sufficiently large the map
$$H^1_\roman{str}(\bold Q_\Sigma/\bold Q,Y^*_n)\rightarrow H^1_\roman{str}(\bold Q_\Sigma/\bold Q,Y^*)\leqno(4.6)$$
is injective. One can check this by replacing these groups by the subgroups\linebreak of $H^1(L,(K/\Cal
O)(\nu)_{\lambda^n})$ and
$H^1(L,(K/\Cal O)(\nu))$ which are unramified outside $\frak p$ and trivial at $\frak p^*$, in a manner similar to the
beginning of the proof of Proposi-tion 4.1. the above map is then injective whenever the connecting homomorphism
$$H^0(L_{\frak p^*},(K/\Cal O)(\nu))\rightarrow H^1(L_{\frak p^*},(K/\Cal O)(\nu)_{\lambda^n})$$
is injective, which holds for sufficiently large $n$.
Now, by Propsition 1.6,
$${\#H^1_\roman{str}(\bold Q_\Sigma/\bold Q,Y_n)\over\#H^1_\roman{str}(\bold Q_\Sigma/\bold
Q,Y^*_n)}=\#H^0(\bold Q_p,(Y^0_n)^*){\#H^0(\bold Q,Y_n)\over\#H^0(\bold Q,Y^*_n)}.\leqno(4.7)$$
Also, $H^0(\bold Q,Y_n)=0$ and a simple calculation shows that
$$\#H^0(\bold Q,Y^*_n)=\cases\mathop{\roman{inf}}\limits_\frak q\#(\Cal O/1-\nu(\frak q))&\roman{if}\ \nu=1\mod\lambda\\
1&\roman{otherwise}\endcases$$
where $\frak q$ runs through a set of primes of $\Cal O_L$ prime to $p\ \!\roman{cond}(\nu)$ of density one. This can be
checked since $Y^*=\roman{Ind}^\bold Q_L(\nu)\mathop{\otimes}\limits_\Cal OK/\Cal O$. So, setting
$$t=\cases\roman{inf}_\frak q\#(\Cal O/(1-\nu(\frak q)))&\roman{if}\ \nu\mod\lambda=1\\
1&\roman{if}\ \nu\mod\lambda\ne1\endcases\leqno(4.8)$$
we get
\noindent(4.9)
\noindent$\#H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,Y)\le{1\over
t}\cdot\prod\limits_{\in\Sigma}\ell_q\cdot\#\roman{Hom}(\roman{Gal}(M_\infty/L(\nu)),(K/\Cal
O)(\nu))^{\roman{Gal}(L(\nu)/L)}$
\noindent where $\ell_q=\#H^0(\bold Q_q,Y^*)$ for $q\ne p,\ell_p=\lim\limits_{n\rightarrow\infty}\#H^0(\bold
Q_p,(Y^0_n)^*).$ This follows from Proposition 4.1, (4.4)-(4.7) and the elementary estimate
$$\#(H^1_{\roman{Se}}(\bold Q_\Sigma/\bold Q,Y)/H^1_\roman{unr}(\bold Q_\Sigma/\bold
Q,Y))\le\prod\limits_{q\in\Sigma-\{p\}}\ell_q,\leqno(4.10)$$
which follows from the fact that $\#H^1(\bold Q^\roman{unr}_q,Y)^{\roman{Gal}(\bold Q^\roman{unr}_q/\bold Q_q)}=\ell_q.$
\eject
Our objective is to compute $H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,V)$ and the main problem is to es-\linebreak timate
$H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,Y)$. By (4.5) this in turn reduces to the problem of estimat-ing
\noindent$\#\roman{Hom}(\roman{Gal}(M_\infty/L(\nu)),(K/\Cal O)(\nu))^{\roman{Gal}(L(\nu)/L)}).$ This order can be
computed\linebreak using the `main conjecture' established by Rubin using ideas of Kolyvagin. (cf.\linebreak [Ru2] and
especially [Ru4]. In the former reference Rubin assumes that the class number of $L$ is prime to $p$.) We could now
derive the result directly from this by referring to [de Sh, Ch.3], but we will recall some of the steps here.
Let $w_\frak f$ denote the number of roots of unity $\zeta$ of $L$ such that $\zeta\equiv1\mod\frak f$ ($\frak f$ an
integral ideal of $\Cal O_L$). We choose an $\frak f$ prime to $p$ such that $w_\frak f=1.$\linebreak Then there is a
grossencharacter $\varphi$ of $L$ satisfying $\varphi((\alpha))=\alpha$ for $\alpha\equiv1\mod\frak f$ (cf. [de Sh,
II.1.4]). According to Weil, after fixing an embedding $\bar\bold Q\hookrightarrow\bar\bold Q_p$ we can asssociate a
$p$-adic character $\varphi_p$ to $\varphi$ (cf. [de Sh, II.1.1 (5)]). We choose an embedding corresponding to a prime
above $\frak p$ and then we find $\varphi_p=\kappa\cdot\chi$ for some $\chi$ of finite order and conductor prime to $p$.
Indeed $\varphi_p$ and $\kappa$ are both unramified at $\frak p^*$ and satisfy $\varphi_p|_{I_\frak p}=\kappa|_{I_\frak
p}=\varepsilon$ where $\varepsilon$ is the cyclotomic character and $I_\frak p$ is an inertia group at $\frak p$. Without
altering $\frak f$ we can even choose\linebreak $\varphi$ so that the order of $\chi$ is prime to $p$. This is by our
hyppothesis that $\kappa$ factored through an extension of the form $\bold Z_p\oplus T$ with $T$ of order prime to $p$.
To see this pick an abelian splitting field of $\varphi_p$ and $\kappa$ whose Galois group has the form $G\oplus G'$ with
$G$ a pro-$p$-group and $G'$ of order prime to $p$. Then we see that $\varphi|_G$ has conductor dividing $\frak f\frak
p^\infty$. Also the only primes which ramify in a $\bold Z_p$-extension lie above $p$ so our hypothesis on $\kappa$
ensures that $\kappa|_G$ has conductor dividing $\frak f\frak p^\infty$. The same is then true of the $p$-part of $\chi$
which therefore has conductor dividing $\frak f$. We can therefore adjust $\varphi$ so that $\chi$ has order prime to $p$
as claimed. We will not however choose $\varphi$ so that $\chi$ is 1 as this would require $\frak f\frak p^\infty$ to be
divisible by $\roman{cond}\ \!\chi$. However we will make the assumption, by altering $\frak f$ if necessary, but still
keeping $\frak f$ prime to $p$, that both $\nu$ and $\varphi_p$ have conductor dividing $\frak f\frak p^\infty$. Thus we
replace $\frak f\frak p^\infty$ by $\roman{l.c.m}.\{f,\roman{cond}\ \!\nu\}.$
The grossencharacter $\varphi$ (or more precisely $\varphi\circ N_{F/L})$ is associated to a (unique) elliptic curve $E$
defined over $F=L(\frak f)$, the ray class field of conductor $\frak f$, with complex multiplication by $\Cal O_L$ and
isomorphic over $\bold C$ to $\bold C/\Cal O_L$ (cf.\linebreak [de Sh, II. Lemma 1.4]). We may even fix a Weierstrass
model of
$E$ over $\Cal O_F$ which has good reduction at all primes above $\frak p$. For each prime $\frak P$ of $F$ above $\frak
p$ we have a formal group $\hat E_\frak P,$ and this is a relative Lubin-Tate group with respect to $F_\frak P$ over
$L_\frak p$ (cf. [de Sh, Ch. II, \S1.10]). We let $\lambda=\lambda_{\hat E_\frak P}$ be the logarithm of this formal
group.
Let $U_\infty$ be the product of the principal local units at the primes above $\frak p$ of $L(\frak f\frak p^\infty)$;
i.e.,
$$U_\infty=\prod\limits_{\frak P|\frak p}U_{\infty,\frak P}\ \ \ \ \roman{where}\ \ \ \ U_{\infty,\frak
P}=\lim_{\longleftarrow} U_n,\frak P,$$
\eject
\noindent each $U_{n,\frak P}$ being the principal local units in $L(\frak f\frak p^n)_\frak P.$ (Note that the primes of
$L(\frak f)$ above $\frak p$ are totally ramified in $L(\frak f\frak p^\infty)$ so we still call them $\{\frak P\}$.) We
wish to define certain homomorphisms $\delta_k$ on $U_\infty$. These were first introduced in [CW] in the case where the
local field $F_\frak P$ is $\bold Q_p$.
Assume for the moment that $F_\frak P$ is $\bold Q_p$. In this case $\hat E_\frak P$ is isomorphic to the Lubin-Tate
group associated to $\pi x+x^p$ where $\pi=\varphi(\frak p)$. Then letting $\omega_n$ be nontrivial roots of
$[\pi^n](x)=0$ chosen so that $[\pi](\omega_n)=\omega_{n-1}$, it was shown in [CW] that to each element
$u=\lim\limits_{\longleftarrow}u_n\in U_{\infty,\frak P}$ there corresponded a unique power series $f_u(T)\in\bold
Z_p[\![T]\!]^\times$ such that $f_u(\omega_n)=u_n$ for $n\ge1$. The definition of $\delta_{k,\frak P}(k\ge1)$ in this
case was then
$$\delta_{k,\frak P}(u)=\bigg({1\over\lambda'(T)}{d\over dT}\bigg)^k\log f_n(T)\bigg|_{T=0}.$$
It is easy to see that $\delta_{k,\frak P}$ gives a homomorphism: $U_\infty\rightarrow U_{\infty,\frak P}\rightarrow\Cal
O_\frak p$ satisfying $\delta_{k,\frak P}(\varepsilon^\sigma)=\theta(\sigma)^k\delta_{k,\frak P}(\varepsilon)$ where
$\theta:\roman{Gal}\Big(\overline F/F\Big)\rightarrow\Cal O^\times_\frak p$ is the character giving the action on
$E[\frak p^\infty]$.
The construction of the power series in [CW] does not extend to the case where the formal group has height $\gt1$ or to
the case where it is defined over an extension of $\bold Q_p$. A more natural approach was developed bt Coleman [Co] which
works in general. (See also [Iw1].) The corresponding generalizations of
$\delta_k$ were given in somewhat greater generality in [Ru3] and then in full generality by de Shalit [de Sh]. We now
summarize these results, thus returning to the general case where $F_\frak P$ is not assumed to be $\bold Q_p$.
