Annals of Mathematics, 141 (1995), 443-551

Pierre de Fermat Andrew John Wiles

Modular elliptic curves and Fermat’s Last Theorem

By Andrew John Wiles*

For Nada, Claire, Kate and Olivia

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatum in duos ejusdem nominis fas est dividere: cujes rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

- Pierre de Fermat ∼ 1637

Abstract. When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s 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 > 2 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.

Introduction

An elliptic curve over 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 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 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 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, 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 ε-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.

*The work on this paper was supported by an NSF grant.

Our approach to the study of elliptic curves is via their associated Galois representations. Suppose that ρ_{p} is the representation of Gal(Q̄/Q) on the p-division points of an elliptic curve over Q, and suppose for the moment that ρ_{3} is irreducible. The choice of 3 is critical because a crucial theorem of Langlands and Tunnell shows that if ρ_{3} is irreducible then it is also modular. We then proceed by showing that under the hypothesis that ρ_{3} is semistable at 3, together with some milder restrictions on the ramification of ρ_{3} at the other primes, every suitable lifting of ρ_{3} is modular. To do this we link the problem, via some novel arguments from commutative algebra, to a class number problem 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 strong but distinct traditions in number theory, the relationship between Galois 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 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 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 elaborated on the ε-conjecture alluded to before. Besides the work of Ribet and others on this problem we draw on some of the more specialized developments 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 ρ_{3} be irreducible at 3 is bypassed by means of an intriguing argument with families of elliptic curves which share a common ρ_{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 addition, this method seems well suited to establishing that all elliptic curves over Q are modular and to generalization to other totally real number fields.

Now we present our methods and results in more detail.

Let f be an eigenform associated to the congruence subgroup Γ_{1}(N) of SL_{2}(Z) of weight k ≥ 2 and character χ. Thus if T_{n} is the Hecke operator associated to an integer n there is an algebraic integer c(n, f) such that T_{n}f = c(n, f)f for each n. We let K_{f} be the number field generated over Q by the {c(n, f)} together with the values of χ and let O_{f} be its ring of integers. For any prime λ of O_{f} let O_{f,λ} be the completion of O_{f} at λ. The following theorem is due to Eichler and Shimura (for k = 2) and Deligne (for k > 2). 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.

Theorem 0.1. For each prime p ∈ Z and each prime λ|p of O_{f} there is a continuous representation

ρ_{f,λ} : Gal(Q̄/Q) ⟶ GL_{2}(O_{f,λ})

which is unramified outside the primes dividing Np and such that for all primes q ∤ Np,

trace ρ_{f,λ}(Frob q) = c(q, f), det ρ_{f,λ}(Frob q) = χ(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 λ-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 λ-adic representations except that the proof may be easier for some λ than for others.

Assume

ρ_{0} : Gal(Q̄/Q) ⟶ GL_{2}(F̄_{p})

is a continuous representation with values in the algebraic closure of a finite field of characteristic p and that det ρ_{0} is odd. We say that ρ_{0} is modular if ρ_{0} and ρ_{f,λ} mod λ are isomorphic over F̄_{p} for some f and λ and some embedding of O_{f}/λ in F̄_{p}. Serre has conjectured that every irreducible ρ_{0} of odd determinant is modular. Very little is known about this conjecture except when the image of ρ_{0} in PGL_{2}(F̄_{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 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{ρ_{f,λ}} = ρ_{f,λ} mod λ is independent of the choice of lattice in K_{f,λ}^{2}.

If O is the ring of integers of a local field (containing Q_{p}) we will say that ρ : Gal(Q̄/Q) ⟶ GL_{2}(O) is a lifting of ρ_{0} if, for a specified embedding of the residue field of O in F̄_{p}, ρ̄ and ρ_{0} are isomorphic over F̄_{p}. Our point of view will be to assume that ρ_{0} is modular and then to attempt to give conditions under which a representation ρ lifting ρ_{0} comes from a modular form in the sense that ρ ≃ ρ_{f,λ} over \overline{K_{f,λ}} for some f, λ. We will restrict our attention to two cases:

(I) ρ_{0} is ordinary (at p) by which we mean that there is a one-dimensional subspace of F̄_{p}^{2}, stable under a decomposition group at p and such that the action on the quotient space is unramified and distinct from the action on the subspace.

(II) ρ_{0} is flat (at p), meaning that as a representation of a decomposition group at p, ρ_{0} is equivalent to one that arises from a finite flat group scheme over Z_{p}, and det ρ_{0} restricted to an inertia group at p is the cyclotomic character.

We say similarly that ρ is ordinary (at p), if viewed as a representation to Q̄_{p}^{2}, there is a one-dimensional subspace of Q̄_{p}^{2} stable under a decomposition group at p and such that the action on the quotient space is unramified.

Let ε : Gal(Q̄/Q) ⟶ Z_{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 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 ρ_{0} is modular.)

Conjecture. Suppose that ρ : Gal(Q̄/Q) ⟶ GL_{2}(O) is an irreducible lifting of ρ_{0} and that ρ is unramified outside of a finite set of primes. There are two cases:

(i) Assume that ρ_{0} is ordinary. Then if ρ is ordinary and det ρ = ε^{k−1}χ for some integer k ≥ 2 and some χ of finite order, ρ comes from a modular form.

(ii) Assume that ρ_{0} is flat and that p is odd. Then if ρ restricted to a decomposition group at p is equivalent to a representation on a p-divisible group, again ρ comes from a modular form.

In case (ii) it is not hard to see that if the form exists it has to be of 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 considering other number fields of Q and by considering groups other than GL_{2}.

We prove two results concerning this conjecture. The first includes the hypothesis that ρ_{0} is modular. Here and for the rest of this paper we will assume that p is an odd prime.

Theorem 0.2. Suppose that ρ_{0} is irreducible and satisfies either (I) or (II) above. Suppose also that ρ_{0} is modular and that

(i) ρ_{0} is absolutely irreducible when restricted to Q(\sqrt{(−1)^{\frac{p−1}{2}} p}).

(ii) If q ≡ −1 mod p is ramified in ρ_{0} then either ρ_{0}|_{D_{q}} is reducible over the algebraic closure where D_{q} is a decomposition group at q or ρ_{0}|_{I_{q}} is absolutely irreducible where I_{q} is an inertia group at q.

Then any representation ρ as in the conjecture does indeed come from a modular form.

The only condition which really seems essential to our method is the requirement that ρ_{0} be modular.

The most interesting case at the moment is when p = 3 and ρ_{0} can be defined over F_{3}. Then since PGL_{2}(F_{3}) ≃ S_{4} every such representation is modular by the theorem of Langlands and Tunnell mentioned above. In particular, every representation into GL_{2}(Z_{3}) whose reduction satisfies the given conditions is modular. We deduce:

Theorem 0.3. Suppose that E is an elliptic curve defined over Q and that ρ_{0} is the Galois action on the 3-division points. Suppose that E has the following properties:

(i) E has good or multiplicative reduction at 3.

(ii) ρ_{0} is absolutely irreducible when restricted to Q(\sqrt{−3} ).

(iii) For any q ≡ −1 mod 3 either ρ_{0}|_{D_{q}} is reducible over the algebraic closure or ρ_{0}|I_{q} is absolutely irreducible.

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) 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). 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 ρ_{0} is reducible. Rather surprisingly, Theorem 0.2 can often be applied in this case 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 we obtain the following theorem:

Theorem 0.4. Suppose that E is a semistable elliptic curve defined over Q. Then E is modular.

More general families of elliptic curves which are modular are given in Chapter 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 ‘Fermat’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 representation 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 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!):

Theorem 0.5. Suppose that u^{p} + v^{p} + w^{p} = 0 with u, v, w ∈ Q and p ≥ 3, then uvw = 0. (Equivalently - there are no nonzero integers a, b, c, n with n > 2 such that a^{n} + b^{n} = c^{n}.)

The second result we prove about the conjecture does not require the assumption that ρ_{0} be modular (since it is already known in this case).

Theorem 0.6. Suppose that ρ_{0} is irreducible and satisfies the hypothesis of the conjecture, including (I) above. Suppose further that

(i) ρ_{0} = Ind_{L}^{Q}κ_{0} for a character κ_{0} of an imaginary quadratic extension L of Q which is unramified at p.

(ii) det ρ_{0}|_{I_{p}} = ω.

Then a representation ρ 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 ℓ-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 ℓ-adic representations. I began with the assumption that the reduction of a given ordinary ℓ-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 ℓ = 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 ℓ 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 ρ_{3}, the mod 3 representation of any given elliptic curve over Q, would necessarily be modular. This enabled me to try inductively to prove that the GL_{2}(Z/3^{n}Z) representation would be modular for each n. At this time I considered only the ordinary case. This led quickly to the study of H^{i}(Gal(F_{∞}/Q), W_{f}) for i = 1 and 2, where F_{∞} is the splitting field of the 𝔪-adic torsion on the Jacobian of a suitable modular curve, 𝔪 being the maximal ideal of a Hecke ring associated to ρ_{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 Gal(Q_{Σ}/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 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 Z_{p}-extension by a similar analysis based on a choice of infinitely many distinct primes q_{i} ≡ 1 mod p^{n_{i}} with n_{i} → ∞ as i → ∞. 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 Σ varies but with the field 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 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 techniques of the first two sections of Chapter 2. This drew in particular on a detailed study of all the congruences between f and other modular forms 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 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 ≡ −1 mod p when varying Σ as the geometric techniques I worked with did not apply in general for primes q ≡ 1 mod p. (This was for much the same reason that the reduction of level argument in [Ri1] is much more difficult when q ≡ 1 mod p.) In all this work I used the more general assumption that ρ_{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 families 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 little work on it. I checked that one could make the appropriate adjustments to the theory in order to describe deformation theories at the minimal level. In the 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 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 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 Z_{p}-extension of Q, and not just the relations up to torsion.

The turning point in this and indeed in the whole proof came in the 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 η-invariant of the appendix) which I saw could be used to test for isomorphisms between Gorenstein rings. A different invariant (the 𝔭/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 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 argument 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 commutative 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 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 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 sufficient in this case and, as explained earlier, when translated into the ringtheoretic 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 estimates described in Chapter 3 were now much stronger, although at that time I was still using primes q ≡ −1 mod p for the argument. The main difficulty was that although I knew how the η-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 𝔭/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, 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 the vanishing of the μ-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, §10]), 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 plausible. 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 breakthrough: I found the argument using families of elliptic curves with a common ρ_{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 construction 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 complete intersection property, as it gave a quick way to exhibit infinite families 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 only used primes q ≡ −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 ≡ 1 mod p he had obtained a version of 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 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 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 old abandoned approach. It was the old idea of picking q_{i}’s with q_{i} ≡ 1mod p^{n_{i}} and n_{i} → ∞ as i → ∞ 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 ηinvariant could be used to count Galois representations. The last step after the 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 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 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.

Table of Contents

Chapter 1 1. Deformations of Galois representations

2. Some computations of cohomology groups

3. Some results on subgroups of GL_{2}(k)

Chapter 2 1. The Gorenstein property

2. Congruences between Hecke rings

3. The main conjectures

Chapter 3 Estimates for the Selmer group

Chapter 4 1. The ordinary CM case

2. Calculation of η

Chapter 5 Application to elliptic curves

Appendix

References

Chapter 1

This chapter is devoted to the study of certain Galois representations. 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 Σ 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 the special primes used in Chapter 3 and [TW].

1. Deformations of Galois representations

Let p be an odd prime. Let Σ be a finite set of primes including p and let Q_{Σ} be the maximal extension of Q unramified outside this set and ∞. Throughout we fix an embedding of \overline{Q}, and so also of Q_{Σ}, in C. We will also fix a choice of decomposition group D_{q} for all primes q in Z. Suppose that k is a finite field characteristic p and that

(1.1)  ρ_{0} : Gal(Q_{Σ}/Q) → GL_{2}(k)

is an irreducible representation. In contrast to the introduction we will assume in the rest of the paper that ρ_{0} comes with its field of definition k. Suppose further that det ρ_{0} is odd. In particular this implies that the smallest field of definition for ρ_{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 ρ_{0} is absolutely irreducible. We consider the deformation [ρ] to GL_{2}(A) of ρ_{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 [ρ] is just a strict equivalence class of homomorphisms ρ : Gal(Q_{Σ}/Q) → GL_{2}(A) such that ρ mod m = ρ_{0}, two such homomorphisms being called strictly equivalent if one can be brought to the other by conjugation by an element of ker : GL_{2}(A) → GL_{2}(k). We often simply write ρ instead of [ρ] for the equivalent class.

We will restrict our choice of ρ_{0} further by assuming that either:

(i) ρ_{0} is ordinary; viz., the restriction of ρ_{0} to the decomposition group D_{p} has (for a suitable choice of basis) the form

(1.2)  ρ_{0}|_{D_{p}} ≈ ( \begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix} )

where χ_{1} and χ_{2} are homomorphisms from D_{p} to k^{∗} with χ_{2} unramified. Moreover we require that χ_{1} ≠ χ_{2}. We do allow here that ρ_{0}|_{D_{p}} be semisimple. (If χ_{1} and χ_{2} are both unramified and ρ_{0}|_{D_{p}} is semisimple then we fix our choices of χ_{1} and χ_{2} once and for all.)

(ii) ρ_{0} is flat at p but not ordinary (cf. [Se1] where the terminology finite is used); viz., ρ_{0}|_{D_{p}} is the representation associated to a finite flat group scheme over Z_{p} but is not ordinary in the sense of (i). (In general when we refer to the flat case we will mean that ρ_{0} is assumed not to be ordinary unless we specify otherwise.) We will assume also that det ρ_{0}|_{I_{p}} = ω where I_{p} is an inertia group at p and ω is the Teichmüller character giving the action on p^{th} roots of unity.

In case (ii) it follows from results of Raynaud that ρ_{0}|_{D_{p}} is absolutely irreducible and one can describe ρ_{0}|_{I_{p}} explicitly. For extending a Jordan-Hölder 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 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 ρ_{0}|_{I_{p}} ⊗_{k} k̄ ≃ ψ_{1} ⊕ ψ_{2} where ψ_{1} and ψ_{2} are the two fundamental characters of

degree 2 (cf. Corollary 3.4.4 of [Ray1]). Since ψ_{1} and ψ_{2} do not extend to characters of Gal(Q̄_{p}/Q_{p}), ρ_{0}|_{D_{p}} must be absolutely irreducible.

We sometimes wish to make one of the following restrictions on the deformations we allow:

(i) (a) Selmer deformations. In this case we assume that ρ_{0} is ordinary, with notion as above, and that the deformation has a representative ρ : Gal(Q_{Σ}/Q) → GL_{2}(A) with the property that (for a suitable choice of basis)

ρ|_{D_{p}} ≈ ( \begin{matrix}χ̃_{1} & ∗ \\ 0 & χ̃_{2} \end{matrix} )

with χ̃_{2} unramified, χ̃ ≡ χ_{2} mod m, and det ρ|_{I_{p}} = εω^{−1}χ_{1}χ_{2} where ε is the cyclotomic character, ε : Gal(Q_{Σ}/Q) → Z_{p}^{∗}, giving the action on all p-power roots of unity, ω is of order prime to p satisfying ω ≡ ε mod p, and χ_{1} and χ_{2} are the characters of (i) viewed as taking values in k^{∗} ↪ A^{∗}.

(i) (b) Ordinary deformations. The same as in (i)(a) but with no condition on the determinant.

(i) (c) Strict deformations. This is a variant on (i) (a) which we only use when ρ_{0}|_{D_{p}} is not semisimple and not flat (i.e. not associated to a finite flat group scheme). We also assume that χ_{1}χ_{2}^{−1} = ω in this case. Then a strict deformation is as in (i)(a) except that we assume in addition that (χ̃_{1}/χ̃_{2})|_{D_{p}} = ε.

(ii) Flat (at p) deformations. We assume that each deformation ρ to GL_{2}(A) has the property that for any quotient A/𝔞 of finite order ρ|_{D_{p}} mod 𝔞 is the Galois representation associated to the Q̄_{p}-points of a finite flat group scheme over Z_{p}.

In each of these four cases, as well as in the unrestricted case (in which we impose no local restriction at p) one can verify that Mazur’s use of Schlessinger’s criteria [Sch] proves the existence of a universal deformation

ρ : Gal(Q_{Σ}/Q) → GL_{2}(R).

In the ordinary and restricted case this was proved by Mazur and in the flat case by Ramakrishna [Ram]. The other cases require minor modifications of Mazur’s argument. We denote the universal ring R_{Σ} in the unrestricted case and R_{Σ}^{se}, R_{Σ}^{ord}, R_{Σ}^{str}, R_{Σ}^{f} in the other four cases. We often omit the Σ 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 O-algebras for O the ring of integers of any local field with residue field k. If we need to record which O we are using we will write R_{Σ,O} etc. It is easy to see that the natural local map of local O-algebras

R_{Σ,O} → R_{Σ} ⊗_{W (k)} 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 :↪ k′ then we have a new deformation problem associated to the representation ρ′_{0} = i ∘ ρ_{0}. There is again a natural map of W (k′)- algebras

R(ρ′_{0}) → R ⊗_{W (k)} W (k′)

which is an isomorphism on Zariski tangent spaces. One can check that this is again an isomorphism by considering the subring R_{1} of R(ρ′_{0}) defined as the subring of all elements whose reduction modulo the maximal ideal lies in k. Since R(ρ′_{0}) is a finite R_{1}-module, R_{1} is also a complete local Noetherian ring with residue field k. The universal representation associated to ρ′_{0} is defined over R_{1} and the universal property of R then defines a map R → R_{1}. So we obtain a section to the map R(ρ′_{0}) → R ⊗_{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 O-algebras tp the restricted cases. In each case we can require A to be an O-algebra and again it is easy to see that

R_{Σ,O}^{·} ≃ R_{Σ}^{·} ⊗_{W (k)} O in each case.

The second generalization concerns primes q ≠ p which are ramified in ρ_{0}. We distinguish three special cases (types (A) and (C) need not be disjoint):

(A) ρ_{0}|_{D_{q}} = (^{χ_{1}} {∗ \atop χ_{2}} ) for a suitable choice of basis, with χ_{1} and χ_{2} unramified, χ_{1}χ_{2}^{−1} = ω and the fixed space of I_{q} of dimension 1,

(B) ρ_{0}|_{I_{q}} = ({χ_{q} \atop 0} {0 \atop 1}), χ_{q} ≠ 1, for a suitable choice of basis,

(C) H^{1}(Q_{q}, W_{λ}) = 0 where W_{λ} is as defined in (1.6).

Then in each case we can define a suitable deformation theory by imposing additional restrictions on those we have already considered, namely:

(A) ρ|_{D_{q}} = (^{ψ_{1}} {∗ \atop ψ_{2}} ) for a suitable choice of basis of A^{2} with ψ_{1} and ψ_{2} unramified and ψ_{1}ψ_{2}^{−1} = ε;

(B) ρ|_{I_{q}} = ({χ_{q} \atop 0} {0 \atop 1}) for a suitable choice of basis (χ_{q} of order prime to p, so the same character as above);

(C) det ρ|_{I_{q}} = det ρ_{0}|_{I_{q}}, i.e., of order prime to p.

Thus if M is a set of primes in Σ distinct from p and each satisfying one of (A), (B) or (C) for ρ_{0}, we will impose the corresponding restriction at each prime in M.

Thus to each set of data D = {·, Σ, O, M} where · is Se, str, ord, flat or unrestricted, we can associate a deformation theory to ρ_{0} provided

(1.3)  ρ_{0} : Gal(Q_{Σ}/Q) → GL_{2}(k)

is itself of type D and O is the ring of integers of a totally ramified extension of W (k); ρ_{0} is ordinary if · is Se or ord, strict if · is strict and flat if · is fl (meaning flat); ρ_{0} is of type M, i.e., of type (A), (B) or (C) at each ramified primes q ≠ p, q ∈ M. We allow different types at different q’s. We will refer to these as the standard deformation theories and write R_{D} for the universal ring associated to D and ρ_{D} for the universal deformation (or even ρ if D is clear from the context).

We note here that if D = (ord, Σ, O, M) and D′ = (Se, Σ, O, M) then there is a simple relation between R_{D} and R_{D′}. Indeed there is a natural map R_{D} → R_{D′} by the universal property of R_{D}, and its kernel is a principal ideal generated by T = ε^{−1}(γ) det ρ_{D}(γ) − 1 where γ ∈ Gal(Q_{Σ}/Q) is any element whose restriction to Gal(Q_{∞}/Q) is a generator (where Q_{∞} is the Z_{p}-extension of Q) and whose restriction to Gal(Q(ζ_{N_{p}})/Q) is trivial for any N prime to p with ζ_{N} ∈ Q_{Σ}, ζ_{N} being a primitive N^{th} root of 1:

(1.4)  R_{D}/T ≃ R′_{D}.

It turns out that under the hypothesis that ρ_{0} is strict, i.e. that ρ_{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 Gal(Q̄_{p}/Q).

Proposition 1.1 (Diamond). Suppose that π : D_{p} → 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 π ≈ ({χ_{1}ε \atop 0} {∗ \atop χ_{2}} ) with χ_{1} and χ_{2} unramified and χ_{1} ≠ χ_{2}. Then the residual representation π̄ is associated to a finite flat group scheme over Z_{p}.

Proof (taken from [Dia, Prop. 6.1]). We may replace π by π ⊗ χ_{2}^{−1} and we let φ = χ_{1}χ_{2}^{−1}. Then π ≅ ({φε \atop 0} {t \atop 1}) determines a cocycle t : D_{p} → M(1) where M is a free A-module of rank one on which D_{p} acts via φ. Let u denote the cohomology class in H^{1}(D_{p}, M(1)) defined by t, and let u_{0} denote its image in H^{1}(D_{p}, M_{0}(1)) where M_{0} = M/𝔪M. Let G = ker φ and let F be the fixed field of G (so F is a finite unramified extension of Q_{p}). Choose n so that p^{n}A = 0. Since H^{2}(G, μ_{p^{r}} → H^{2}(G, μ_{p^{s}}) is injective for r ≤ s, we see that the natural map of A[D_{p}/G]-modules H^{1}(G, μ_{p^{n}} ⊗_{Z_{p}} M) → H^{1}(G, M(1)) is an isomorphism. By Kummer theory, we have H^{1}(G, M(1)) ≅ F^{×}/(F^{×})^{p^{n}} ⊗_{Z_{p}} M as D_{p}-modules. Now consider the commutative diagram

H^{1}(G, M(1))^{D_{p}} ⟶^{∼}((F^{×}/(F^{×})^{p^{n}} ⊗_{Z_{p}} M)^{D_{p}}⟶M^{D_{p}}
↓  ↓  ↓ ,
H^{1}(G, M_{0}(1)) ⟶^{∼} (F^{×}/(F^{×})^{p}) ⊗_{F_{p}} M_{0} ⟶ M_{0}

where the right-hand horizontal maps are induced by v_{p} : F^{×} → Z. If φ ≠ 1, then M^{D_{p}} ⊂ 𝔪M, so that the element res u_{0} of H^{1}(G, M_{0}(1)) is in the image of (O_{F}^{×}/(O_{F}^{×})^{p}) ⊗_{F_{p}} M_{0}. But this means that π̄ is “peu ramifié” in the sense of [Se] and therefore π̄ comes from a finite flat group scheme. (See [E1, (8.20].)

Remark. Diamond also observes that essentially the same proof shows that if π : Gal(Q̄_{q}/Q_{q}) → GL_{2}(A), where A is a complete local Noetherian ring with residue field k, has the form π|_{I_{q}} ≅ ({1 \atop 0} {∗ \atop 1}) with π̄ ramified then π is of type (A).

Globally, Proposition 1.1 says that if ρ_{0} is strict and if D = (Se, Σ, O, M) and D′ = (str, Σ, O, M) then the natural map R_{D} → R_{D′} is an isomorphism.

In each case the tangent space of R_{D} may be computed as in [Ma1]. Let λ be a uniformizer for O and let U_{λ} ≃ k^{2} be the representation space for ρ_{0}. (The motivation for the subscript λ will become apparent later.) Let V_{λ} be the representation space of Gal(Q_{Σ}/Q) on Adρ_{0} = Hom_{k}(U_{λ}, U_{λ}) ≃ M_{2}(k). Then there is an isomorphism of k-vector spaces (cf. the proof of Prop. 1.2 below)

(1.5)  Hom_{k}(m_{D}/(m_{D}^{2}, λ), k) ≃ H_{D}^{1}(Q_{Σ}/Q, V_{λ})

where H_{D}^{1}(Q_{Σ}/Q, V_{λ}) is a subspace of H^{1}(Q_{Σ}/Q, V_{λ}) which we now describe and m_{D} is the maximal ideal of R_{C}alD. It consists of the cohomology classes which satisfy certain local restrictions at p and at the primes in M. We call m_{D}/(m_{D}^{2}, λ) the reduced cotangent space of R_{D}.