To an element $u=\lim\limits_{\longleftarrow}u_n\in U_\infty$ we can associate a power series $f_{u,\frak
P}(T)\in\mathbreak\Cal O_\frak P[\![T]\!]^\times$ where $\Cal O_\frak P$ is the ring of integers of $F_\frak P$; see [de
Sh, Ch. II \S4.5]. (More precisely $f_{u,\frak P}(T)$ is the $\frak P$-component of the power series described there.) For
$\frak P$ we will choose the prime above $\frak p$ corresponding to our chosen embedding $\overline{\bold
Q}\hookrightarrow\overline{\bold Q}_p.$ This power series satisfies $u_{n,\frak P}=(f_{u,\frak P})(\omega_n)$ for all
$n\gt0,n\equiv0(d)$ where $d=[F_\frak P:L_\frak p]$ and $\{\omega_n\}$ is chosen as before as an inverse system of
$\pi^n$ division points of $\hat E_\frak P$. We define a homomorphism $\delta_k:U_\infty\rightarrow\Cal O_\frak P$ by
$$\delta_k(u):=\delta_{k,\frak P}(u)=\Bigg({1\over\lambda'_{\widehat E_\frak P}(T)}{d\over dT}\Bigg)^k\log
f_{u,\frak P}(T)\Bigg|_{T=0}.\leqno(4.11)$$
Then
$$\delta_k(u^\tau)=\theta(\tau)^k\delta_k(u)\ \ \ \ \roman{for}\ \tau\in\roman{Gal}(\bar F/F)\leqno(4.12)$$
where $\theta$ again denotes the action on $E[\frak p^\infty]$. Now $\theta=\varphi_p$ on $\roman{Gal}(\bar F/F).$ We
actually want a homomorphism on $u_\infty$ with a transformation property corresponding to $\nu$ on all of
$\roman{Gal}(\bar L/L).$ Observe that $\nu=\varphi^2_p$ on $\roman{Gal}(\bar F/F)$. Let $S$\linebreak
\eject
\noindent be a set of coset
representatives for $\roman{Gal}(\bar L/L)/\roman{Gal}(\bar L/F)$ and define
$$\Phi_2(u)=\sum_{\sigma\in S}\nu^{-1}(\sigma)\delta_2(u^\sigma)\in\Cal O_\frak P[\nu].\leqno(4.13)$$
Each term is independent of the choice of coset representative by (4.8) and it is easily checked that
$$\Phi_2(u^\sigma)=\nu(\sigma)\Phi_2(u).$$
It takes integral values in $\Cal O_\frak P[\nu].$ Let $U_\infty(\nu)$ denote the product of the groups\linebreak of local
principal units at the primes above $\frak p$ of the field $L(\nu)$ (by which we mean projective limis of local principal
units as before). Then $\Phi_2$ factors through $U_\infty(\nu)$ and thus defines a continuous homomorphism
$$\Phi_2:U_\infty(\nu)\rightarrow\bold C_p.$$
Let $\Cal C_\infty$ be the group of projective limits of elliptic units in $L(\nu)$ as defined in [Ru4]. Then we have a
crucial theorem of Rubin (cf. [Ru4], [Ru2]), proved using the ideas of Kolyvagin:
\vskip6pt
{\smc Theorem} 4.2. {\it There is an equality of characteristic ideals as $\Lambda=\mathbreak\bold
Z_p[[\roman{Gal}(L(\nu)/L)]]$-modules}:
$$\roman{char}_\wedge(\roman{Gal}(M_\infty/L(\nu)))=\roman{char}_\wedge(U_\infty(\nu)/\overline{\Cal C}_\infty).$$
Let $\nu_0=\nu\mod\lambda.$ For any $\bold Z_p[\roman{Gal}(L(\nu_0)/L)]$-module $X$ we write $X^{(\nu_0)}$ for the
maximal quotient of $X\mathop{\otimes}\limits_{\bold Z_p}\Cal O$ on which the action of $\roman{Gal}(L(\nu_0)/L)$ is via
the Teichm\"uller lift of $\nu_0$. Since $\roman{Gal}(L(\nu)/L)$ decomposes into a direct product of a pro-$p$ group and
a group of order prime to $p$,
$$\roman{Gal}(L(\nu)/L)\simeq\roman{Gal}(L(\nu)/L(\nu_0))\times\roman{Gal}(L(\nu_0)/L),$$
we can also consider any $\bold Z_p[[\roman{Gal}(L(\nu)/L)]]$-module also as a $\bold
Z_p[\roman{Gal}(L(\nu_0)/L)]$-module. In particular $X^{(\nu_0)}$ is a module over $\bold
Z_p[\roman{Gal}(L(\nu_0)/L)]^{(\nu_0)}\simeq\Cal O$. Also $\Lambda^{\nu_0)}\simeq\Cal O[[T]].$
Now according to results of Iwasawa ([Iw2, \S12], [Ru2, Theorem 5.1]), $U_\infty(\nu)^{(\nu_0)}$ is a free
$\Lambda^{(\nu_0)}$-module of rank one. We extend $\Phi_2\ \Cal O$-linearly to $U_\infty(\nu)\otimes_{\bold Z_p}\Cal O$
and it then factors through $U_\infty(\nu)^{(\nu_0)}.$ Suppose that $u$ is a\linebreak generator of
$U_\infty(\nu)^{(\nu_0)}$ and
$\beta$ an element of $\bar\Cal C_\infty^{(\nu_0)}$. Then $f(\gamma-1)u=\beta$ for some $f(T)\in\Cal O[[T]]$ and $\gamma$
a topological generator of $\roman{Gal}(L(\nu)/L(\nu_0)).$ Computing\linebreak $\Phi_2$ on both $u$ and $\beta$ gives
$$f(\nu(\gamma)-1)=\phi_2(\beta)/\Phi_2(u).\leqno(4.14)$$
Next we let $e(\frak a)$ be the projective limit of elliptic units in $\lim\limits_{\longleftarrow}L^\times_{\frak f\frak
p^n}$ for $\frak a$ some ideal prime to $6\frak f\frak p$ described in [de Sh, Ch. II,\S4.9]. Then by the proposition of
Chapter II, \S2.7 of [de Sh] this is a $12^\roman{th}$ power in $\lim\limits_{\longleftarrow}L_{\frak f\frak
p^n}^\times.$ We\linebreak
\eject
\noindent let $\beta_1=\beta(\frak a)^{1/12}$ be the projection of $e(\frak a)^{1/12}$ to $U_\infty$ and take
$\beta=\roman{Norm}\ \!\beta_1$ where the norm is from $L_{\frak f\frak p^\infty}$ to $L(\nu)$. A generalization of the
calculation in [CW] which may be found in [de Sh, Ch. II, \S4.10] shows that
$$\Phi_2(\beta)=(\roman{root\ of\ unity})\Omega^{-2}(N\frak a-\nu(\frak a))L_\frak f(2,\bar\nu)\in\Cal
O_\frak P[\nu]\leqno(4.15)$$
where $\Omega$ is a basis for the $\Cal O_L$-module of periods of our chosen Weierstrass model\linebreak of $E_{/F}$.
(Recall that this was chosen to have good reduction at primes above $\frak p$.\linebreak The periods are those of the
standard Neron differential.) Also $\nu$ here should be interpreted as the grossencharacter whose associated $p$-adic
character, via the chosen embedding $\overline{\bold Q}\hookrightarrow\overline{\bold Q}_p$, is $\nu$, and $\overline\nu$
is the complex conjugate of $\nu$.
The only restrictions we have placed on $\frak f$ are that (i) $\frak f$ is prime to $\frak p$;\linebreak (ii) $w_\frak
f=1$; and (iii) $\roman{cond}\ \!\nu|\frak f\frak p^\infty$. Now let $\frak f_0\frak p^\infty$ be the conductor of $\nu$
with
$\frak f_0$ prime to $p$. We show now that we can choose $\frak f$ such that $L_\frak f(2,\overline\nu)/L_{\frak
f_0}(2,\overline\nu)$ is a $p$-adic unit unless $\nu_0=1$ in which case we can choose it to be $t$ as defined in (4.4).
We can clearly choose $L_\frak f(2,\overline\nu)/L_{\frak f_0}(2,\overline\nu)$ to be a unit if $\nu_0\not=1,$ as
$\overline\nu(\frak q)\nu(\frak q)=\roman{Norm}\ \!\frak q^2$ for any ideal $\frak q$ prime to $\frak f_0\frak p$. Note
that if $\nu_0=1$ then also $p=3$. Also if $\nu_0=1$ then we see that
$$\inf\limits_\frak q\#\Big\{\Cal O/\{L_{\frak f_0\frak q}(2,\overline\nu)/L_{\frak f_0}(2,\overline\nu)\}\Big\}=t$$
since $\overline\nu\varepsilon^{-2}=\nu^{-1}.$
We can compute $\Phi_2(u)$ by choosing a special local unit and showing that $\Phi_2(u)$ is a $p$-adic unit, but it is
sufficient for us to know that it is integral. Then since $\roman{Gal}(M_\infty/L(\nu))$ has no finite
$\Lambda$-submodule (by a result of Greenberg; see [Gre2, end of \S4]) we deduce from Theorem 4.2, (4.14) and (4.15) that
$$\matrix{\#\roman{Hom}(\roman{Gal}(M_\infty/L(\nu)),(K/\Cal O)(\nu))^{\roman{Gal}(L(\nu)/L)}}\\
\\
&\hskip-1.8in{\le\cases\#\Cal
O/\Omega^{-2}L_{\frak f_0}(2,\bar\nu)&\roman{if}\ \nu_0\ne1\\
(\#\Cal O/\Omega^{-2}L_{\frak f_0}(2,\bar\nu))\cdot t&\roman{if}\ \nu_0=1.\endcases}\endmatrix$$
Combining this with (4.9) gives:
$$\#H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,Y)\le\#\Big(\Cal O/\Omega^{-2}L_{\frak
f_0}(2,\bar\nu)\Big)\cdot\prod\limits_{q\in\Sigma}\ell_q$$
where $\ell_q=\#H^0(\bold Q_q,Y^*)$ (for $q\ne p$), $\ell_p=\#H^0(\bold Q_p,(Y^0)^*)$.
Since $V\simeq Y\oplus(K/\Cal O)(\psi)\oplus K/\Cal O$ we need also a formula for
$$\#\ker\Big\{H^1(\bold Q_\Sigma/\bold Q,(K/\Cal O)(\psi)\oplus K/\Cal O)\rightarrow H^1(\bold Q_p^\roman{unr},(K/\Cal
O)(\psi)\oplus K/\Cal O)\Big\}.$$
This is easily computed to be
$$\#(\Cal O/h_L)\cdot\prod_{q\in\Sigma-\{p\}}\ell_q\leqno(4.16)$$
where $\ell_q=\#H^0(\bold Q_q,((K/\Cal O)(\psi)\oplus K/\Cal O)^*)$ and $h_L$ is the class number of $\Cal O_L$.
Combining these gives:
\vskip6pt
{\smc Proposition} 4.3.
$$\#H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,V)\le\#(\Cal O/\Omega^{-2}L_{\frak f_0}(2,\overline\nu))\cdot\#(\Cal
O/h_L)\cdot\prod_{q\in\Sigma}\ell_q$$
{\it where} $\ell_q=\#H^0(\bold Q_q,V^*)$ ({\it for} $q\ne p$), $\ell_p=\#H^0(\bold Q_p,(Y^0)^*).$
\
\
\centerline{\bf2. Calculation of $\eta$}
\
We need to calculate explicitly the invariants $\eta_{\Cal D,f}$ introduced in Chapter 2, \S3 in a special case. Let
$\rho_0$ be an irreducible representation as in (1.1). Suppose that $f$ is a newform of weight 2 and level $N,\lambda$ a
prime of $\Cal O_f$ above $p$ and $\rho_{f,\lambda}$ a deformation of $\rho_0$. Let $\frak m$ be the kernel of the
homomorphism $\bold T_1(N)\rightarrow\Cal O_f/\lambda$ arising from $f$. We write $T$ for $\bold T_1(N)_\frak
m\mathop{\otimes}\limits_{W(k_\frak m)}\Cal O$, where $\Cal O=\Cal O_{f,\lambda}$ and $k_\frak m$ is\linebreak the
residue field of
$\frak m$. Assume that $p\nmid N$. We assume here that $k$ is the residue field of $\Cal O$ and that it is chosen to
contain $k_\frak m$. Then by Corollary 1 of Theorem 2.1, $\bold T_1(N)_\frak m$ is Gorenstein andit follows that $T$ is
also a Gorenstein $\Cal O$-algebra (see the discussion following (2.42)). So we can use perfect pairings (the second one
$T$-bilinear)
$$\Cal O\times\Cal O\rightarrow\Cal O,\ \ \ \langle\ ,\ \rangle:T\times T\rightarrow\Cal O$$
to define an invariant $\eta$ of $T$. If $\pi:T\rightarrow\Cal O$ is the natural map, we set $(\eta)=(\hat\pi(1))$ where
$\hat\pi$ is the adjoint of $\pi$ with respect to the pairings. It is well-defined as an ideal of $T$, depending only on
$\pi$. Furthermore, as we noted in Chapter 2, \S3, $\pi(\eta)=\langle\eta,\eta\rangle$ up to a unit in $\Cal O$ and as
noted in the appendix $\eta=\roman{Ann}\ \!\frak p=T[\frak p]$ where $\frak p=\ker\pi$. We now give an explicit formula
for $\eta$ developed by Hida (cf. [Hi2] for a survey of his earlier results) by interpreting $\langle\ ,\ \rangle$ in
terms of the cup product pairing on the cohomology of $X_1(N)$, and then in terms of the Petersson inner product of $f$
with itself. The following account (which does not require the CM hypothesis) is adapted from [Hi2] and we refer there
for more details.
Let
$$(\ ,\ ):H^1(X_1(N),\Cal O_f)\times H^1(X_1(N),\Cal O_f)\rightarrow\Cal O_f\leqno(4.17)$$
be the cup product pairing with $\Cal O_f$ as coefficients. (We sometimes drop the $\Cal C$ from $X_1(N)_{/\bold C}$ or
$J_1(N)_{/\bold C}$ if the context makes it clear that we are referring to the complex manifolds.) In particular
$(t_*x,y)=(x,t^*y)$ for all $x,y$ and for each standard Hecke correspondence $t$. We use the action of $t$ on
$H^1(X_1(N),\Cal O_f)$ given by $x\mapsto t^*x$ and simply write $tx$ for $t^*x$. This is the same\linebreak
\eject
\noindent as the action
induced by $t_*\in\bold T_1(N)$ on $H^1(J_1(N),\Cal O_f)\simeq H^1(X_1(N),\Cal O_f).$ Let $\frak p_f$ be the minimal
prime of
$\bold T_1(N)\otimes\Cal O_f$ associated to $f$ (i.e., the kernel of $\bold T_1(N)\otimes\Cal O_f\rightarrow\Cal O_f$
given by $t_l\otimes\beta\mapsto\beta c_t(f)$ where $tf=c_t(f)f)$, and let
$$L_f=H^1(X_1(N),\Cal O_f)[\frak p_f].$$
If $f=\Sigma a_nq^n$ let $f^\rho=\Sigma\bar a_nq^n$. Then $f^\rho$ is again a newform and we define $L_{f^\rho}$ by
replacing $f$ by $f^\rho$ in the definition of $L_f$. (Note here that $\Cal O_f=\Cal O_{f^\rho}$ as these rings are the
integers of fields which are either totally real or CM by a result of Shimura. Actually this is not essential as we could
replace $\Cal O_f$ by any ring of integers containing it.) Then the pairing $(\ ,\ )$ induces another by restriction
$$(\ ,\ ):L_f\times L_{f^\rho}\rightarrow\Cal O_f.\leqno(4.18)$$
Replacing $\Cal O$ (and the $\Cal O_f$-modules) by the localization of $\Cal O_f$ at $p$ (if necessary) we can assume
that $L_f$ and $L_{f^\rho}$ are free of rank 2 and direct summands as $\Cal O_f$-modules of the respective cohomology
groups. Let $\delta_1,\delta_2$ be a basis of $L_f$. Then also $\bar\delta_1,\bar\delta_2$ is a basis of
$L_{f^\rho}=\overline{L_f}.$ Here complex conjugation acts on $H^1(X_1(N),\Cal O_f)$ via its action on $\Cal O_f$. We can
then verify that
$$(\boldsymbol\delta,\bar{\boldsymbol\delta}):=\det(\delta_i,\bar\delta_j)$$
is an element of $\Cal O_f$ (or its localization at $p$) whose image in $\Cal O_{f,\lambda}$ is given by $\pi(\eta^2)$
(unit). To see this, consider a modified pairing $\langle\ ,\ \rangle$ defined by
$$\langle x,y\rangle=(x,w_\zeta y)\leqno(4.19)$$
where $w_\zeta$ is defined as in (2.4). Then $\langle tx,y\rangle=\langle x,ty\rangle$ for all $x,y$ and Hecke operators
$t$. Furthermore
$$\det\langle\delta_i,\delta_j\rangle=\det(\delta_i,w_\zeta\delta_j)=c\det(\delta_i\overline\delta_j)$$
for some $p$-adic unit $c$ (in $\Cal O_f)$. This is because $w_\zeta(L_{f^\rho})=L_f$ and $w_\zeta(L_f)=L_{f^\rho}.$ (One
can check this, foe example, using the explicit bases described\linebreak below.) Moreover, by Theorem 2.1,
$$\aligned H^1(X_1(N),\bold Z)\otimes_{\bold T_1(N)}\bold T_1(N)_\frak m&\simeq\bold T_1(N)^2_\frak m,\\
H^1(X_1(N),\Cal O_f)\otimes_{\bold T_1(N)\otimes\Cal O_f}T&\simeq T^2.\endaligned$$
Thus (4.18) can be viewed (after tensoring with $\Cal O_{f,\lambda}$ and modifying it as in (4.19)) as a perfect pairing
of $T$-modules and so this serves to compute $\pi(\eta^2)$ as explained earlier (the square coming from the fact that we
have a rank 2 module).
To give a more useful expression for $(\boldsymbol\delta,\bar{\boldsymbol\delta})$ we observe that $f$ and
$\overline{f^\rho}$ can be viewed as elements of $H^1(X_1(N),\bold C)\simeq H^1_\roman{DR}(X_1(N),\bold C)$ via $f\mapsto
f(z)dz,\overline{f^\rho}\mapsto\overline{f^\rho}d\overline z$. Then $\{f,\overline{f^\rho}\}$ form a basis for
$L_f\otimes_{\Cal O_f}\bold C.$ Similarly $\{\bar f,f^\rho\}$ form a basis\linebreak
\eject
\noindent for $L_{f^\rho}\otimes_{\Cal O_f}\bold C.$ Define the vectors
$\boldsymbol\omega_1=(f,\overline{f^\rho}),\boldsymbol\omega_2=(\bar f,f^\rho)$ and write\linebreak
$\boldsymbol\omega_1=C\boldsymbol\delta$ and $\boldsymbol\omega_2=\bar C\bar{\boldsymbol\delta}$ with $C\in M_2(\bold
C)$. Then writing $f_1=f,f_2=\overline{f^\rho}$ we set\linebreak
$$(\boldsymbol\omega,\bar{\boldsymbol\omega}):=\det((f_i,\overline{f_j}))=(\boldsymbol\delta,\bar{\boldsymbol\delta})\det(C\bar
C).$$
Now $(\boldsymbol\omega,\bar{\boldsymbol\omega})$ is given explicitly in terms of the (non-normalized) Petersson inner
product $\langle\ ,\ \rangle$:
$$(\boldsymbol\omega,\bar{\boldsymbol\omega})=-4\langle f,f\rangle^2$$
where $\langle f,f\rangle=\int_{\frak H/\Gamma_1(N)}f\bar f\ \!dx\ \!dy.$
To compute $\det(C)$ we consider integrals over classes in $H_1(X_1(N),\Cal O_f).$ By Poincar'e duality there exist
classes $c_1,c_2$ in $H_1(X_1(N),\Cal O_f)$ such that\linebreak $\det(\int_{c_j}\delta_i)$ is a unit in $\Cal O_f$. Hence
$\det C$ generates the same $\Cal O_f$-module as is generated by $\Big\{\det\Big(\int_{c_j}f_i\Big)\Big\}$ for all such
choices of classes $(c_1,c_2)$ and with $\{f_1,f_2\}=\{f,\overline{f^\rho}\}$. Letting $u_f$ be a generator of the $\Cal
O_f$-module
$\Big\{\det\Big(\int_{c_j}f_i\Big)\Big\}$ we have the following formula of Hida:
\vskip6pt
{\smc Proposition} 4.4. $\pi(\eta^2)=\langle f,f\rangle^2/u_f\bar u_f\times($ {\it unit in} $\Cal O_{f,\lambda})$.