We begin with p. First we may write (since p ≠ 2), as k[Gal(Q_{Σ}/Q)]- modules,

(1.6) V_{λ} = W_{λ} ⊕ k, where W_{λ} = {f ∈ Hom_{k}(U_{λ}, U_{λ}) : tracef = 0}
≃ (Sym^{2} ⊗ det^{−1})ρ_{0}

and k is the one-dimensional subspace of scalar multiplications. Then if ρ_{0} is ordinary the action of D_{p} on U_{λ} induces a filtration of U_{λ} and also on W_{λ} and V_{λ}. Suppose we write these 0 ⊂ U_{λ}^{0} ⊂ U_{λ}, 0 ⊂ W_{λ}^{0} ⊂ W_{λ}^{1} ⊂ W_{λ} and 0 ⊂ V_{λ}^{0} ⊂ V_{λ}^{1} ⊂ V_{λ}. Thus U_{λ}^{0} is defined by the requirement that D_{p} act on it via the character χ_{1} (cf. (1.2)) and on U_{λ}/U_{λ}^{0} via χ_{2}. For W_{λ} the filtrations are defined by

\begin{aligned}W_{λ}^{1} & = \{f ∈ W_{λ} : f(U_{λ}^{0}) ⊂ U_{λ}^{0}\}, \\ W_{λ}^{0} & = \{f ∈ W_{λ}^{1} : f = 0 on U_{λ}^{0}\}, \end{aligned}

and the filtrations for V_{λ} are obtained by replacing W by V. We note that these filtrations are often characterized by the action of D_{p}. Thus the action of D_{p} on W_{λ}^{0} is via χ_{1}/χ_{2}; on W_{λ}^{1}/W_{λ}^{0} it is trivial and on Q_{λ}/W_{λ}^{1} it is via χ_{2}/χ_{1}. These determine the filtration if either χ_{1}/χ_{2} is not quadratic or ρ_{0}|_{D_{p}} is not semisimple. We define the k-vector spaces

V_{λ}^{ord} = {f ∈ V_{λ}^{1} : f = 0 in Hom(U_{λ}/U_{λ}^{0}, U_{λ}/U_{λ}^{0})},
H_{Se}^{1}(Q_{p}, V_{λ}) = ker{H^{1}(Q_{p}, V_{λ}) → H^{1}(Q_{p}^{unr}, V_{λ}/W_{λ}^{0})},
H_{ord}^{1}(Q_{p}, V_{λ}) = ker{H^{1}(Q_{p}, V_{λ}) → H^{1}(Q_{p}^{unr}, V_{λ}/V_{λ}^{ord})},
H_{str}^{1}(Q_{p}, V_{λ}) = ker{H^{1}(Q_{p}, V_{λ}) → H^{1}(Q_{p}, W_{λ}/W_{λ}^{0}) ⊕ H^{1}(Q_{p}^{unr}, k)}.

In the Selmer case we make an analogous definition for H_{Se}^{1}(Q_{p}, W_{λ}) by replacing V_{λ} by W_{λ}, 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}(Q_{p}, V_{λ}) → Ext_{k[D_{p}]}^{1}(U_{λ}, U_{λ})

where the extensions are computed in the category of k-vector spaces with local Galois action. Then H_{f}^{1}(Q_{p}, V_{λ}) is defined as the k-subspace of H^{1}(Q_{p}, V_{λ}) which is the inverse image of Ext_{fl}^{1}(G, G), the group of extensions in the category of finite flat commutative group schemes over Z_{p} killed by p, G being the (unique) finite flat group scheme over Z_{p} associated to U_{λ}. 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 ∈ M we have three cases (A), (B), (C). In case (A) there is a filtration by D_{q} entirely analogous to the one for p. We write this 0 ⊂ W_{λ}^{0,q} ⊂ W_{λ}^{1,q} ⊂ W_{λ} and we set

H_{D_{q}}^{1}(Q_{q}, V_{λ}) = \{ \begin{matrix}ker : H^{1}(Q_{q}, V_{λ} \\ → H^{1}(Q_{q}, W_{λ}/W_{λ}^{0,q}) ⊕ H^{1}(Q_{q}^{unr}, k) in case (A) \\ ker : H^{1}(Q_{q}, V_{λ}) \\ → H^{1}(Q_{q}^{unr}, V_{λ}) in case (B) or (C). \end{matrix}

Again we make an analogous definition for H_{D_{q}}^{1}(Q_{q}, W_{λ}) by replacing V_{λ} by W_{λ} and deleting the last term in case (A). We now define the k-vector space H_{D}^{1}(Q_{Σ}/Q, V_{λ}) as

H_{D}^{1}(Q_{Σ}/Q, V_{λ}) = {α ∈ H^{1}(Q_{Σ}/Q, V_{λ}) : α_{q} ∈ H_{D_{q}}^{1}(Q_{q}, V_{λ}) for all q ∈ M,
α_{q} ∈ H_{∗}^{1}(Q_{p}, V_{λ})}

where ∗ is Se, str, ord, fl or unrestricted according to the type of D. A similar definition applies to H_{D}^{1}(Q_{Σ}/Q, W_{λ}) if · is Selmer or strict.

Now and for the rest of the section we are going to assume that ρ_{0} arises from the reduction of the λ-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 Σ, and that there ia a prime λ of O_{f} such that ρ_{0} = ρ_{f,λ} mod λ. Here 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 ⊇ O_{f,λ}/λ 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 ρ_{0} as being of type D then D is defined using O-algebras where O ⊇ O_{f,λ} is an unramified extension whose residue field is k. (Although this condition is unnecessary, it is convenient to use λ as the uniformizer for O.) Finally we assume that ρ_{f,λ} itself is of type D. Again this is a slight abuse of terminology as we are really considering the extension of scalars ρ_{f,λ} ⊗_{O_{f,λ}} O and not ρ_{f,λ} 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 ρ_{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_{D} → O of O-algebras which takes the universal deformation to (the class of) ρ_{f,λ}. Let 𝔭_{D} = ker : R_{D} → O. Let K be the field of fractions of O and let U_{f} = (K/O)^{2} with the Galois action taken from ρ_{f,λ}. Similarly, let V_{f} = Adρ_{f,λ} ⊗_{O} K/O ≃ (K/O)^{4} with the adjoint representation so that

V_{f} ≃ W_{f} ⊕ K/O

where W_{f} has Galois action via Sym^{2}ρ_{f,λ} ⊗ det ρ_{f,λ}^{−1} and the action on the second factor is trivial. Then if ρ_{0} is ordinary the filtration of U_{f} under the Adρ action of D_{p} induces one on W_{f} which we write 0 ⊂ W_{f}^{0} ⊂ W_{f}^{1} ⊂ W_{f}. Often to simplify the notation we will drop the index f from W_{f}^{1}, V_{f} etc. There is also a filtration on W_{λ^{n}} = {ker λ^{n} : W_{f} → W_{f}} given by W_{λ^{n}}^{i} = W^{λ^{n}} ∩ W^{i} (compatible with our previous description for n = 1). Likewise we write V_{λ^{n}} for {ker λ^{n} : V_{f} → V_{f}}.

We now explain how to extend the definition of H_{D}^{1} to give meaning to H_{D}^{1}(Q_{Σ}/Q, V_{λ^{n}}) and H_{D}^{1}(Q_{Σ}/Q, V) and these are O/λ^{n} and O-modules, respectively. In the case where ρ_{0} is ordinary the definitions are the same with V_{λ^{n}} or V replacing V_{λ} and O/λ^{n} or K/O replacing k. One checks easily that as O-modules

(1.7)  H_{D}^{1}(Q_{Σ}/Q, V_{λ^{n}}) ≃ H_{D}^{1}(Q_{Σ}/Q, V)_{λ^{n}},

where as usual the subscript λ^{n} denotes the kernel of multiplication by λ^{n}. This just uses the divisibility of H^{0}(Q_{Σ}/Q, V) and H^{0}(Q_{p}, W/W^{0}) in the strict case. In the Selmer case one checks that for m > n the kernel of

H^{1}(Q_{p}^{unr}, V_{λ^{n}}/W_{λ^{n}}^{0}) → H^{1}(Q_{p}^{unr}, V_{λ^{m}}/W_{λ^{m}}^{0})

has only the zero element fixed under Gal(Q_{p}^{unr}/Q_{p}) and the ord case is similar. Checking conditions at q ∈ M is dome with similar arguments. In the Selmer and strict cases we make analogous definitions with W_{λ^{n}} in place of V_{λ^{n}} and W in place of V and the analogue of (1.7) still holds.

We now consider the case where ρ_{0} is flat (but not ordinary). We claim first that there is a natural map of O-modules

(1.8)  H^{1}(Q_{p}, V_{λ_{n}}) → Ext_{O[D_{p}]}^{1}(U_{λ^{m}}, U_{λ^{n}})

for each m ≥ n where the extensions are of O-modules with local Galois action. To describe this suppose that α ∈ H^{1}(Q_{p}, V_{λ^{n}}). Then we can associate to α a representation ρ_{α} : Gal(Q̄_{p}/Q_{p}) → GL_{2}(O_{n}[ε]) (where O_{n}[ε] = O[ε]/(λ^{n}ε, ε^{2})) which is an O-algebra deformation of ρ_{0} (see the proof of Proposition 1.1 below). Let E = O_{n}[ε]^{2} where the Galois action is via ρ_{α}. Then there is an exact sequence

0 ⟶ εE/λ^{m} ⟶ E/λ^{m} ⟶ (E/ε)/λ^{m} ⟶ 0
|≀  |≀
U_{λ^{n}}  U_{λ^{m}}

and hence an extension class in Ext^{1}(U_{λ^{m}}, U_{λ^{n}}). One checks now that (1.8) is a map of O-modules. We define H_{f}^{1}(Q_{p}, V_{λ^{n}}) to be the inverse image of Ext_{fl}^{1}(U_{λ^{n}}, U_{λ_{n}}) under (1.8), i.e., those extensions which are already extensions in the category of finite flat group schemes Z_{p}. Observe that Ext_{fl}^{1}(U_{λ^{n}}, U_{λ^{n}}) ∩ Ext_{O[D_{p}]}^{1}(U_{λ^{n}}, U_{λ^{n}}) is an O-module, so H_{f}^{1}(Q_{p}, V_{λ^{n}}) is seen to be an O-sub-module of H^{1}(Q_{p}, V_{λ_{n}}). We observe that our definition is equivalent to requiring that the classes in H_{f}^{1}(Q_{p}, V_{λ^{n}}) map under (1.8) to Ext_{fl}^{1}(U_{λ^{m}}, U_{λ^{n}}) for all m ≥ n. For if e_{m} is the extension class in Ext^{1}(U_{λ^{m}}, U_{λ^{n}}) then e_{m} ↪ e_{n} ⊕U_{λ^{m}} as Galois-modules and we can apply results of [Ray1] to see that e_{m} comes from a finite flat group scheme over Z_{p} if e_{n} does.