\vskip6pt
Now we restrict to the case where $\rho_0=\roman{Ind}^\bold Q_L\kappa_0$ for some imaginary\linebreak quadratic field $L$
which is unramified at $p$ and some $k^\times$-valued character $\kappa_0$\linebreak of $\roman{Gal}(\bar L/L).$ We assume
that
$\rho_0$ is irreducible, i.e., that $\kappa_0\ne\kappa_{0,\sigma}$ where
$\kappa_{0,\sigma}(\delta)=\kappa_0(\sigma^{-1}\delta\sigma)$ for any $\sigma$ representing the nontrivial coset
of\linebreak
$\roman{Gal}(\bar L/\bold Q)/\roman{Gal}(\bar L/L).$ In addition we wish to assume that $\rho_0$ is ordinary and
$\det\rho_0|_{I_p}=\omega.$ In particular $p$ splits in $L$. These conditions imply that, if $\frak p$ is a prime of $L$
above $p\kappa_0(\alpha)\equiv\alpha^{-1}\mod\frak p$ on $U_\frak p$ after possible replacement of $\kappa_0$ by
$\kappa_{0,\sigma}$. Here the $U_\frak p$ are the units of $L_\frak p$ and since $\kappa_0$ is a character, the
restriction of $\kappa_0$ to an inertia group $I_\frak p$ induces a homomorphism on $U_\frak p$. We assume now that
$\frak p$ is fixed and $\kappa_0$ chosen to satisfy this congruence. Our choice of\linebreak $\kappa_0$ will imply that
the grossencharacter introduced below has conductor prime to $p$.
We choose a (primitive) grossencharacter $\varphi$ on $L$ together with an em-bedding $\overline{\bold
Q}\hookrightarrow\overline{\bold Q}_p$ corresponding to the prime $\frak p$ above $p$ such that the induced $p$-adic
character $\varphi_p$ has the properties:
\vskip6pt
\hskip-10pt(i) $\varphi_p\mod\overline p=\kappa_0$ ($\overline p=$ maximal ideal of $\overline{\bold Q}_p$).
\vskip6pt
\hskip-12.5pt(ii) $\varphi_p$ factors through an abelian extension isomorphic to $\bold Z_p\oplus T$ with $T$ of
\hskip5pt finite order prime to $p$.
\vskip6pt
\hskip-15pt(iii) $\varphi((\alpha))=\alpha$ for $\alpha\equiv1(\frak f)$ for some integral ideal $\frak f$ prime to $p$.
\vskip6pt
\noindent To obtain $\varphi$ it is necessary first to define $\varphi_p$. Let $M_\infty$ denote the maximal abelian
extension of $L$ which is unramified outside $\frak p$. Let $\theta:\roman{Gal}(M_\infty/L)\rightarrow\overline{\bold
Q_p}^\times$ be any character which factors through a $\bold Z_p$-extension and induces the\linebreak
\eject
\noindent homomorphism
$\alpha\mapsto\alpha^{-1}$ on $U_{\frak p,1}\mapsto\roman{Gal}(M_\infty/L)$ where $U_{\frak p,1}=\{u\in U_\frak
p:u\equiv1(\frak p)\}.$ Then set $\varphi_p=\kappa_0\theta$, and pick a grossencharacter $\varphi$ such that
$(\varphi)_p=\varphi_p.$ Note that our choice of $\varphi$ here is not necessarily intended to be the same as the choice
of grossencharacter in Section 1.
Now let $\frak f_\varphi$ be the conductor of $\varphi$ and let $F$ be the ray class field of con-ductor $\frak
f_\varphi\bar\frak f_\varphi$. Then over $F$ there is an elliptic curve, unique up to isomorphism, with complex
multiplication by $\Cal O_L$ and period lattice free, of rank one over $\Cal O_L$ and with associated grossencharacter
$\varphi\circ N_{F/L}$. The curve $E_{/F}$ is the extension\linebreak of scalars of a unique elliptic curve $E_{/F^+}$
where
$F^+$ is the subfield of $F$ of\linebreak index 2. (See [Sh1, (5.4.3)].) Over $F^+$ this elliptic curve has only the
$p$-power isogenies of the form $\pm p^m$ for $m\in\bold Z$. To see this observe that $F$ is unramified at $p$ and
$\rho_0$ is ordinary so that the only isogenies of degree $p$ over $F$ are the ones that correspond to division by
$\ker\frak p$ and
$\ker\frak p'$ where $\frak p\frak p'=(p)$ in $L$. Over $F^+$ these two subgroups are interchanged by complex
conjugation, which gives the assertion. We let $E/_{\Cal O_{F^+,(p)}}$ denote a Weierstrass model over $\Cal
O_{F^+,(p)}$, the localization of $\Cal O_{F^+}$ at $p$, with good reduction at the primes above $p$. Let $\omega_E$ be a
Neron differential of
$E_{/\Cal O_{F^+,(p)}}.$ Let $\Omega$ be a basis for the $\Cal O_L$-module of periods of $\omega_E.$ Then
$\overline\Omega=u\cdot\Omega$ for some $p$-adic unit in $F^\times$.
According to a theorem of Hecke, $\varphi$ is associated to a cusp form $f_\varphi$ in such\linebreak a way that the
$L$-series
$L(s,\varphi)$ and $L(s,f_\varphi)$ are equal (cf. [Sh4, Lemma 3]). Moreover since $\varphi$ was assumed primitive,
$f=f_\varphi$ is a newform. Thus the integer $N=\roman{cond}\ \!f=|\Delta_{L/\bold Q}|\roman{Norm}_{L/\bold
Q}(\roman{cond}\ \!\varphi)$ is prime to $p$ and there is a homomorphism
$$\psi_f:\bold T_1(N)\twoheadrightarrow R_f\subset\Cal O_f\subset\Cal O_\varphi$$
satisfying $\psi_f(T_l)=\varphi(\frak c)+\varphi(\hat\frak c)$ if $l=\frak c\hat\frak c$ in $L,\ (l\nmid N)$ and
$\psi_f(T_l)=0$ if $l$ is inert\linebreak in $L\ (l\nmid N)$. Also $\psi_f(l\langle
l\rangle)=\varphi((l))\psi(l)$ where $\psi$ is the quadratic character\linebreak associated to $L$. Using the embedding of
$\bar{\bold Q}$ in $\bar{\bold Q}_p$ chosen above we get a prime $\lambda$ of $\Cal O_f$ above $p$, a maximal ideal
$\frak m$ of $\bold T_1(N)$ and a homomorphism\linebreak $\bold T_1(N)_\frak m\rightarrow\Cal O_{f,\lambda}$ such that the
associated representation $\rho_{f,\lambda}$ reduces to\linebreak $\rho_0\mod\lambda$.
Let $\frak p_0=\ker\psi_f:\bold T_1(N)\rightarrow\Cal O_f$ and let
$$A_f=J_1(N)/\frak p_0J_1(N)$$
be the abelian variety associated to $f$ by Shimura. Over $F^+$ there is an isogeny
$$A_{f/F^+}\sim(E_{/F^+})^d$$
where $d=[\Cal O_f:\bold Z]$ (see [Sh4, Th. 1]). To see this one checks that the $p$-adic Ga-lois representation
associated to the Tate modules on each side are equivalent\linebreak to $(\roman{Ind}^{F^+}_F\varphi_o)\otimes_{\bold
Z_p}K_{f,p}$ where $K_{f,p}=\Cal O_f\otimes\bold Q_p$ and where $\varphi_p:\roman{Gal}(\overline F/F)\rightarrow\bold
Z^\times_p$ is the $p$-adic character associated to $\varphi$ and restricted to $F$. (one compares
$\roman{trace}(\roman{Frob}\ \!\ell)$ in the two representations for $\ell\nmid Np$ and $\ell$ split completely in $F^+$;
cf. the discussion after Theorem 2.1 for the representation on $A_f.$)
\eject
Now pick a nonconstant map
$$\pi:X_1(N)_{/F^+}\rightarrow E_{/F^+}$$
which factors through $A_{f/F^+}$. Let $M$ be the composite of $F^+$ and the normal closure of $K_f$ viewed in $\bold C$.
Let $\omega_E$ be a Neron differential of $E_{/\Cal O_{F^+,(p)}}.$ Extending scalars to $M$ we can write
$$\pi^*\omega_E=\sum_{\sigma\in\roman{Hom}(K_f,\bold C)}a_\sigma\omega_{f^\sigma},\ \ \ \ a_\sigma\in M$$
where $\omega_{f^\sigma}=\sum\limits_{n=1}^\infty a_n(f^\sigma)q^n{dq\over q}$ for each $\sigma$. By suitably choosing $\pi$
we can assume that $a_{\roman{id}}\ne0.$ Then there exist $\lambda_i\in\Cal O_M$ and $t_i\in\bold T_1(N)$ such that
$$\sum\lambda_it_i\pi^*\omega_E=c_1\omega_f\ \ \ \ \roman{for\ some}\ \ c_1\in M.$$
We consider the map
$$\pi':H_1(X_1(N)_{/\bold C},\bold Z)\otimes\Cal O_{M,(p)}\rightarrow H_1(E_{/\bold C},\bold Z)\otimes\Cal
O_{M,(p)}\leqno(4.20)$$
given by $\pi'=\sum\lambda_i(\pi\circ t_i)$. Even if $\pi'$ is not surjective we claim that the image of $\pi'$ always has
the form $H_1(E_{/\bold C},\bold Z)\otimes a\Cal O_{M,(p)}$ for some $a\in\Cal O_M$. This is because tensored with $\bold
Z_p\ \pi'$ can be viewed as a $\roman{Gal}(\overline{\bold Q}/F^+)$-equivariant map of $p$-adic Tate-modules, and the omly
$p$-power isogenies on $E_{/F^+}$ have the form $\pm p^m$ for some $m\in\bold Z$. It follows that we can factor $\pi'$ as
$(1\otimes a)\circ\alpha$ for some other surjective $\alpha$
$$\alpha:H_1(X_1(N)_{/\bold C},\bold Z)\otimes\Cal O_M\rightarrow H^1(E_{/\bold C},\bold Z)\otimes\Cal O_M,$$
now allowing $a$ to be in $\Cal O_{M,(p)}$. Now define $\alpha^*$ on $\Omega^1_{E/\bold C}$ by $\alpha^*=\sum
a^{-1}\lambda_it_i\circ\pi^*$ where $\pi^*:\Omega^1_{E/\bold C}\rightarrow\Omega^1_{J_1(N)/\bold C}$ is the map induced by
$\pi$ and $t_i$ has the usual action on $\Omega^1_{J_1(N)/\bold C}$. Then $\alpha^*(\omega_E)=c\omega_f$ for some $c\in M$
and
$$\int_\gamma\alpha^*(\omega_E)=\int_{\alpha(\gamma)}\omega_E\leqno(4.21)$$
for any class $\gamma\in H_1(X_1(N)_{/\bold C},\Cal O_M)$. We note that $\alpha$ (on homology as in\linebreak (4.20)) also
comes from a map of abelian varieties $\alpha:J_1(N)_{/F^+}\otimes_{\bold Z}\Cal O_M\rightarrow E_{/F^+}\otimes_\bold Z\Cal
O_M$ although we have not used this to define $\alpha^*$.