In the flat (non-ordinary) case ρ_{0}|_{I_{p}} is determined by Raynaud’s results as mentioned at the beginning of the chapter. It follows in particular that, since ρ_{0}|_{D_{p}} is absolutely irreducible, V (Q_{p} = H^{0}(Q_{p}, V) is divisible in this case (in fact V (Q_{p}) ≃ KT/O). This H^{1}(Q_{p}, V_{λ^{n}}) ≃ H^{1}(Q_{p}, V)_{λ^{n}} and hence we can define

H_{f}^{1}(Q_{p}, V) = ⋃_{n=1}^{∞} H_{f}^{1}(Q_{p}, V_{λ^{n}}),

and we claim that H_{f}^{1}(Q_{p}, V)_{λ^{n}} ≃ H_{f}^{1}(Q_{p}, V_{λ^{n}}). To see this we have to compare representations for m ≥ n,

ρ_{n,m} : Gal(Q̄_{p}/Q_{p}) ⟶ GL_{2}(O_{n}[ε]/λ^{m})
∥  ↓φ_{m,n}
ρ_{m,m} : Gal(Q̄_{p}/Q_{p}) ⟶ GL_{2}(O_{m}[ε]/λ^{m})

where ρ_{n,m} and ρ_{m,m} are obtained from α_{n} ∈ H^{1}(Q_{p}, V X_{λ^{n}}) and im(α_{n}) ∈ H^{1}(Q_{p}, V_{λ^{m}}) and φ_{m,n} : a + bε → a + λ^{m−n}bε. By [Ram, Prop 1.1 and Lemma 2.1] if ρ_{n,m} comes from a finite flat group scheme then so does ρ_{m,m}. Conversely φ_{m,n} is injective and so ρ_{n,m} comes from a finite flat group scheme if ρ_{m,m} does; cf. [Ray1]. The definitions of H_{D}^{1}(Q_{Σ}/Q, V_{λ^{n}}) and H_{D}^{1}(Q_{Σ}/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 ρ_{0}|_{I_{p}} to see that H^{1}(Q_{p}, V) is divisible. For it is enough to check that H^{2}(Q_{p}, V_{λ}) = 0 and this follows by duality from the fact that H^{0}(Q_{p}, V_{λ}^{∗}) = 0 where V_{λ}^{∗} = Hom(V_{λ}, μ_{p}) and μ_{p} is the group of p^{th} roots of unity. (Again this follows from the explicit form of ρ_{0}|_{D_{p}}.) Much subtler is the fact that H_{f}^{1}(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

Hom_{O}(𝔭_{R}/𝔭_{R}^{2}, K/O) ≃ H_{f}^{1}(Q_{p}, V)

where R is the universal local flat deformation ring for ρ_{0}|_{D_{p}} and O-algebras. (This exists by Theorem 1.1 of [Ram] because ρ_{0}|_{D_{p}} is absolutely irreducible.) Since R ≃ R^{fl} ⊗_{W (k)} O where R^{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_{f}^{1}(Q_{p}, V) then follows. We refer to [Ram] for more details about R^{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 D = (ord, Σ, W (k), ϕ) with some restriction on χ_{1}, χ_{2} in i(a)) in [MT, Prop 25].

Proposition 1.2. Suppose that ρ_{f,λ} is a deformation of ρ_{0} of type D = (·, Σ, O, M) with O an unramified extension of O_{f,λ}. Then as O-modules

Hom_{O}(𝔭_{D}/𝔭_{D}^{2}, K/O) ≃ H_{D}^{1}(Q_{Σ}/Q, V).

Remark. The isomorphism is functorial in an obvious way if one changes D to a larger D′.

Proof. We will just describe the Selmer case with M = ϕ as the other cases use similar arguments. Suppose that α is a cocycle which represents a cohomology class in H_{Se}^{1}(Q_{Σ}/Q, V_{λ^{n}}). Let O_{n}[ε] denote the ring O[ε]/(λ^{n}ε, ε^{2}). We can associate to α a representation

ρ_{α} : Gal(Q_{Σ}/Q) → GL_{2}(O_{n}[ε])

as follows: set ρ_{α}(g) = α(g)ρ_{f,λ}(g) where ρ_{f,λ}(g), a priori in GL_{2}(O), is viewed in GL_{2}(O_{n}[ε]) via the natural mapping O → O_{n}[ε]. Here a basis for O^{2} is chosen so that the representation ρ_{f,λ} on the decomposition group D_{p} ⊂ Gal(Q_{Σ}/Q) has the upper triangular form of (i)(a), and then α(g) ∈ V_{λ^{n}} is viewed in GL_{2}(O_{n}[ε]) by identifying

V_{λ_{n}} ≃ \{ ( \begin{matrix}1 + yε & xε \\ zε & 1 − tε \end{matrix} ) \} = \{ker : GL_{2}(O_{n}[ε]) → GL_{2}(O)\}.

Then

W_{λ^{n}}^{0} = \{ \binom{1 xε}{1} \},

\begin{aligned}W_{λ^{n}}^{1} & = \{ \binom{1 + yε xε}{1 − yε} \}, \\ W_{λ^{n}} & = \{ ( \begin{matrix}1 + yε & xε \\ zε & 1 − yε \end{matrix} ) \}, \end{aligned}

and

V_{λ^{n}}^{1} = \{ \binom{1 + yε xε}{1 − tε} \}.

One checks readily that ρ_{α} is a continuous homomorphism and that the deformation [ρ_{α}] is unchanged if we add a coboundary to α.

We need to check that [ρ_{α}] is a Selmer deformation. Let H = Gal(Q̄_{p}/Q_{p}^{unr}) and G = Gal(Q_{p}^{unr}/Q_{p}). Consider the exact sequence of O[G]- modules

0 → (V_{λ^{n}}^{1}/W_{λ^{n}}^{0})^{H} → (V_{λ^{n}}/W_{λ^{n}}^{0})^{H} → X → 0

where X is a submodule of (V_{λ^{n}}/V_{λ^{n}}^{1})^{H}. Since the action of _{p} on V_{λ^{n}}/V_{λ^{n}}^{1} is via a character which is nontrivial mod λ (it equals χ_{2}χ_{1}^{−1} mod λ and χ_{1} ≢ χ_{2}), we see that X^{G} = 0 and H^{1}(G, X) = 0. Then we have an exact diagram of O-modules

0

↓

H^{1}(G, (V_{λ^{n}}^{1}/W_{λ^{n}}^{0})^{H}) ≃ H^{1}(G, (V_{λ^{n}}/W_{λ^{n}}^{0})^{H})

↓

H^{1}(Q_{p}, V_{λ^{n}}/W_{λ^{n}}^{0})

↓

H^{1}(Q_{p}^{unr}, V_{λ^{n}}/W_{λ^{n}}^{0})^{G}.

By hypothesis the image of α is zero in H^{1}(Q_{p}^{unr}, V_{λ^{n}}/W_{λ^{n}}^{0})^{G}. Hence it is in the image of H^{1}(G, (V_{λ^{n}}^{1}/W_{λ^{n}}^{0})^{H}). Thus we can assume that it is represented in H^{1}(Q_{p}, V_{λ^{n}}/W_{λ^{n}}^{0}) by a cocycle, which maps G to V_{λ^{n}}^{1}/W_{λ^{n}}^{0}; i.e., f(D_{p}) ⊂ V_{λ^{n}}^{1}/W_{λ^{n}}^{0}, f(I_{p}) = 0. The difference between f and the image of α is a coboundary {σ ↦ σμ̄ − μ̄} for some u ∈ V_{λ^{n}}. By subtracting the coboundary {σ ↦ σu − u} from α globally we get a new α such that α = f as cocycles mapping G to V_{λ^{n}}^{1}/W_{λ^{n}}^{0}. Thus α(D_{p}) ⊂ V_{λ^{n}}^{1}, α(I_{p}) ⊂ W_{λ^{n}}^{0} and it is now easy to check that [ρ_{α}] is a Selmer deformation of ρ_{0}.

Since [ρ_{α}] is a Selmer deformation there is a unique map of local Oalgebras φ_{α} : R_{D} → O_{n}[ε] inducing it. (If M ≠ ϕ we must check the other conditions also.) Since ρ_{α} ≡ ρ_{f,λ} mod ε we see that restricting φ_{α} to 𝔭_{D} gives a homomorphism of O-modules,

φ_{α} : 𝔭_{D} → ε.O/λ^{n}

such that φ_{α}(𝔭_{D}^{2}) = 0. Thus we have defined a map φ : α → φ_{α},

φ : H_{Se}^{1}(Q_{Σ}/Q, V_{λ^{n}}) → Hom_{O}(𝔭_{D}/𝔭_{D}^{2}, O/λ^{n}).

It is straightforward to check that this is a map of O-modules. To check the injectivity of φ suppose that φ_{α}(𝔭_{D}) = 0. Then φ_{α} factors through R_{D}/𝔭_{D} ≃ O and being an O-algebra homomorphism this determines φ_{α}. Thus [ρ_{f,λ}] = [ρ_{α}]. If A^{−1}ρ_{α}A = ρ_{f,λ} then A mod ε is seen to be central by Schur’s lemma and so may be taken to be I. A simple calculation now shows that α is a coboundary.

To see that φ is surjective choose

Ψ ∈ Hom_{O}(𝔭_{D}/𝔭_{D}^{2}, O/λ^{n}).

Then ρ_{Ψ} : Gal(Q_{Σ}/Q) → GL_{2}(R_{D}/(𝔭_{D}^{2}, ker Ψ)) is induced by a representative of the universal deformation (chosen to equal ρ_{f,λ} when reduced mod 𝔭_{D}) and we define a map α_{Ψ} : Gal(Q_{Σ}/Q) → V_{λ^{n}} by

α_{Ψ}(g) = ρ_{Ψ}(g)ρ_{f,λ}(g)^{−1} ∈ \{ \begin{matrix}1 + 𝔭_{D}/(𝔭_{D}^{2}, ker Ψ) & 𝔭_{D}/(𝔭_{D}^{2}, ker Ψ) \\ 𝔭_{D}/(𝔭_{D}^{2}, ker Ψ) & 1 + 𝔭_{D}/(𝔭_{D}^{2}, ker Ψ) \end{matrix} \} ⊆ V_{λ^{n}}

where ρ_{f,λ}(g) is viewed in GL_{2}(R_{D}/(𝔭_{D}^{2}, ker Ψ)) via the structural map O → R_{D} (R_{D} being an O-algebra and the structural map being local because of the existence of a section). The right-hand inclusion comes from

\begin{aligned}𝔭_{D}/(𝔭_{D}^{2}, ker Ψ) & ↪^{Ψ} & O/λ^{n} & →^{∼} & (O/λ^{n}) · ε \\ & & 1 & ↦ & ε. \end{aligned}

Then α_{Ψ} is really seen to be a continuous cocycle whose cohomology class lies in H_{Se}^{1}(Q_{Σ}/Q, V_{λ^{n}}). Finally φ(α_{Ψ}) = Ψ. Moreover, the constructions are compatible with change of n, i.e., for V_{λ^{n}} ↪V_{λ^{n+1}} and λ:O/λ^{n} ↪ O/λ^{n+1}. □

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_{F}^{1} 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 ρ_{f,λ} we obtain also a lattice T ≃ O^{4} with Galois action via Ad ρ_{f,λ}. Let V = T ⊗_{Z_{p}} Q_{p} be associated vector space and identify V with V/T. Let pr : V → V be the natural projection and define cohomology modules by

H_{F}^{1} (Q_{p}, V) = ker : H^{1}(Q_{p}, V) → H^{1}(Q_{p}, V ⊗_{Q_{p}} B_{crys}),
H_{F}^{1} (Q_{p}, V) = pr(H_{F}^{1} (Q_{p}, V)) ⊂ H^{1}(Q_{p}, V),
H_{F}^{1} (Q_{p}, V_{λ^{n}}) = (j_{n})^{−1}(H_{F}^{1} (Q_{p}, V)) ⊂ H^{1}(Q_{p}, V_{λ^{n}}),

where j_{n} : V_{λ^{n}} → V is the natural map and the two groups in the definition of H_{F}^{1} (Q_{p}, V) are defined using continuous cochains. Similar definitions apply to V^{∗} = Hom_{Q_{p}}(V, Q_{p}(1)) and indeed to any finite-dimensional continuous p-adic representation space. The reader is cautioned that the definition of H_{F}^{1} (Q_{p}, V_{λ^{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 ρ = ρ_{f,λ}, Ad ρ, etc. Henceforth we will only use the notation H_{F}^{1} (Q_{p}, −) when the underlying vector space is crystalline.

Proposition 1.3. (i) If ρ_{0} is flat but ordinary and ρ_{f,λ} is associated to a p-divisible group then for all n

H_{f}^{1}(Q_{p}, V_{λ^{n}}) = H_{F}^{1} (Q_{p}, V_{λ^{n}}).

(ii) If ρ_{f,λ} is ordinary, det ρ_{f,λ}|_{I_{p}} = ε and ρ_{f,λ} is associated to a p-divisible

group, then for all n,

H_{F}^{1} (Q_{p}, V_{λ^{n}}) ⊆ H_{Se}^{1}(Q_{p}, V_{λ^{n}}.

Proof. Beginning with (i), we define H_{f}^{1}(Q_{p}, V) = {α ∈ H^{1}(Q_{p}, V) : κ(α/λ^{n}) ∈ H_{f}^{1}(Q_{p}, V) for all n} where κ : H^{1}(Q_{p}, V) → H^{1}(Q_{p}, V). Then we see that in case (i), H_{f}^{1}(Q_{p}, V) is divisible. So it is enough to how that

H_{F}^{1} (Q_{p}, V) = H_{f}^{1}(Q_{p}, V).