We claim now that $c\in\Cal O_{M,(p)}$. We can compute $\alpha^*(\omega_E)$ by considering $\alpha^*(\omega_E\otimes1)=\sum
t_i\pi^*\otimes a^{-1}\lambda_i$ on $\Omega^1_{E/F^+}\otimes\Cal O_M$ and then mapping the image in\linebreak
$\Omega^1_{J_1(N)/F^+}\otimes\Cal O_M$ to $\Omega^1_{J_1(N)/F^+}\otimes_{\Cal O_{F^+}}\Cal O_M=\Omega^1_{J_1(N)/M}.$ Now let
us write $\Cal O_1$ for $\Cal O_{F^+,(p)}$. Then there are isomorphisms
$$\Omega^1_{J_1(N)_{/\Cal O_1}\otimes\Cal O_2}\mathop{\longrightarrow}\limits^{s_1\atop\sim}\roman{Hom}(\Cal
O_M,\Omega^1_{J_1(N)_{/\Cal O_1}})\mathop{\longrightarrow}\limits^{s_2\atop\sim}\Omega^1_{J_1(N)_{/\Cal
O_1}}\otimes\delta^{-1}$$
\eject
\noindent where $\delta$ is the different of $M/\bold Q$. The first isomorphism can be described as follows. Let
$e(\gamma):J_1(N)\rightarrow J_1(N)\otimes\Cal O_M$ for $\gamma\in\Cal O_M$ be the map $x\mapsto x\otimes\gamma$. Then
$t_1(\omega)(\delta)=e(\gamma)^*\omega$. Similar identifications occur for $E$ in place of $J_1(N)$. So to check that
$\alpha^*(\omega_E\otimes1)\in\Omega^1_{J_1(N)_{/\Cal O_1}}\otimes\Cal O_M$ it is enough to observe that by its construction
$\alpha$ comes from a homomorphism $J_1(N)_{/\Cal O_1}\otimes\Cal O_M\rightarrow E_{/\Cal O_1}\otimes\Cal O_M.$ It follows
that we can compare the periods of $f$ and of $\omega_E$.
For $f^\rho$ we use the fact that $\overline{\int_\gamma f^\rho\ \!dz}=\int_{\gamma^c}f\ \!dz$ where $c$ is the $\Cal
O_M$-linear map on homology coming from complex conjugation on the curve. We deduce:
\vskip6pt
{\smc Proposition} 4.5. $u_f={1\over4\pi^2}\Omega^2.(1/p$-{\it adic integer})){\it.}
\vskip6pt
We now give an expression for $\langle f_\varphi,f_\varphi\rangle$ in terms of the $L$-function of $\varphi$. This was first
observed by Shimura [Sh2] although the precise form we want was given by Hida.
\vskip6pt
{\smc Proposition} 4.6.
$$\langle f_\varphi,f_\varphi\rangle={1\over16\pi^3}N^2\Bigg\{\prod_{q|N\atop q\not\in S_\varphi}\Big(1-{1\over
q}\Big)\Bigg\}L_N(2,\varphi^2\bar{\hat\chi})L_N(1,\psi)$$
{\it where $\chi$ is the character of $f_\varphi$ and $\hat\chi$ its restriction to $L$};
{\it $\psi$ is the quadratic character associated to $L$};
{\it $L_N(\ \ \ )$ denotes that the Euler factors for primes dividing $N$ have been removed};
{\it $S_\varphi$ is the set of primes $q|N$ such that $q=\frak q\frak q'$ with $\frak q\nmid\roman{cond}\ \!\varphi$ and
$\frak q,\frak q'$ primes of $L$}, {\it not necessarily distinct.}
\vskip6pt
{\it Proof.} One begins with a formula of Petterson that for an eigenform of weight 2 on $\Gamma_1(N)$ says
$$\langle f,f\rangle=(4\pi)^{-2}\Gamma(2)\Big({1\over3}\Big)\pi[\roman{SL}_2(\bold
Z):\Gamma_1(N)\cdot(\pm1)]\cdot\roman{Res}_{s=2}D(s,f,f^\rho)$$
where $D(s,f,f^\rho)=\sum\limits_{n=1}^\infty|a_n|^2n^{-s}$ if $f=\sum\limits_{n=1}^\infty a_nq^n$ (cf. [Hi3, (5.13)]). One
checks that, removing the Euler factors at primes dividing $N$,
$$D_N(s,f,f^\rho)=L_N(s,\varphi^2\bar{\hat\chi})L_N(s-1,\psi)\zeta_{\bold Q,N}(s-1)/\zeta_{\bold Q,N}(2s-2)$$
by using Lemma 1 of [Sh3]. For each Euler factor of $f$ at a $q|N$ of the form $(1-\alpha_qq^{-s})$ we get also an Euler
factor in $D(s,f,f^\rho)$ of the form $(1-\alpha_q\bar\alpha_qq^{-s})$. When $f=f_\varphi$ this can only happen for a split
prime $\frak q$ where $\frak q'$ divides the\linebreak conductor of $\varphi$ but $\frak q$ does not, or for a ramified prime
$\frak q$ which does not divide\linebreak the conductor of $\varphi$. In this case we get a term $(1-q^{1-s})$ since
$|\varphi(\frak q)|^2=q.$
Putting together the propositions of this section we now have a formula for $\pi(\eta)$ as defined at the beginning of this
section. Actually it is more convenient\linebreak
\eject
\noindent to give a formula for $\pi(\eta_M)$, an invariant defined in the same way
but with
$\bold T_1(M)_{\frak m_1}\otimes_{W(k_{\frak m_1})}\Cal O$ replacing $\bold T_1(N)_\frak m\otimes_{W(k_\frak m)}\Cal O$
where $M=pM_0$ with $p\nmid M_0$ and $M/N$ is of the form
$$\prod_{q\in S_\varphi}q\cdot\prod_{q\nmid N\atop q|M_0}q^2.$$
Here $\frak m_1$ is defined by the requirements that $\rho_{\frak m_1}=\rho_0,U_q\in\frak m$ if $q|M(q\ne p)$\linebreak and
there is an embedding (which we fix) $k_{\frak m_1}\hookrightarrow k$ over $k_0$ taking $U_p\rightarrow\alpha_p$ where
$\alpha_p$ is the unit eigenvalue of $\roman{Frob}\ \!p$ in $\rho_{f,\lambda}$. So if $f'$ is the eigenform obtained from
$f$ by `removing the Euler factors' at $q|(M/N)(q\ne p)$ and\linebreak removing the non-unit Euler factor at $p$ we have
$\eta_M={\hat\pi(1)}$ where
$\pi:T_1=\bold T_1(M)_{\frak m_1}\mathop{\otimes}\limits_{W(k_{\frak m_1)}}\Cal O\rightarrow\Cal O$ corresponds to $f'$ and
the adjoint is taken with respect\linebreak to perfect pairings of $T_1$ and $\Cal O$ with themselves as $\Cal O$-modules,
the first one assumed $T_1$-bilinear.
Property (ii) of $\varphi_p$ ensures that $M$ is as in (2.24) with $\Cal D=(\roman{Se},\Sigma,\Cal O,\phi)$ where $\Sigma$ is
the set of primes dividing $M$. (Note that $S_\varphi$ is precisely the set of primes $q$ for which $n_q=1$ in the notation
of Chapter 2, \S3.) As in Chapter 2, \S3 there is a canonical map
$$R_{\Cal D}\rightarrow\bold T_{\Cal D}\simeq\bold T_1(M)_{\frak m_1}\mathop{\otimes}\limits_{W(k_{\frak m_1)}}\Cal O$$
which is surjective by the arguments in the proof of Proposition 2.15. Here we are considering a slightly more general
situation than that in Chapter 2, \S3 as we are allowing $\rho_0$ to be induced from a character of $\bold Q(\sqrt{-3})$. In
this special case we define $\bold T_\Cal D$ to be $\bold T_1(M)_{\frak m_1}\mathop{\otimes}\limits_{W(k_{\frak m_1)}}\Cal
O$. The existence of the map in (4.22) is proved as in Chapter 2, \S3. For the surjectivity, note that for each $q|M$ (with
$q\ne p$) $U_q$ is zero in $\bold T_\Cal D$ as $U_q\in\frak m_1$ for each such $q$ so that we can apply Remark 2.8. To see
that $U_p$ is in the image of $R_\Cal D$ we use that it is the eigenvalue of $\roman{Frob}\ \!p$ on the unique unramified
quotient which is free of rank one in the representation $\rho$ described after the corollaries to Theorem 2.1 (cf. Theorem
2.1.4 of [Wi1]). To verify this one checks that $\bold T_\Cal D$ is reduced or alternatively one can apply the method of
Remark 2.11. We deduce that $U_p\in\bold T^{\roman tr}_\Cal D$, the $W(k_{\frak m_1})$-subalgebra of $\bold T_1(M)_{\frak
m_1}$ generated by the traces, and it follows then that it is in the image of $R_\Cal D$. We also need to give a definition
of $\bold T_\Cal D$ where $\Cal D=(\roman{ord},\Sigma,\Cal O,\phi)$ and $\rho_0$ is induced from a character of $\bold
Q(\sqrt{-3}).$ For this we use (2.31).