We have to compare two constructions associated to a nonzero element α of H^{1}(Q_{p}, V). The first is to associate an extension

(1.9)  0 → V → E →^{δ} K → 0

of K-vector spaces with commuting continuous Galois action. If we fix an e with δ(e) = 1 the action on e is defined by σe = e + α̂(σ) with α̂ a cocycle representing α. The second construction begins with the image of the subspace ⟨α⟩ in H^{1}(Q_{p}, V). By the analogue of Proposition 1.2 in the local case, there is an O-module isomorphism

H^{1}(Q_{p}, V) ≃ Hom_{O}(𝔭_{R}/𝔭_{R}^{2}, K/O)

where R is the universal deformation ring of ρ_{0} viewed as a representation of Gal(Q̄_{p}/Q) on O-algebras and 𝔭_{R} is the ideal of R corresponding to 𝔭_{D} (i.e., its inverse image in R). Since α ≠ 0, associated to ⟨α⟩ is a quotient 𝔭_{R}/(𝔭_{R}^{2}, 𝔞) of 𝔭_{R}/𝔭_{R}^{2} which is a free O-module of rank one. We then obtain a homomorphism

ρ_{α} : Gal(Q̄_{p}/Q_{p}) → GL_{2}(R/(𝔭_{R}^{2}, 𝔞))

induced from the universal deformation (we pick a representation in the universal class). This is associated to an O-module of rank 4 which tensored with K gives a K-vector space E′ ≃ (K)^{4} which is an extension

(1.10)  0 → U → E′ → U → 0

where U ≃ K^{2} has the Galis representation ρ_{f,λ} (viewed locally).

In the first construction α ∈ H_{F}^{1} (Q_{p}, 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 Q_{p} replaces K in (1.9). On the other hand ⟨α⟩ ⊆ H_{f}^{1}(Q_{p}, V) if and only if the second construction can be made through R^{fl}, or equivalently if and only if E′ is the representation associated to a p-divisible group. A priori, the representation associated to ρ_{α} 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 ρ_{α} comes from a p-divisible group. For more details on R^{fl}, the universal flat deformation ring of the local representation ρ_{0}, see [Ram].) Now the extension E′ comes from a p-divisible group if and only if it is crystalline; cf. [Fo, §6]. 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 V as Hom_{K}(U, U) and let

X = ker : {Hom_{K}(U, U) ⊗ U → U}

where the map is the natural one f ⊗ w ↦ f(w). (All tensor products in this proof will be as K-vector spaces.) Then as K[D_{p}]-modules

E′ ≃ (E ⊗ 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 from standard properties of crystalline representations that if E is crystalline, so is E ⊗ U and also E′. Conversely, we can recover E from E′ as follows. Consider E′ ⊗ U ≃ (E ⊗ U ⊗ U)/(X ⊗ U). Then there is a natural map φ : E ⊗ (det) → E′ ⊗ U induced by the direct sum decomposition U ⊗ U ≃ (det) ⊕ Sym^{2}U. Here det denotes a 1-dimensional vector space over K with Galois action via det ρ_{f,λ}. Now we claim that φ is injective on V ⊗ (det). For if f ∈ V then φ(f) = f ⊗ (w_{1} ⊗ w_{2} − w_{2} ⊗ w_{1}) where w_{1}, w_{2} are a basis for U for which w_{1} ∧ w_{2} = 1 in det ≃ K. So if φ(f) ∈ X ⊗ U then

f(w_{1}) ⊗ w_{2} − f(w_{2}) ⊗ w_{1} = 0 in U ⊗ U.

But this is false unless f(w_{1}) = f(w_{2}) = 0 whence f = 0. So φ is injective on V ⊗ det and if φ itself were not injective then E would split contradicting α ≠ 0. So φ is injective and we have exhibited E ⊗(det) as a subrepresentation of E′ ⊗ 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_{Se}^{1}(Q_{p}, V_{λ^{n}}) = j_{n}^{−1}(H_{Se}^{1}(Q_{p}, V)) (this

was already used in (1.7)). We next have to show that H_{F}^{1} (Q_{p},V) ⊆ H_{Se}^{1}(Q_{p},V) where the latter is defined by

H_{Se}^{1}(Q_{p}, V) = ker : H^{1}(Q_{p}, V) → H^{1}(Q_{p}^{unr}, V/V^{0})

with V^{0} the subspace of V on which I_{p} acts via ε. But this follows from the computations in Corollary 3.8.4 of [BK]. Finally we observe that

pr(H_{Se}^{1}(Q_{p}, V)) ⊆ H_{Se}^{1}(Q_{p}, V)

although the inclusion may be strict, and

pr(H_{F}^{1} (Q_{p}, V)) = H_{F}^{1} (Q_{p}, V)

by definition. This completes the proof. □

These groups have the property that for s ≥ r,

(1.11)  H^{1}(Q_{p}, V_{λ}^{r}) ∩ j_{r,s}^{−1}(H_{F}^{1} (Q_{p}, V_{λ^{s}})) = H_{F}^{1} (Q_{p}, V_{λ^{r}})

where j_{r,s} : V_{λ^{r}} → V_{λ^{s}} is the natural injection. The same holds for V_{λ^{r}}^{∗} and V_{λ^{s}}^{∗} in place of V_{λ^{r}} and V_{λ^{s}} where V_{λ^{r}}^{∗} is defined by

V_{λ^{r}}^{∗} = Hom(V_{λ^{r}}, μ_{p^{r}})

and similarly for V_{λ^{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 ⊂ V be a Galois stable lattice so that T ≃ O^{4}. Define

H_{F}^{1} (Q_{p}, T) = i^{−1}(H_{F}^{1} (Q_{p}, V))

under the natural inclusion i : T ↪ V, and likewise for the dual lattice T^{∗} = Hom_{Z_{p}}(V, (Q_{p}/Z_{p})(1)) in V^{∗}. (Here V^{∗} = Hom(V, Q_{p}(1)); throughout this paper we use M^{∗} to denote a dual of M with a Cartier twist.) Also write pr_{n} : T → T/λ^{n} for the natural projection map, and for the mapping it induces on cohomology.

Proposition 1.4. If ρ_{f,λ} is associated to a p-divisible group (the ordinary case is allowed) then

(i) pr_{n}(H_{F}^{1} (Q_{p}, T)) = H_{F}^{1} (Q_{p}, T/λ^{n}) and similarly for T^{∗}, T^{∗}/λ^{n}.

(ii) H_{F}^{1} (Q_{p}, V_{λ^{n}}) is the orthogonal complement of H_{F}^{1} (Q_{p}, V_{λ^{n}}^{∗}) under Tate local duality between H^{1}(Q_{p}, V_{λ^{n}}) and H^{1}(Q_{p}, V_{λ^{n}}^{∗}) and similarly for W_{λ^{n}} and W_{λ^{n}}^{∗} replacing V_{λ^{n}} and V_{λ^{n}}^{∗}.

More generally these results hold for any crystalline representation V′ in place of V and λ′ a uniformizer in K′ where K′ is any finite extension of Q_{p} with K′ ⊂ End_{Gal(\overline{Q}_{p}/Q_{p})}V′.

Proof. We first observe that pr_{n}(H_{F}^{1} (Q_{p}, T)) ⊂ H_{F}^{1} (Q_{p}, T/λ^{n}). Now from the construction we may identify T/λ^{n} with V_{λ^{n}}. A result of Bloch-Kato ([BK, Prop. 3.8]) says that H_{F}^{1} (Q_{p}, V) and H_{F}^{1} (Q_{p}, V^{∗}) are orthogonal complements under Tate local duality. It follows formally that H_{F}^{1} (Q_{p}, V_{λ^{n}}^{∗}) and pr_{n}(H_{F}^{1} (Q_{p}, T)) are orthogonal complements, so to prove the proposition it is enough to show that

(1.12)  #H_{F}^{1} (Q_{p}, V_{λ^{n}}^{∗})#H_{F}^{1} (Q_{p}, V_{λ^{n}}) = #H^{1}(Q_{p}, V_{λ^{n}}).

Now if r = dim_{K} H_{F}^{1} (Q_{p}, V) and s = dim_{K} H_{F}^{1} (Q_{p}, V^{∗}) then

(1.13)  r + s = dim_{K} H^{0}(Q_{p}, V) + dim_{K} H^{0}(Q_{p}, V^{∗}) + dim_{K} V.

¿From the definition,

(1.14) #H_{F}^{1} (Q_{p}, V_{λ^{n}}) = #(O/λ^{n})^{r} · # ker{H^{1}(Q_{p}, V_{λ^{n}}) → H^{1}(Q_{p}, V)}.

The second factor is equal to #{V (Q_{p})/λ^{n}V (Q_{p})}. When we write V (Q_{p})^{div} for the maximal divisible subgroup of V (Q_{p}) this is the same as

#(V (Q_{p})/V (Q_{p})^{div})/λ^{n} = #(V (Q_{p})/V (Q_{p})^{div})_{λ^{n}}
= #V (Q_{p})_{λ^{n}}/#(V (Q_{p})^{div})_{λ^{n}}.

Combining this with (1.14) gives

(1.15) #H_{F}^{1} (Q_{p}, V_{λ^{n}}) = #(O/λ^{n})^{r}
· #H^{0}(Q_{p}, V_{λ^{n}})/#(O/λ^{n})^{dim_{K}H^{0}(Q_{p},V)}.

This, together with an analogous formula for #H_{F}^{1} (Q_{p}, V_{λ^{n}}^{∗}) and (1.13), gives

#H_{F}^{1} (Q_{p}, V^{λ^{n}})#H_{F}^{1} (Q_{p}, V_{λ^{n}}^{∗}) = #(O/λ^{n})^{4} · #H^{0}(Q_{p}, V_{λ^{n}})#H^{0}(Q_{p}, V_{λ^{n}}^{∗}).

As #H^{0}(Q_{p}, V^{∗}λ^{n}) = #H^{2}(Q_{p}, V_{λ^{n}}) the assertion of (1.12) now follows from the formula for the Euler characteristic of V_{λ^{n}}.

The proof for W_{λ^{n}}, or indeed more generally for any crystalline representation, is the same. □

We also give a characterization of the orthogonal complements of H_{Se}^{1}(Q_{p}, W_{λ^{n}}) and H_{Se}^{1}(Q_{p}, V_{λ^{n}}), under Tate’s local duality. We write these duals as H_{Se^{∗}}^{1}(Q_{p}, W_{λ^{n}}^{∗}) and H_{Se^{∗}}^{1}(Q_{p}, V_{λ^{n}}^{∗}) respectively. Let

φ_{w} : H^{1}(Q_{p}, W_{λ^{n}}^{∗}) → (Q_{p}, W_{λ^{n}}^{∗}/(W_{λ^{n}}^{∗})^{0})

be the natural map where (W_{λ^{n}}^{∗})^{i} is the orthogonal complement of W_{λ^{n}}^{1−i} in W_{λ^{n}}^{∗}, and let X_{n,i} be defined as the image under the composite map

\begin{aligned}X_{n,i} = im : Z_{p}^{×}/(Z_{p}^{×})^{p^{n}} ⊗ O/λ^{n} & → H^{1}(Q_{p}, μ_{p^{n}} ⊗ O/λ^{n}) \\ & → H^{1}(Q_{p}, W_{λ^{n}}^{∗}/(W_{λ^{n}}^{∗})^{0}) \end{aligned}

where in the middle term μ_{p^{n}} ⊗ O/λ^{n} is to be identified with (W_{λ^{n}}^{∗})^{1}/(W_{λ^{n}}^{∗})^{0}. Similarly if we replace W_{λ^{n}}^{∗} by V_{λ^{n}}^{∗} we let Y_{n,i} be the image of Z_{p}^{×}/(Z_{p}^{×})^{p^{n}} ⊗ (O/λ^{n})^{2} in H^{1}(Q_{p}, V_{λ^{n}}^{∗}/(W_{λ^{n}}^{∗})^{0}), and we replace φ_{w} by the analogous map φ_{v}.

Proposition 1.5.

\begin{aligned}H_{Se^{∗}}^{1}(Q_{p}, W_{λ^{n}}^{∗}) & = φ_{w}^{−1}(X_{n,i}), \\ H_{Se^{∗}}^{1}(Q_{p}, V_{λ^{n}}^{∗}) & = φ_{v}^{−1}(Y_{n,i}). \end{aligned}

Proof. This can be checked by dualizing the sequence

0 → H_{Str}^{1}(Q_{p}, W_{λ^{n}}) → H_{Se}^{1}(Q_{p}, W_{λ^{n}})
→ ker : {H^{1}(Q_{p}, W_{λ^{n}}/(W_{λ^{n}})^{0}) → H^{1}(Q_{p}^{unr}, W_{λ^{n}}/(W_{λ^{n}})^{0}},

where H_{str}^{1}(Q_{p}, W_{λ^{n}}) = ker : H^{1}(Q_{p}, W_{λ^{n}}) → H^{1}(Q_{p}, W_{λ^{n}}/(W_{λ^{n}})^{0}). The first term is orthogonal to ker : H^{1}(Q_{p}, W_{λ^{n}}^{∗}) → H^{1}(Q_{p}, W_{λ^{n}}^{∗}/(W_{λ^{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}(Q_{p}, μ_{p^{n}}) × H^{1}(Q_{p}, Z/p^{n}) → Z/p^{n}

the orthogonal complement of the unramified homomorphisms is the image of the units Z_{p}^{×}/(Z_{p}^{×})^{p^{n}} → H^{1}(Q_{p}, μ_{p^{n}}). The proof for V_{λ^{n}} is essentially the same. □

2. Some computations of cohomology groups

We now make some comparisons of orders of cohomology groups using 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 = ∏ L_{q} ⊆ ∏_{p∈Σ} H^{1}(Q_{q}, X)

is a subgroup, where X is a finite module for Gal(Q_{Σ}/Q) of p-power order. We define L^{∗} to be the orthogonal complement of L under the perfect pairing (local Tate duality)

∏_{q∈Σ} H^{1}(Q_{q}, X) × ∏_{q∈Σ} H^{1}(Q_{q}, X^{∗}) → Q_{p}/Z_{p}

where X^{∗} = Hom(X, μ_{p^{∞}}). Let

λ_{X} : H^{1}(Q_{Σ}/Q, X) → ∏_{q∈Σ} H^{1}(Q_{q}, X)

be the localization map and similarly λ_{X^{∗}} for X^{∗}. Then we set

H_{L}^{1}(Q_{Σ}/Q, X) = λ_{X}^{−1}(L), H_{L^{∗}}^{1}(Q_{Σ}/Q, X^{∗}) = λ_{X^{∗}}^{−1}(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 ∈ Σ.

Proposition 1.6.

#H_{L}^{1}(Q_{Σ}/Q, X)/#H_{L^{∗}}^{1}(Q_{Σ}/Q, X^{∗}) = h_{∞} ∏_{q∈Σ} h_{q}

where

\{ \begin{matrix}h_{q} & = #H^{0}(Q_{q}, X^{∗})/[H^{1}(Q_{q}, X) : L_{q}] \\ h_{∞} & = #H^{0}(R, X^{∗})#H^{0}(Q, X)/#H^{0}(Q, X^{∗}). \end{matrix}

Proof.AdaptingtheexactsequenceproofofPoitouandTate(cf.[Mi2,Th.4.20]) we get a seven term exact sequence

0 ⟶ H_{L}^{1}(Q_{Σ}/Q, X) ⟶ H^{1}(Q_{Σ}/Q, X) ⟶ ∏_{q∈Σ} H^{1}(Q_{q}, X)/L_{q}
↓
∏_{q∈Σ} H^{2}(Q_{q}, X) ⟵ H^{2}(Q_{Σ}/Q, X) ⟵ H_{L^{∗}}^{1}(Q_{Σ}/Q, X^{∗})^{∧}

| → H^{0}(Q_{Σ}/Q, X^{∗})^{∧} ⟶ 0,

where M^{∧} = Hom(M, Q_{p}/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 proposition. We repeat that in the above proposition X can be arbitrary of p-power order. □

We wish to apply the proposition to investigate H_{D}^{1}. Let D = (·, Σ, O, M) be a standard deformation theory as in Section 1 and define a corresponding group L_{n} = L_{D,n} by setting

L_{n,q} = \{ \begin{matrix}H^{1}(Q_{q}, V_{λ^{n}}) & for q ≠ p and q ∉ M \\ H_{D_{q}}^{1}(Q_{q}, V_{λ^{n}}) & for q ≠ p and q ∈ M \\ H.^{1}(Q_{p}, V_{λ^{n}}) & for q = p. \end{matrix}

Then H_{D}^{1}(Q_{Σ}/Q, V_{λ^{n}}) = H_{L^{n}}^{1}(Q_{Σ}/Q, V_{λ^{n}}) and we also define

H_{D^{∗}}^{1}(Q_{Σ}/Q, V_{λ^{n}}^{∗}) = H_{L_{n}^{∗}}^{1}(Q_{Σ}/Q, V_{λ^{n}}^{∗}).

We will adopt the convention implicit in the above that if we consider Σ′ ⊃ Σ then H_{D}^{1}(Q_{Σ′}/Q, V_{λ^{n}}) places no local restriction on the cohomology classes at primes q ∈ Σ′ − Σ. Thus in H_{D^{∗}}^{1}(Q_{Σ′}/Q, V_{λ^{n}}^{∗}) we will require (by duality) that the cohomology class be locally trivial at q ∈ Σ′ − Σ.

We need now some estimates for the local cohomology groups. First we consider an arbitrary finite Gal(Q_{Σ}/Q)-module X:

Proposition 1.7. If q ∉ Σ, and X is an arbitrary finite Gal(Q_{Σ}/Q)- module of p-power order,

#H_{L′}^{1}(Q_{Σ∪q}/Q, X)/#H_{L}^{1}(Q_{Σ}/Q, X) ≤ #H^{0}(Q_{q}, X^{∗})

where L′_{ℓ} = L_{ℓ} for ℓ ∈ Σ and L′_{q} = H′(Q_{q}, X).

Proof. Consider the short exact sequence of inflation-restriction:

0→H_{L}^{1}(Q_{Σ}/Q, X)→H_{L′}^{1}(Q_{Σ∪q}/Q, X)→Hom(Gal(Q_{Σ∪q}/Q_{Σ}), X)^{Gal(Q_{Σ}/Q)}

↓  ↓

∩

H^{1}(Q_{q}^{unr}, X)^{Gal(Q_{q}^{unr}/Q_{q})} →^{∼}H^{1}(Q_{q}^{unr}, X)^{Gal(Q_{q}^{unr}/Q_{q})}

The proposition follows when we note that

#H^{0}(Q_{q}, X^{∗}) = #H^{1}(Q_{q}^{unr}, X)^{Gal(Q_{q}^{unr}/Q_{q})}. □

Now we return to the study of V_{λ^{n}} and W_{λ^{n}}.
Proposition 1.8. If q ∈ M (q ≠ p) and X = V_{λ^{n}} then h_{q} = 1.

Proof. This is a straightforward calculation. For example if q is of type (A) then we have

L_{n,q} = ker{H^{1}(Q_{q}, V_{λ^{n}}) → H^{1}(Q_{q}, W_{λ^{n}}/W_{λ^{n}}^{0}) ⊕ H^{1}(Q_{q}^{unr}, O/λ^{n})}.

Using the long exact sequence of cohomology associated to

0 → W_{λ^{n}}^{0} → W_{λ^{n}} → W_{λ^{n}}/W_{λ^{n}}^{0} → 0

one obtains a formula for the order of L_{n,q} in terms of #H^{1}(Q_{q}, W_{λ^{n}}), #H^{i}(Q_{q}, W_{λ^{n}}/W_{λ^{n}}^{0}) etc. Using local Euler characteristics these are easily reduced to ones involving H^{0}(Q_{q}, W_{λ^{n}}^{∗}) etc. and the result follows easily. □

The calculation of h_{p} is more delicate. We content ourselves with an inequality in some cases.

Proposition 1.9. (i) If X = V_{λ^{n}} then
h_{p}h_{∞} = #(O/λ)^{3n}#H^{0}(Q_{p}, V_{λ^{n}}^{∗})/#H^{0}(Q, V_{λ^{n}}^{∗})

in the unrestricted case.

(ii) If X = V_{λ^{n}} then

h_{p}h_{∞} ≤ #(O/λ)^{n}#H^{0}(Q_{p}, (V_{λ^{n}}^{ord})^{∗})/#H^{0}(Q, W_{λ^{n}}^{∗})

in the ordinary case.

(iii) If X = V_{λ^{n}} or W_{λ^{n}} then h_{p}h_{∞} ≤ #H^{0}(Q_{p}, (W_{λ^{n}}^{0})^{∗})/#H^{0}(Q, W_{λ^{n}}^{∗}) in the Selmer case.

(iv) If X = V_{λ^{n}} or W_{λ^{n}} then h_{p}h_{∞} = 1 in the strict case.

(v) If X = V_{λ^{n}} then h_{p}h_{∞} = 1 in the flat case.

(vi) If X = V_{λ^{n}} or W_{λ^{n}} then h_{p}h_{∞} = 1/#H^{0}(Q, V_{λ^{n}}^{∗}) if L_{n,p} = H_{F}^{1} (Q_{p}, X) and ρ_{f,λ} arises from an ordinary p-divisible group.

Proof. Case (i) is trivial. Consider then case (ii) with X = V_{λ^{n}.} We have a long exact sequence of cohomology associated to the exact sequence:

(1.16)  0 → W_{λ^{n}}^{0} → V_{λ^{n}} → V_{λ^{n}}/W_{λ^{n}}^{0} → 0.

In particular this gives the map u in the diagram

H^{1}(Q_{p}, V_{λ^{n}})

u

| ↓ ⧹ ⧹ δ

⧹

⧹ ↘

1 → Z = H^{1}(Q_{p}^{unr}/Q_{p},(V_{λ^{n}}/W_{λ^{n}}^{0})^{H}) → H^{1}(Q_{p},V_{λ^{n}}/W_{λ^{n}}^{0}) → H^{1}(Q_{p}^{unr},V_{λ^{n}}/W_{λ^{n}}^{0})^{G} → 1

where G = Gal(Q_{p}^{unr}/Q_{p}), H = Gal(Q̄_{p}/Q_{p}^{unr}) and δ is defined to make the triangle commute. Then writing h_{i}(M) for #H^{1}(Q_{p}, M) we have that #Z = h_{0}(V_{λ^{n}}/W_{λ^{n}}^{0}) and #im δ ≥ (#im u)/(#Z). A simple calculation using the long exact sequence associated to (1.16) gives

(1.17)  #im u = \frac{h_{1}(V_{λ^{n}}/W_{λ^{n}}^{0})h_{2}(V_{λ^{n}})}{h_{2}(W_{λ^{n}}^{0})h_{2}(V_{λ^{n}}/W_{λ^{n}}^{0})}.

Hence

[H^{1}(Q_{p}, V_{λ^{n}}) : L_{n,p}] = #imδ ≥ #(O/λ)^{3n}h_{0}(V_{λ^{n}}^{∗})/h_{0}(W_{λ^{n}}^{0})^{∗}.

The inequality in (iii) follows for X = V_{λ^{n}} and the case X = W_{λ^{n}} is similar. Case (ii) is similar. In case (iv) we just need #im u which is given by (1.17) with W_{λ^{n}} replacing V_{λ^{n}}. In case (v) we have already observed in Section 1 that Raynaud’s results imply that #H^{0}(Q_{p}, V_{λ^{n}}^{∗}) = 1 in the flat case. Moreover #H_{f}^{1}(Q_{p}, V_{λ^{n}}) can be computed to be #(O/λ)^{2n} from

H_{f}^{1}(Q_{p}, V_{λ^{n}}) ≃ H_{f}^{1}(Q_{p}, V)_{λ^{n}} ≃ Hom_{O}(𝔭_{R}/𝔭_{R}^{2}, K/O)_{λ^{n}}

where R is the universal local flat deformation ring of ρ_{0} for O-algebras. Using the relation R ≃ R^{fl} ⊗_{W (k)} O where R^{fl} is the corresponding ring for W (k)-

algebras, and the main theorem of [Ram] (Theorem 4.2) which computes R^{fl}, we can deduce the result.