Now we take
$$M=Np\prod_{q\in S_\varphi}q.$$
\eject
\noindent The arguments in the proof of Theorem 2.17 show that
$$\pi(\eta_M)\ \roman{is\ divisible\ by}\ \pi(\eta)(\alpha^2_p-\langle p\rangle)\cdot\prod_{q\in S_\varphi}(q-1)$$
where $\alpha_p$ is the unit eigenvalue of $\roman{Frob}\ \!p$ in $\rho_{f,\lambda}$. The factor at $p$ is given by remark
2.18 and at $q$ it comes from the argument of Proposition 2.12 but with $H=H'=1$. Combining this with Propositions 4.4, 4.5,
and 4.6, we have that
$$\pi(\eta_M)\ \roman{is\ divisible\ by}\
\Omega^{-2}L_N\Big(2,\varphi^2\bar{\hat\chi}\Big){L_N(1,\phi)\over\pi}(\alpha^2_p-\langle
p\rangle)\prod_{q|N}(q-1).\leqno(4.23)$$
We deduce:
\vskip6pt
{\smc Theorem} 4.7. $\#(\Cal O/\pi(\eta_M))=\#H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,V).$
\vskip6pt
{\it Proof.} As explained in Chapter 2, \S3 it is sufficient to prove the inequality $\#(\Cal
O/\pi(\eta_M))\ge\#H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,V)$ as the opposite one is immediate. For this it suffices to
compare (4.23) with Proposition 4.3. Since
$$L_N(2,\bar\nu)=L_N(2,\nu)=L_N(2,\varphi^2\bar{\hat\chi})$$
(note that the right-hand term is real by Proposition 4.6) it suffices to air up the Euler factors at $q$ for $q|N$ in
(4.23) and in the expression for the upper bound of $\#H^1_\roman{Se}(\bold Q_\Sigma/\bold Q,V).\hfill\qed$
\vskip6pt
We now deduce the main theorem in the CM case using the method of Theorem 2.17.
\vskip6pt
{\smc Theorem} 4.8. {\it Suppose that $\rho_0$ as in} (1.1) {\it is an irreducible represen-\linebreak tation of odd determinant
such that $\rho_0=\roman{Ind}^\bold Q_L\kappa_0$ for a character $\kappa_0$ of an imaginary quadratic extension $L$ of $\bold Q$
which is unramified at $p$. Assume also that}:
\vskip6pt
(i) $\det\rho_0\Big|_{I_p}=\omega$;
\vskip6pt
(ii) $\rho_0$ {\it is ordinary.}
\vskip6pt
\noindent{\it Then for every $\Cal D=(\cdot,\Sigma,\Cal O,\phi)$ uch that $\rho_0$ os of type $\Cal D$ with
$\cdot=\roman{Se}$ or $\roman{ord}$},
$$R_\Cal D\simeq\bold T_\Cal D$$
{\it and $\bold T_\Cal D$ is a complete intersection.}
\vskip6pt
{\smc Corollary}. {\it For any $\rho_0$ as in the theorem suppose that
$$\rho:\roman{Gal}(\bar{\bold Q}/\bold Q)\rightarrow\roman{GL}_2(\Cal O)$$ is a continuous representation with values in the
ring of integers of a local field}, {\it unramified outside a finite set of primes}, {\it satisfying $\bar\rho\simeq\rho_0$
when viewed as representations to $\roman{GL}_2(\bar{\bold F}_p)$. Suppose further that}:
\eject
\hskip-12pt(i) $\rho\Big|_{D_p}$ {\it is ordinary};
\
\hskip-15pt(ii) $\det\rho\Big|_{I_p}=\chi\varepsilon^{k-1}$ {\it with $\chi$ of finite order}, $k\ge2${\it.}
\
\noindent{\it Then $\rho$ is associated to a modular form of weight $k$.}
\
\
\centerline{\bf Chapter 5}
\
\
In this chapter we prove the main results about elliptic curves and especially show how to remove the hypothesis that the
representation associated to the 3-division points should be irreducible.
\
\
\centerline{\bf Application to elliptic curves}
\
The key result used is the following theorem of Langlands and Tunnell, extending earlier results of Hecke in the case where
the projective image is dihedral.
\
{\smc Theorem 5.1} (Langlands-Tunnell). {\it Suppose that $\rho:\roman{Gal}(\bar\bold Q/\bold
Q)\rightarrow\mathbreak\roman{GL}_2(\bold C)$ is a continuous irreducible representation whose image is finite and solvable.
Suppose further that $\det\rho$ is odd. Then there exists a weight one newform $f$ such that $L(s,f)=L(s,\rho)$ up to
finitely many Euler factors.}
\
Langlands actually proved in [La] a much more general result without restriction on the determinant or the number field
(which in our case is $\bold Q$). However in the crucial case where the image in $\roman{PGL}_2(\bold C)$ is $S_4$, the
result was only obtained with an additional hypothesis. This was subsequently removed by Tunnell in [Tu].
\
Suppose then that $$\rho_0:\roman{Gal}(\bar\bold Q/\bold Q)\rightarrow\roman{GL_2(\bold F_3)}$$ is an irreducible
representation of odd determinant. We now show, using the theorem, that this representation is modular in the sense that over
$\bar\bold F_3,\mathbreak \rho_0\approx\rho_{g,\mu}\mod\mu$ for some pair $(g,\mu)$ with $g$ some newform of weight 2 (cf.
[Se,\linebreak
\S5.3]). There exists a representation $$i:\roman{GL}_2(\bold F_3)\hookrightarrow\roman{GL}_2\Big(\bold
Z\Big[\sqrt{-2}\Big]\Big)\subset\roman{GL}_2(\bold C).$$ By composing $i$ with an automorphism of $\roman{GL}_2(\bold F_3)$
if necessary we can assume that $i$ induces the identity on reduction $\mod\big(1+\sqrt{-2}\big)$. So if we
consider\linebreak
\eject
\noindent$i\circ\rho_0:\roman{Gal}(\bar\bold Q/\bold Q)\rightarrow\roman{GL}_2(\bold C)$ we obtain an irreducible
representation which is easily seen to be odd and whose image is solvable. Applying the theorem we\linebreak find a newform
$f$ of weight one associated to this representation. Its eigenvalues lie in $\bold Z\big[\sqrt{-2}\big].$ Now pick a modular
form $E$ of weight one such that $E\equiv1(3).$ For example, we can take $E=6E_{1,\chi}$ where $E_{1,\chi}$ is the Eisenstein
series with Mellin transform given by $\zeta(s)\zeta(s,\chi)$ for $\chi$ the quadratic character associated to $\bold
Q(\sqrt{-3})$. Then $fE\equiv f\mod3$ and using the Deligne-Serre lemma ([DS, Lemma 6.11]) we can find an eigenform $g'$ of
weight 2 with the same eigenvalues as $f$ modulo a prime $\mu'$ above $(1+\sqrt{-2}).$ There is a newform $g$ of weight 2
which has the same eigenvalues as $g'$ for almost all $T_l$'s, and we replace $(g',\mu')$ by $(g,\mu)$ for some prime $\mu$
above $(1+\sqrt{-2}).$ Then the pair $(g,\mu)$ satisfies our requirements for a suitable choice of $\mu$ (compatible with
$\mu'$).
\
We can apply this to an elliptic curve $E$ defined over $\bold Q$ by considering $E[3].$ We now show how in studying elliptic
curves our restriction to irreducible representations in the deformation theory can be circumvented.
\
{\smc Theorem 5.2.} {\it All semistable elliptic curves over $\bold Q$ are modular.}
\
{\it Proof.} Suppose that $E$ is a semistable elliptic curve over $\bold Q$. Assume\linebreak first that the representation
$\bar\rho_{E,3}$ on $E[3]$ is irreducible. Then if $\rho_0=\bar\rho_{E,3}$ restricted to $\roman{Gal}(\bar{\bold Q}/\bold
Q(\sqrt{-3}))$ were not absolutely irreducible, the image of the restriction would be abelian of order prime to 3. As the
semistable hypothesis implies that all the inertia groups outside 3 in the splitting field of $\rho_0$ have order dividing 3
this means that the splitting field of $\rho_0$ is unramified outside 3. However, $\bold Q(\sqrt{-3})$ has no nontrivial
abelian extensions unramified outside 3 and of order prime to 3. So $\rho_0$ itself would factor through an abelian extension
of $\bold Q$ and this is a contradiction as $\rho_0$ is assumed odd and irreducible. So $\rho_0$ restricted to
$\roman{Gal}(\bar{\bold Q}/\bold Q(\sqrt{-3}))$ is absolutely irreducible and $\rho_{E,3}$ is then modular by Theorem 0.2
(proved at the end of Chapter 3). By Serre's isogeny theorem, $E$ is also modular (in the sense of being a factor of the
Jacobian of a modular curve).
So assume now that $\bar\rho_{E,3}$ is reducible. Then we claim that the representation $\bar\rho_{E,5}$ on the 5-division
points is irreducible. This is because $X_0(15)(\bold Q)$ has only four rational points besides the cusps and these correspond
to non-semistable curves which in any case are modular; cf. [BiKu, pp. 79-80]. If we knew that $\bar\rho_{E,5}$ was modular we
could now prove the theorem in the same way we did knowing that $\bar\rho_{E,3}$ was modular once we observe that
$\bar\rho_{E,5}$ restricted to $\roman{Gal}(\bar{\bold Q}/\bold Q(\sqrt5))$ is absolutely irreducible. This irreducibility
follows a similar argument to the one for $\bar\rho_{E,3}$ since the only nontrivial abelian extension of $\bold Q(\sqrt5)$
unramified outside 5 and of order prime to 5 is $\bold Q(\zeta_5)$ which is abelian over $\bold Q$. Alternatively, it is
enough to check that there are no elliptic curves\linebreak $E$ for which $\bar\rho_{E,5}$ is an induced representation over
$\bold Q(\sqrt5)$ and $E$ is semistable\linebreak
\eject
\noindent at 5. This can be checked in the supersingular case using the description of $\bar\rho_{E,5}|_{D_5}$ (in particular
it is induced from a character of the unramified quadratic extension of $\bold Q_5$ whose restriction to inertia is the
fundamental character of\linebreak level 2) and in the ordinary case it is straightforward.