We now prove (vi). From the definitions

#H_{F}^{1} (Q_{p}, V_{λ^{n}}) = \{ \begin{matrix}(#O/λ^{n})^{r}#H^{0}(Q_{p}, W_{λ^{n}}) & if ρ_{f,λ}|_{D_{p}} does not split \\ (#O/λ^{n})^{r} & if ρ_{f,λ}|_{D_{p}} splits \end{matrix}

where r = dim_{K} H_{F}^{1} (Q_{p}, 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. □

3. Some results on subgroups of GL_{2}(k)

We now give two group-theoretic results which will not be used until 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 ρ_{0} as in (1.1) so that im ρ_{0} is a subgroup of GL_{2}(k) and det ρ_{0} is assumed odd.

Lemma 1.10. If im ρ_{0} has order divisible by p then:

(i) It contains an element γ_{0} of order m ≥ 3 with (m, p) = 1 and γ_{0} trivial on any abelian quotient of im ρ_{0}.

(ii) It contains an element ρ_{0}(σ) with any prescribed image in the Sylow 2-subgroup of (im ρ_{0})/(im ρ_{0})′ and with the ratio of the eigenvalues not equal to ω(σ). (Here (im ρ_{0})′ denotes the derived subgroup of (im ρ_{0}).)

The same results hold if the image of the projective representation ρ̃_{0} associated to ρ_{0} is isomorphic to A_{4}, S_{4} or A_{5}.

Proof. (i) Let G = im ρ_{0} and let Z denote the center of G. Then we have a surjection G′ → (G/Z)′ where the ′ denotes the derived group. By Dickson’s classification of the subgroups of GL_{2}(k) containing an element of order p, (G/Z) is isomorphic to PGL_{2}(k′) or PSL_{2}(k′) for some finite field k′ of characteristic p or possibly to A_{5} when p = 3, cf. [Di, §260]. In each case we can find, and then lift to G′, an element of order m with (m, p) = 1 and m ≥ 3, except possibly in the case p = 3 and PSL_{2}(F_{3}) ≃ A_{4} or PGL_{2}(F_{3}) ≃ 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 ρ_{0} is assumed absolutely irreducible, G = im ρ_{0} has no fixed line. We claim that the same then holds for the derived group G′ For otherwise since G′ ◁ G we could obtain a second fixed line by taking ⟨gv⟩ where ⟨v⟩ is the original fixed line and g is a suitable element of G. Thus G′ would be contained in the group of diagonal matrices for a suitable basis and it would be central in which case G would be abelian or its normalizer in 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 GL_{2}(k) containing an element of order p the image of im ρ_{0} in PGL_{2}(k) is isomorphic to PGL_{2}(k′) or 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 PSL_{2}(F_{3}) when p = 3. It follows that p ∤ [G : G′] except in this one case which we treat separately. So assuming now that p ∤ [G : G′] we see that G′ contains a nontrivial unipotent element u. Since G′ has no fixed line there must be another noncommuting unipotent element v in G′. Pick a basis for ρ_{0}|_{G′} consisting of their fixed vectors. Then let τ be an element of Gal(Q_{Σ}/Q) for which the image of ρ_{0}(τ) in G/G′ is prescribed and let ρ_{0}(τ) = ({a \atop c} {b \atop d}). Then

δ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) \binom{1 sα}{1} \binom{1}{rβ 1}

has det (δ) = det ρ_{0}(τ) and trace δ = sα(raβ + c) + brβ + a + d. Since p ≥ 3 we can choose this trace to avoid any two given values (by varying s) unless raβ + c = 0 for all r. But raβ + c cannot be zero for all r as otherwise a = c = 0. So we can find a δ for which the ratio of the eigenvalues is not ω(τ), det(δ) being, of course, fixed.

Now suppose that im ρ_{0} does not have order divisible by p but that the associated projective representation ρ̃_{0} has image isomorphic to S_{4} or A_{5}, so necessarily p ≠ 3. Pick an element τ such that the image of ρ_{0}(τ) in G/G′ is any prescribed class. Since this fixes both det ρ_{0}(τ) and ω(τ) we have to show that we can avoid at most two particular values of the trace for τ. To achieve this we can adapt our first choice of τ by multiplying by any element og G′. So pick σ ∈ G′ as in (i) which we can assume in these two cases has order 3. Pick a basis for ρ_{0}, by expending scalars if necessary, so that σ ↦ (^{α} _{α^{−1}} ). Then one checks easily that if ρ_{0}(τ) = ({a \atop c} {b \atop d}) we cannot have the traces of all of τ, στ and σ^{2}τ lying in a set of the form {∓t} unless a = d = 0. However we can ensure that ρ_{0}(τ) does not satisfy this by first multiplying τ 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 PSL_{2}(F_{3}) ≃ 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 ρ_{0}(σ) 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 {∓I} (such elements necessary have determinant 1 and order prime to p) and we know that #G′ > 2 as was noted in the proof of part (i). □

Remark. By a well-known result on the finite subgroups of PGL_{2}(\overline{F}_{p}) this lemma covers all ρ_{0} whose images are absolutely irreducible and for which ρ̃_{0} is not dihedral.

Let K_{1} be the splitting field of ρ_{0}. Then we can view W_{λ} and W_{λ}^{∗} as Gal(K_{1}(ζ_{p})/Q)-modules. We need to analyze their cohomology. Recall that we are assuming that ρ_{0} is absolutely irreducible. Let ρ̃_{0} be the associated projective representation to PGL_{2}(k).

The following proposition is based on the computations in [CPS].

Proposition 1.11. Suppose that ρ_{0} is absolutely irreducible. Then

H^{1}(K_{1}(ζ_{p})/Q, W_{λ}^{∗}) = 0.

Proof. If the image of ρ_{0} has order prime to p the lemma is trivial. The subgroups of GL_{2}(k) containing an element of order p which are not contained in a Borel subgroup have been classified by Dickson [Di, §260] or [Hu, II.8.27]. Their images inside PGL_{2}(k′) where k′ is the quadratic extension of k are conjugate to PGL_{2}(F) or 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}(ζ_{p})/Q, W_{λ}^{∗}) ≠ 0. Then by considering the inflation-restriction sequence with respect to the normal subgroup Gal(K_{1}(ζ_{p})/K_{1}) we see that ζ_{p} ∈ K_{1}. Next, since the representation is (absolutely) irreducible, the center Z of Gal(K_{1}/Q) is contained in the diagonal matrices and so acts trivially on W_{λ}. So by considering the inflationrestriction sequence with respect to Z we see that Z acts trivially on ζ_{p} (and on W_{λ}^{∗}). So Gal(Q(ζ_{p})/Q) is a quotient of Gal(K_{1}/Q)/Z. This rules out all cases when p ≠ 3, and when p = 3 we only have to consider the case where the image of the projective representation is isomporphic as a group to PGL_{2}(F) for some finite field of characteristic 3. (Note that S_{4} ≃ PGL_{2}(F_{3}).)

Extending scalars commutes with formation of duals and H^{1}, so we may assume without loss of generality F ⊆ k. If p = 3 and #F > 3 then H^{1}(PSL_{2}(F), W_{λ}) = 0 by results of [CPS]. Then if ρ̃_{0} is the projective representation associated to ρ_{0} suppose that g^{−1}im ρ̃_{0}g = PGL_{2}(F) and let H = gPSL_{2}(F)g^{−1}. Then W_{λ} ≃ W_{λ}^{∗} over H and

(1.18)  H^{1}(H, W_{λ}) ⊗_{F} F̄ ≃ H^{1}(g^{−1}Hg, g^{−1}(W_{λ} ⊗_{F} F̄)) = 0.

We deduce also that H^{1}(im ρ_{0}, W_{λ}^{∗}) = 0.

Finally we consider the case where F = F_{3}. I am grateful to Taylor for the following argument. First we consider the action of PSL_{2}(F_{3}) on W_{λ} explicitly by considering the conjugation action on matrices {A ∈ M_{2}(F_{3}) : trace A = 0}. One sees that no such matrix is fixed by all the elements of order 2, whence

H^{1}(PSL_{2}(F_{3}), W_{λ}) ≃ H^{1}(Z/3, (W_{λ})^{C_{2}×C_{2}}) = 0

where C_{2}×C_{2} denotes the normal subgroup of order 4 in PSL_{2}(F_{3}) ≃ A_{4}. Next we verify that there is a unique copy of A_{4} in PGL_{2}(F̄_{3}) up to conjugation. For suppose that A, B ∈ GL_{2}(F̄_{3}) are such that A^{2} = B^{2} = I with the images of A, B representing distinct nontrivial commuting elements of PGL_{2}(F̄_{3}). We can choose A = ({1 \atop 0} {0 \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 PGL_{2}(\overline{F}_{3}) we have B = ({0 \atop a} {−a^{−1} \atop 0}) 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 PGL_{2}(F_{3}) is its own centralizer so it has index at most 6 in its normalizer N. Since N/⟨A, B⟩ ≃ S_{3} there is a unique subgroup of N in which ⟨A, B⟩ has index 3 whence the image of the embedding of A_{4} in PGL_{2}(F̄_{3}) is indeed unique (up to conjugation). So arguing as in (1.18) by extending scalars we see that H^{1}(im ρ_{0}, W_{λ}^{∗}) = 0 when F = F_{3} also. □

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].

Lemma 1.12. Suppose that ρ_{0} is absolutely irreducible and that

(a) ρ̃_{0} is dihedral (the case where the image is Z/2 × Z/2 is allowed),

(b) ρ_{0}|_{L} is absolutely irreducible where L = Q(\sqrt{(−1)^{(p−1)/2}p}).

Then for any positive integer n and any irreducible Galois stable subspace X of W_{λ} ⊗ k̄ there exists an element σ ∈ Gal(Q̄/Q) such that

(i) ρ̃_{0}(σ) ≠ 1,

(ii) σ fixes Q(ζ_{p^{n}}),

(iii) σ has an eigenvalue 1 on X.

Proof. If ρ̃_{0} is dihedral then ρ_{0} ⊗ k̄ = Ind_{H}^{G}χ for some H of index 2 in G, where G = Gal(K_{1}/Q). (As before, K_{1} is the splitting field of ρ_{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_{λ} ⊗ k̄ ≃ δ ⊕ Ind_{H}^{G}(χ/χ′) where δ is the quadratic character G → G/H and χ′ is the conjugate of χ by any element of G − H. Note that χ ≠ χ′ since H has nontrivial image in PGL_{2}(k̄).

To find a σ such that δ(σ) = 1 and conditions (i) and (ii) hold, observe that M(ζ_{p^{n}}) is abelian where M is the quadratic field associated to δ. So conditions (i) and (ii) can be satisfied if ρ̃_{0} is non-abelian. If ρ̃_{0} is abelian (i.e., the image has the form Z/2 × Z/2), then we use hypothesis (b). If Ind_{H}^{G}(χ/χ′) is irreducible over k̄ then W_{λ}⊗k̄ 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 = Ind_{H}^{G}(χ/χ′) ⊗ k̄ is absolutely irreducible then pick any σ ∈ G − H. This satisfies (i) and can be made to satisfy (ii) if (b) holds. Finally, since σ ∈ G−H we see that σ has trace zero and σ^{2} = 1 in its action on X. Thus it has an eigenvalue equal to 1. □

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 Gorenstein property whose proof is rather technical, depending on a characteristic 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 η-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 equality of the η-invariant for the Hecke rings and the 𝔭/𝔭^{2}-invariant for the deformation rings. In Chapter 2, Section 2, we compute the change in the η-invariant and in Chapter 1, Section 1, we estimated the change in the 𝔭/𝔭^{2}- invariant.

1. The Gorenstein property

For any positive integer N let X_{1}(N) = X_{1}(N)_{/Q} be the modular curve over Q corresponding to the group Γ_{1}(N) and let J_{1}(N) be its Jacobian. Let T_{1}(N) be the ring of endomorphisms of J_{1}(N) which is generated over Z by the standard Hecke operators {T_{l} = T_{l∗} for l ∤ N, U_{q} = U_{q∗} for q|N, ⟨a⟩ = ⟨a⟩_{∗} for (a, N) = 1}. For precise definitions of these see [MW1, Ch. 2,§5]. In particular if one identifies the cotangent space of J_{1}(N)(C) with the space of cusp forms of weight 2 on Γ_{1}(N) then the action induced by T_{1}(N) is the usual one on cusp forms. We let Δ = {⟨a⟩ : (a, N) = 1}.