Consider the twisted form $X(\rho)_{/\bold Q}$ of $X(5)_{/\bold Q}$ defined as follows. Let $X(5)_{/\bold Q}$ be the
(geometrically disconnected) curve whose non-cuspidal points classify elliptic curves with full level 5 structure and let the
twisted curve be defined by the cohomology class (even homomorphism) in
$$H^1(\roman{Gal}(L/\bold Q),\ \ \ \roman{Aut}\ !X(5)_{/L})$$
given by $\bar\rho_{E,5}:\roman{Gal}(L/\bold Q)\longrightarrow\roman{GL}_2(\bold Z/5\bold Z)\subseteq\roman{Aut}\ \!X(5)_{/L}$
where $L$ denotes the splitting field of $\bar\rho_{E,5}$. Then $E$ defines a rational point on $X(\rho)_{/\bold Q}$ and hence
also of an irreducible component of it which we denote $C$. This curve $C$ is smooth as $X(\rho)_{/\bar{\bold
Q}}=X(5)_{/\bar{\bold Q}}$ is smooth. It has genus zero since the same is true of the irreducible components of
$X(5)_{/\bar{\bold Q}}$.
A rational point on $C$ (necessarily non-cuspidal) corresponds to an elliptic curve $E'$ over $\bold Q$ with an isomorphism
$E'[5]\simeq E[5]$ as Galois modules (cf. [DR,\linebreak VI, Prop. 3.2]). We claim that we can choose such a point with the two
properties that (i) the Galois representation $\bar\rho_{E',3}$ is irreducible and (ii) $E'$ (or a quadtratic twist)has
semistable reduction at 5. The curve $E'$ (or a quadratic twist) will then satisfy all the properties needed to apply Theorem
0.2. (For the primes $q\ne5$ we just use the fact that $E'$ is semistable at $q\Longleftrightarrow\#\bar\rho_{E',5}(I_q)|5.$)
So $E'$ will be modular and hence so too will $\bar\rho_{E',5}.$
To pick a rational point on $C$ satisfying (i) and (ii) we use the Hilbert irreducibility theorem. For, to ensure condition
(i) holds, we only have to eliminate the possibility that the image of $\bar\rho_{E',3}$ is reducible. But this corresponds to
$E'$ being the image of a rational point on an irreducible covering of $C$ of degree\linebreak 4. Let $\bold Q(t)$ be the
function field of $C$. We have therefore an irreducible poly-nomial $f(x,t)\in\bold Q(t)[x]$ of degree $\gt1$ and we need to
ensure that for many values $t_0$ in $\bold Q,\ f(x,t_0)$ has no rational solution. Hilbert's theorem ensures\linebreak that
there exists a $t_1$ such that $f(x,t_1)$ is irreducible. Then we pick a prime $p_1\ne5$ such that $f(x,t_1)$ has no root mod $
p_1$. (This is easily achieved using the\linebreak \v Cebotarev density theorem; cf. [CF, ex. 6.2, p. 362].) So finally we pick
any
$t_0\in\bold Q$ which is
$p_1$-adically close to $t_1$ and also 5-adically close to the original value of $t$ giving $E$. This last condition ensures
that $E'$ (corresponding to $t_0$)\linebreak or a quadratic twist has semistable reduction at 5. To see this, observe that
since
$j_E\ne0,1728$, we can find a family $E(j):y^2=x^3-g_2(j)x-g_3(j)$ with rational functions $g_2(j),g_3(j)$ which are finite at
$j_E$ and with the $j$-invariant of\linebreak $E(j_0)$ equal to $j_0$ whenever the $g_i(j_0)$ are finite. Then $E$ is given by
a quadratic\linebreak twist of $E(j_E)$ and so after a change of functions of the form $g_2(j)\mapsto
u^2g_2(j),\mathbreak g_3(j)\mapsto u^3g_3(j)$ with $u\in\bold Q^\times$ we can assume that $E(j_E)=E$ and that the equation
$E(j_E)$ is minimal at 5. Then for
$j'\in\bold Q$ close enough 5-adically to $j_E$\linebreak
\eject
\noindent the equation $E(j')$ is still minimal and semistable at 5, since a criterion for this, for an integral model, is that
either $\roman{ord}_5(\triangle(E(j')))=0$ or $\roman{ord}_5(c_4(E(j')))=0.$ So up to a quadratic twist $E'$ is also
semistable.\hfill$\qed$
\
This kind of argument can be applied more generally.
\
{\smc Theorem 5.3.} {\it Suppose that $E$ is an elliptic curve defined over $\bold Q$ with the following properties}:
\
(i) $E$ {\it has good or multiplicative reduction at} 3, 5,
(ii) {\it For $p=3,5$ and for any prime $q\!\equiv-1\!\!\mod p$ either $\bar\rho_{E,p}|_{D_q}$ is reducible\linebreak over
$\bar\bold F_p$ or $\bar\rho_{E,p}|{I_q}$ is irreducible over $\bar\bold F_p$.
\
\noindent Then $E$ is modular.}
\
{\it Proof.} the main point to be checked is that one can carry over condi-\linebreak tion (ii) to the new curve $E'$. For
this we use that for any odd prime $p\ne q$, $$\bar\rho_{E,p}|_{D_q}\ \roman{is\ absolutely\ irreducible\ and\ }
\bar\rho_{E,p}|_{I_q}\ \roman{is\ absolutely\ reducible}$$ $$\roman{and}\ 3\nmid\#\bar\rho_{E,p}(I_q)$$ $$\Updownarrow$$
\hskip10pt$E$ acquires good reduction over an abelian 2-power extension of
\hskip48pt$\bold Q^\roman{unr}_q$ but not over an abelian extension of $\bold Q_q$.
\
Suppose then that $q\equiv-1(3)$ and that $E'$ does not satisfy condition (ii) at $q$ (for $p=3$). Then we claim that also
$3\nmid\#\bar\rho_{E',3}(I_q).$ For otherwise $\bar\rho_{E',3}(I_q)$ has its normalizer in $\roman{GL}_2(\bold F_3)$
contained in a Borel, whence $\bar\rho_{E',3}(D_q)$ would be reducible which contradicts our hypothesis. So using the above
equivalence we deduce, by passing via $\bar\rho_{E',5}\simeq\bar\rho_{E,5},$ that $E$ also does not satisfy hypothesis (ii) at
$p=3$.
We also need to ensure that $\bar\rho_{E',3}$ is absolutely irreducible over $\bold Q(\sqrt{-3}\ ).$ This we can do by
observing that the property that the image of $\bar\rho_{E',3}$ lies in the\linebreak Sylow 2-subgroup of $\roman{GL}_2(\bold
F_3)$ implies that $E'$ is the image of a rational point on a certain irreducible covering of $C$ of nontrivial degree. We
can then argue in the same way we did in the previous theorem to eliminate the possibility that $\bar\rho_{E',3}$ was
reducible, this time using two separate coverings to ensure that the image of $\bar\rho_{E',3}$ is neither reducible nor
contained in a Sylow 2-subgroup.
Finally one also has to show that if both $\bar\rho_{E,5}$ is irreducible and $\bar\rho_{E,3}$ is induced from a character
of $\bold Q(\sqrt{-3}\ )$ then $E$ is modular. (The case where\linebreak both were reducible has already been considered.)
Taylor has pointed out that curves satisfying both these conditions are classified by the non-cuspidal\linebreak rational
points on a modular curve isomorphic to $X_0(45)/W_9,$ and this is an elliptic curve isogenous to $X_0(15)$ with rank zero
over $\bold Q$. The non-cuspidal rational points correspond to modular elliptic curves of conductor 338.\hfill$\square$
\eject
\
\centerline{\bf Appendix}
\
\centerline{\bf Gorenstein rings and local complete intersections}
\
{\smc Proposition 1.} {\it Suppose that $\Cal O$ is a complete discrete valuation ring\linebreak and that
$\varphi:S\rightarrow T$ is a surjective local $\Cal O$-algebra homomorphism between com-\linebreak plete local
Noetherian $\Cal O$-algebras. Suppose further that $\frak p_T$ is a prime ideal of $T$ such that $T/\frak
p_T\mathop{\longrightarrow}\limits^{{}\atop\scriptstyle\sim}\Cal O$ and let $\frak p_S=\varphi^{-1}(\frak p_T)$. Assume that}
\vskip6pt(i) $T\simeq\Cal O[\![x_1,\dots,x_r]\!]/(f_1,\dots,f_{r-u})$ {\it where $r$ is the size of a minimal set
of\linebreak $\Cal O$-generators of} $\frak p_T/\frak p_T^2,$
\vskip6pt(ii) $\varphi$ {\it induces an isomorphism $\frak
p_S/\frak p_S^2\mathop{\longrightarrow}\limits^{{}\atop\scriptstyle\sim}\frak p_T/\frak p_T^2$ and that these are finitely
generated $\Cal O$-modules whose free part has rank $u$.
\vskip6pt\noindent Then $\varphi$ is an isomorphism.
Proof. } First we consider the case where $u=0$. We may assume that the generators $x_1,\dots,x_r$ lie in $\frak p_T$ by
subtracting their residues in $T/\frak p_T\mathop{\longrightarrow}\limits^{{}\atop\scriptstyle\sim}\Cal O.$ By (ii) we may
also write $$S\simeq\Cal O[\![x_1,\dots,x_r]\!]/(g_1,\dots,g_s)$$ with $s\ge r$ (by allowing repetitions if necessary) and
$\frak p_S$ generated by the images of $\{x_1\dots,x_r\}.$ Let $\frak p=(x_1,\dots,x_r)$ in $\Cal[\![x_1,\dots,x_r]\!].$
Writing $f_i\equiv\mathbreak\Sigma a_{ij}x_j\mod\frak p^2$ with $a_{ij}\in\Cal O,$ we see that the Fitting ideal as an $\Cal
O$-module of $\frak p_T/\frak p_T^2$ is given by $$F_\Cal O(\frak p_T/\frak p_T^2)=\det(a_{ij})\in\Cal O$$ and that this is
nonzero by the hypothesis that $u=0.$ Similarly, if each\linebreak $g_i\equiv\Sigma b_{ij}x_j\mod\frak p^2,$ then $$F_\Cal
O(\frak p_S/\frak p_S^2)=\{\det(b_{ij}):i\in I,\#I=r,I\subseteq\{1,\dots,s\}\}.$$ By (ii) again we see that
$\det(a_{ij})=\det(b_{ij})$ as ideals of $\Cal O$ for some choice $I_0$\linebreak of $I$. After renumbering we may assume
that $I_0=\{1,\dots,r\}.$ Then each $g_i\mathbreak (i=1,\dots,r)$ can be written $g_i=\Sigma r_{ij}f_i$ for some
$r_{ij}\in\Cal[\![x_1,\dots,x_r]\!]$ and we have $$\det(b_{ij})\equiv\det(r_{ij})\cdot\det(a_{ij})\mod\frak p.$$ Hence
$\det(r_{ij})$ is a unit, whence $(r_{ij})$ is an invertible matrix. Thus the $f_i$'s can be expressed in terms of the
$g_i$'s and so $S\simeq T$.
We can extend this to the case $u\ne0$ by picking $x_1,\dots,x_{r-u}$ so that they generate $(\frak p_T/\frak
p_T^2)^\roman{tors}.$ Then we can write each $f_i\equiv\sum_{i=1}^{r-u}a_{ij}x_j\mod\frak p^2$ and likewise for the $g_i$'s.
The argument is now just as before but applied to the Fitting ideals of $(\frak p_T/\frak
p_T^2)^\roman{tors}.$\hfill$\square$
\eject
For the next proposition we continue to assume that $\Cal O$ is a complete discrete valuation ring. Let $T$ be a local $\Cal
O$-algebra which as a module is finite and free over $\Cal O$. In addition, we assume the existence of an isomorphism of
$T$-modules $T\mathop{\longrightarrow}\limits^{{}\atop\scriptstyle\sim}\roman{Hom}_\Cal O(T,\Cal O).$ We call a local $\Cal
O$-algebra which is finite and free and satisfies this extra condition a Gorenstein $\Cal O$-algebra (cf. \S5 of [Ti1]). Now
suppose that $\frak p$ is a prime ideal of $T$ such that $T/\frak p\simeq\Cal O.$
Let $\beta:T\rightarrow T/\frak p\simeq\Cal O$ be the natural map and define a principal ideal of $T$\linebreak by
$$(\eta_T)=(\hat\beta(1))$$ where $\hat\beta:\Cal O\rightarrow T$ is the adjoint of $\beta$ with respect to perfect $\Cal
O$-pairings on $\Cal O$\linebreak and $T$, and where the pairing of $T$ with itself is $T$-bilinear. (By a perfect\linebreak
pairing on a free $\Cal O$-module $M$ of finite rank we mean a pairing $M\times M\rightarrow\Cal O$\linebreak such that both
the induced maps $M\!\rightarrow\!\roman{Hom}_\Cal O(M,\Cal O)$ are isomorphisms. \!When\linebreak $M=T$ we are thus
requiring that this be an isomorphism of $T$-modules also.) The ideal $(\eta_T)$ is independent of the pairing. Also
$T/\eta_T$ is torsion-free as an $\Cal O$-module, as can be seen by applying $\roman{Hom}(\ ,\Cal O)$ to the sequence
$$0\rightarrow\frak p\rightarrow T\rightarrow\Cal O\rightarrow0,$$ to obtain a homomorphism
$T/\eta_T\hookrightarrow\roman{Hom}(\frak p,\Cal O).$ This also shows that $(\eta_T)=\roman{Ann}\frak p.$
If we let $l(M)$ denote the length of an $\Cal O$-module $M$, then $$l(\frak p/\frak p^2)\ge l(\Cal O/\overline{\eta_T})$$
(where we write $\overline{\eta_T}$ for $\beta(\eta_T))$ because $\frak p$ is a faithful $T/\eta_T$-module. (For a brief
account of the relevant properties of Fitting ideals see the appendix to [MW1].) Indeed, writing $F_R(M)$ for the Fitting
ideal of $M$ as an $R$-module, we have $$F_{T/\eta_T}(\frak p)=0\Rightarrow F_T(\frak p)\subset(\eta_T)\Rightarrow
F_{T/\frak p}(\frak p/\frak p^2)\subset(\overline{\eta_T})$$ and we then use the fact that the length of an $\Cal O$-module
$M$ is equal to the length of $\Cal O/F_\Cal O(M)$ as $\Cal O$ is a discrete valuation ring. In particular when $\frak
p/\frak p^2$ is a torsion $\Cal O$-module then $\overline\eta_T\ne0.$
We need a criterion for a Gorenstein $\Cal O$-algebra to be a complete inter-section. We will say that a local $\Cal
O$-algebra $S$ which is finite and free over\linebreak $\Cal O$ is a complete intersection over $\Cal O$ if there is an $\Cal
O$-algebra isomorphism\linebreak $S\simeq\Cal O[\![x_1,\dots,x_r]\!]/(f_1,\dots,f_r)$ for some $r$. Such a ring is
necessarily a Gorenstein $\Cal O$-algebra and $\{f_1,\dots,f_r\}$ is necessarily a regular sequence. That (i)
$\Rightarrow$\linebreak(ii) in the following proposition is due to Tate (see A.3, conclusion 4, in the appendix in [M Ro].)
\vskip6pt
{\smc Proposition 2.} {\it Assume that $\Cal O$ is a complete discrete valuation ring and that $T$ is a local Gorenstein
$\Cal O$-algebra which is finite and free over $\Cal O$ and\linebreak}
\eject
\noindent{\it that $\frak p_T$ is a prime ideal of $T$ such that $T/\frak p_T\cong\Cal O$ and $\frak p_T/\frak p^2_T$ is a
torsion $\Cal O$-module. Then the following two conditions are equivalent}:\vskip6pt
\hskip-10pt(i) $T$ {\it is a complete intersection over $\Cal O$.}\vskip6pt
\hskip-13pt(ii) $l(\frak p_T/\frak p_T^2)=l(\Cal O/\overline{\eta_T})$ {\it as $\Cal O$-modules.}\vskip6pt
\
{\it Proof.} To prove that (ii) $\Rightarrow$ (i), pick a complete intersection $S$ over $\Cal O$ (so assumed finite and
flat over $\Cal O$) such that $\alpha\!:\!S\!\!\twoheadrightarrow\!\!T$ and such that $\frak p_S/\frak p_S^2\simeq\frak
p_T/\frak p_T^2$\linebreak where $\frak p_S=\alpha^{-1}(\frak p_T).$ The existence of such an $S$ seems to be well known (cf.
[Ti2, \S6]) but here is an argument suggested by N. Katz and H. Lenstra (independently).
Write $T=\Cal O[x_1,\dots,x_r]/(f_1,\dots,f_s)$ with $\frak p_T$ the image in $T$ of $\frak p=(x_1,\dots,x_r).$ Since $T$ is
local and finite and free over $\Cal O$, it follows that also $T\simeq\Cal O[\![x_1,\dots,x_r]\!]/(f_1,\dots,f_s).$ We can
pick $g_1,\dots,g_r$ such that $g_i=\Sigma a_{ij}f_j$ with $a_{ij}\in\Cal O$ and such that $$(f_1,\dots,f_s,\frak
p^2)=(g_1,\dots,g_r,\frak p^2).$$ We then modify $g_1,\dots,g_r$ by the addition of elements $\{\alpha_i\}$ of
$(f_1,\dots,f_s)^2$ and set $(g_1'=g_1+\alpha_1,\dots,g_r'=g_r+\alpha_r).$ Since $T$ is finite over $\Cal O$, there exists
an $N$ such that for each $i,x_i^N$ can be written in $T$ as a polynomial $h_i(x_1,\dots,x_r)$ of\linebreak total degree less
than $N$. We can assume also that $N$ is chosen greater than the total degree of $g_i$ for each $i$. Set
$\alpha_i=(x_i^N-h_i(x_1,\dots,x_r))^2.$ Then set $S=\Cal O[\![x_1,\dots,x_r]\!]/(g_1',\dots,g_r').$ Then $S$ is finite over
$\Cal O$ by construction and also\linebreak $\dim(S)\le1$ since $\dim(S/\lambda)=0$ where $(\lambda)$ is the maximal ideal of
$\Cal O$. It follows that $\{g_1',\dots,g_r'\}$ is a regular sequence and hence that $\roman{depth}(S)=\dim(S)=1.$ In
particular the maximal $\Cal O$-torsion submodule of $S$ is zero since it is also a finite length $S$-submodule of $S$.
Now $\Cal O/(\bar\eta_S)\simeq\Cal O/(\bar\eta_T),$ since $l(\Cal O/(\bar\eta_S))=l(\frak p_S/\frak p^2_S)$ by (i)
$\Rightarrow$ (ii) and $l(\Cal O/(\hat\eta_T))=l(\frak p_T/\frak p^2_T)$ by hypothesis. Pick isomorphisms
$$T\simeq\roman{Hom}_\Cal O(T,\Cal O),\ S\simeq\roman{Hom}_\Cal O(S,\Cal O)$$ as $T$-modules and $S$-modules, respectively.
The existence of the latter for complete intersections over $\Cal O$ is well known; cf. conclusion 1 of Theorem A.3 of [M
Ro]. Then we have a sequence of maps, in which $\hat\alpha$ and $\hat\beta$ denote the adjoints with respect to these
isomorphisms: $$\Cal
O\mathop{\longrightarrow}\limits^{{}\atop\scriptstyle{\hat\beta}}T\mathop{\longrightarrow}
\limits^{{}\atop\scriptstyle{\hat\alpha}}S\mathop{\longrightarrow}\limits^{{}\atop\scriptstyle\alpha}T\mathop{\longrightarrow}
\limits^{{}\atop\scriptstyle\beta}\Cal O.$$ One checks that $\hat\alpha$ is a map of $S$-modules ($T$ being given an
$S$-action via $\alpha$) and in particular that $\alpha\circ\hat\alpha$ is multiplication by an element $t$ of $T$. Now
$(\beta\circ\hat\beta)=(\bar\eta_T)$ in $\Cal O$ and $(\beta\circ\alpha)\circ(\widehat{\ \beta\circ\alpha\
})=(\bar\eta_S)$ in
$\Cal O$. As $(\bar\eta_S)=(\bar\eta_T)$ in $\Cal O$, we have that $t$ is a unit $\mod\frak p_T$ and hence that
$\alpha\circ\hat\alpha$ is an isomorphism. It follows\linebreak
\eject
\noindent that $S\simeq T$, as otherwise $S\simeq\ker\alpha\oplus\roman{im}\hat\alpha$ is a nontrivial decomposition as
$S$-modules, which contradicts $S$ being local.\hfill$\square$
\
{\it Remark.} Lenstra has made an important improvement to this proposi-\linebreak tion by showing that replacing
$\bar\eta_T$ by $\beta(\roman{ann}\ \frak p)$ gives a criterion valid for all local $\Cal O$-algebra which are finite and
free over $\Cal O$, thus without the Gorenstein hypothesis.
\
{\eightpoint{\smc Princeton University, Princeton, NJ}
\
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\
\centerline{(Received October 14, 1994)}}
\
\
\vfill\eject
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