The group (Z/NZ)^{∗} acts naturally on X_{1}(N) via Δ and for any subgroup H ⊆ (Z/NZ)^{∗} we let X_{H}(N) = X_{H}(N)_{/Q} be the quotient X_{1}(N)/H. Thus for H = (Z/NZ)^{∗} we have X_{H}(N) = X_{0}(N) corresponding to the group Γ_{0}(N). In Section 2 it will sometimes be convenient to assume that H decomposes as a product H = ∏ H_{q} in (Z/NZ)^{∗} ≃ ∏(Z/q^{r}Z)^{∗} where the product is over the distinct prime powers dividing N. We let J_{H}(N) denote the Jacobian 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 T_{H}(N) and sometimes, if H and N are clear from the context, we addreviate this to T.

Let p be a prime ≥ 3. Let 𝔪 be a maximal ideal of T = T_{H}(N) with p ∈ 𝔪. Then associated to 𝔪 there is a continuous odd semisimple Galois representation ρ_{𝔪},

(2.1)  ρ_{𝔪} : Gal(\overline{Q}/Q) → GL_{2}(T/𝔪)

unramified outside Np which satisfies

trace ρ_{𝔪}(Frob q) = T_{q}, det ρ_{𝔪}(Frob q) = ⟨q⟩q

for each prime q ∤ Np. Here Frob q denotes a Frobenius at q in Gal(\overline{Q}/Q). The representation ρ_{𝔪} is unique up to isomorphism. If p ∤ N (resp. p|N) we say that 𝔪 is ordinary if T_{p} ∉ 𝔪 (resp. U_{p} ∉ 𝔪). This implies (cf., for example, theorem 2 of [Wi1]) that for our fixed decomposition group D_{p} at p,

ρ_{𝔪}|_{D_{p}} ≈ ( \begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix} )

for a suitable choic of basis, with χ_{2} unramified and χ_{2}(Frob p) = T_{p} mod 𝔪 (resp. equal to U_{p}). In particular ρ_{𝔪} is ordinary in the sense of Chapter 1 provided χ_{1} ≠ χ_{2}. We will say that 𝔪 is D_{p}-distinguished if 𝔪 is ordinary and χ_{1} ≠ χ_{2}. (In practice χ_{1} is usually ramified so this imposes no extra condition.) We caution the reader that if ρ_{𝔪} is ordinary in the sense of Chapter 1 then we can only conclude that 𝔪 is D_{p}-distinguished if p ∤ N.

Let T_{𝔪} denote the completion of T at 𝔪 so that T_{𝔪} is a direct factor of the complete semi-local ring T_{p} = T⊗Z_{p}. Let D be the points of the associated 𝔪-divisible group

D = J_{H}(N)(\overline{Q})_{𝔪} ≃ J_{H}(N)(\overline{Q})_{p^{∞}} ⊗_{T_{p}} T_{𝔪}.

It is known that D̂ = Hom_{Z_{p}}(D, Q_{p}/Z_{p}) is a rank 2 T_{𝔪}-module, i.e., that D̂ ⊗_{Z_{p}} Q_{p} ≃ (T_{𝔪} ⊗_{Z_{p}} Q_{p})^{2}. Briefly it is enough to show that H^{1}(X_{H}(N), C) is

free of rank 2 over T ⊗ C and this reduces to showing that S_{2}(Γ_{H}(N), C), the space of cusp forms of weight 2 on Γ_{H}(N), is free of rank 1 over T ⊗ C. One shows then that if {f_{1}, …, f_{r}} is a complete set of normalized newforms in S_{2}(Γ_{H}(N), C) of levels m_{1}, …, m_{r} then if we set d_{i} = N/m_{i}, the form f = Σf_{i}(d_{i}z) is a basis vector of S_{2}(Γ_{H}(N), C) as a T ⊗ C-module.

If 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 D by Pontrjagin duals of rank 1 T_{𝔪}-modules (in the sense explained above)

(2.2)  0 → D^{0} → D → D^{E} → 0

where D^{0} is stable under D_{p} and the induced action on 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_{p}x + p⟨p⟩ = 0 in T_{𝔪} if p ∤ N. We can describe D^{0} and D^{E} as follows. Pick a σ ∈ I_{p} which induces a generator of Gal(Q_{p}(ζ_{Np^{∞}})/Q_{p}(ζ_{Np})). Let ε : D_{p} → Z_{p}^{×} be the cyclotomic character. Then D^{0} = ker(σ − ε(σ))^{div}, the kernel being taken inside D and ‘div’ meaning the maximal divisible subgroup. Although in [Wi1] this filtration is given only for a factor A_{f} of J_{1}(N) it is easy to deduce the result for J_{H}(N) itself. We note that this filtration is defined without reference to characteristic p and also that if 𝔪 is D_{p}-distinguished, D^{0} (resp. D^{E}) can be described as the maximal submodule on which σ − χ̃_{1}(σ) is topologically nilpotent for all σ ∈ Gal(\overline{Q}_{p}/Q_{p}) (resp. quotient on which σ − χ̃_{2}(σ) is topologically nilpotent for all σ ∈ Gal(\overline{Q}_{p}/Q_{p})), where χ̃_{i}(σ) is any lifting of χ_{i}(σ) to T_{𝔪}.

The Weil pairing ⟨ , ⟩ on J_{H}(N)(\overline{Q})_{p^{M}} satisfies the relation ⟨t_{∗}x, y⟩ = ⟨x, t^{∗}y⟩ for any Hecke operator t. It is more convenient to use an adapted pairing defined as follows. Let w_{ζ}, for ζ a primitive N^{th} root of 1, be the involution of X_{1}(N)_{/Q(ζ)} defined in [MW1, p. 235]. This induces an involution of X_{H}(N)_{/Q(ζ)} also. Then we can define a new pairing [ , ] by setting (for a fixed choice of ζ)

(2.3)  [x, y] = ⟨x, w_{ζ}y⟩.

Then [t_{∗}x, y] = [x, t_{∗}y] for all Hecke operators t. In particular we obtain an induced pairing on 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, §9]. However some of the cases we need are not covered in these accounts and we will present these here.

Theorem 2.1. (i) If p ∤ N and ρ_{𝔪} is irreducible then

J_{H}(N)(\overline{Q})[𝔪] ≃ (T/𝔪)^{2}.

(ii) If p ∤ N and ρ_{𝔪} is irreducible and 𝔪 is D_{p}-distinguished then

J_{H}(Np)(\overline{Q})[𝔪] ≃ (T/𝔪)^{2}.

(In case (ii) 𝔪 is a maximal ideal of T = T_{H}(Np).)

Corollary 1. In case (i), J_{H} { \atop (N)(\overline{Q})}_{𝔪} ≃ T_{𝔪}^{2} and Ta_{𝔪}(J_{H}(N)(\overline{Q})) ≃

T_{𝔪}^{2}.

In case (ii), J_{H} { \atop (Np)(\overline{Q})}_{𝔪} ≃ T_{𝔪}^{2} and Ta_{𝔪}(J_{H}(Np)(\overline{Q})) ≃ T_{𝔪}^{2} (where T_{𝔪} = T_{H}(Np)_{𝔪}).

Corollary 2. In either of cases (i) or (ii) T_{𝔪} is a Gorenstein ring.

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 Z_{p}-algebra as defined in the appendix. For finite flat local Z_{p}-algebras the notions of Gorenstein ring and Gorenstein Z_{p}-algebra are the same.) Here Ta_{𝔪}(J_{H}(N)(\overline{Q})) = Ta_{p}(J_{H}(N)(\overline{Q})) ⊗_{T_{p}} T_{𝔪} is the 𝔪-adic Tate module of

J_{H}(N).

We should also point out that although Corollary 1 gives a representation from the 𝔪-adic Tate module

ρ = ρ_{T_{𝔪}} : Gal(\overline{Q}/Q) → GL_{2}(T_{𝔪})

this can be constructed in a much more elementary way. (See [Ca3] for another argument.) For, the representation exists with T_{𝔪} ⊗ Q replacing T_{𝔪} when we use the fact that Hom(Q_{p}/Z_{p}, D)⊗Q was free of rank 2. A standard argument using the Eichler-Shimura relations implies that this representation ρ′ with values in GL_{2}(T_{𝔪} ⊗ Q) has the property that

trace ρ′(Frob ℓ) = T_{ℓ}, det ρ′(Frob ℓ) = ℓ⟨ℓ⟩

for all ℓ ∤ Np. We can normalize this representation by picking a complex conjugation c and choosing a basis such that ρ′(c)=({1 \atop 0} {0 \atop −1}), and then by picking a τ for which ρ′(τ) = ({a_{τ} \atop c_{τ}} {b_{τ} \atop d_{τ}} ) with b_{τ} c_{τ} ≢ 0(𝔪) and by rescaling the basis so that b_{τ} = 1. (Note that the explicit description of the traces shows that if ρ_{𝔪} is also normalized so that ρ_{𝔪}(c) = ({1 \atop 0} {0 \atop −1}) then b_{ρ}c_{τ} mod 𝔪 = b_{τ,𝔪}c_{τ,𝔪} where ρ_{𝔪}(τ) = ({a_{τ,𝔪} \atop c_{τ,𝔪}} {b_{τ,𝔪} \atop d_{τ,𝔪}} ). The existence of a τ such that b_{τ} c_{τ} ≢ 0(𝔪) comes from the irreducibility of ρ_{𝔪}.) With this normalization one checks that ρ′ actually takes values in the (closed) subring of T_{𝔪} generated over 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 ρ by the traces and determinants (Theorem 1 of [Ca3]). One can also often interpret the U_{q} operators in terms of ρ for q|N using the π_{q} ≃ π(σ_{q}) theorem of Langlands (cf. [Ca1]) and the U_{q} operator in case (ii) using Theorem 2.1.4 of [Wi1].

Proof (of theorem). The important technique for proving such multiplicityone results is due to Mazur and is based on the q-expansion principle in characteristic p. Since the kernel of J_{H}(N)(\overline{Q}) → J_{1}(N)(\overline{Q}) is an abelian group on which Gal(\overline{Q}/Q) acts through an abelian extension of Q, the intersection with ker 𝔪 is trivial when ρ_{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 Γ_{0}(N) in [Ri1, Th. 5.2] which summarizes Mazur’s argument. The case of Γ_{1}(N) is similar (cf. [E1, Th. 9.2]).

Now consider case (ii). Let Δ_{(p)} = {⟨a⟩ : a ≡ 1(N)} ⊆ Δ. Let us first assume that Δ_{(p)} is nontrivial mod 𝔪, i.e., that δ−1∉𝔪 for some δ ∈Δ_{(p)}. This case is essentially covered in [Ti1] (and also in [Gro]). We briefly review the argument for use later. Let K = Q_{p}(ζ_{p}), ζ_{p} being a primitive p^{th} root of unity, and let O be the ring of integers of the completion of the maximal unramified extension of K. Using the fact that Δ_{(p)} is nontrivial mod 𝔪 together with Proposition 4, p. 269 of [MW1] we find that

J_{1}(Np)_{𝔪/O}^{ét}(\overline{F}_{p}) ≃ (Pic^{0}Σ_{1}^{ét} × Pic^{0}Σ_{1}^{μ})_{𝔪}(\overline{F}_{p})

where the notation is taken from [MW1] loc. cit. Here Σ_{1}^{ét} and Σ_{1}^{μ} are the two smooth irreducible components of the special fibre of the canonical model of X_{1}(Np)_{/O} described in [MW1, Ch. 2]. (The smoothness in this case was proved in [DR].) Also J_{1}(Np)_{𝔪/O}^{ét} denotes the canonical étale quotient of the 𝔪-divisible group over O. This makes sense because J_{1}(Np)_{𝔪} does extend to a p-divisible group over O (again by a theorem of Deligne and Rapoport [DR] and because Δ_{(p)} is nontrivial mod 𝔪). It is ordinary as follows from (2.2) when we use the main theorem of Tate ([Ta]) since D^{0} and D^{E} clearly correspond to ordinary p-divisible groups.

Now the q-expansion principle implies that dim_{\overline{F}_{p}} X[𝔪′] ≤ 1 where

X = {H^{0}(Σ_{1}^{μ}, Ω^{1}) ⊕ H^{0}(Σ_{1}^{ét}, Ω^{1})}

and 𝔪′ is defined by embedding T/𝔪 ↪ \overline{F}_{p} and setting 𝔪′ = ker : T⊗\overline{F}_{p} → \overline{F}_{p} under the map t ⊗ a ↦ at mod 𝔪. Also T acts on Pic^{0}Σ_{1}^{μ} × Pic^{0}Σ_{1}^{ét}, the abelian variety part of the closed fibre of the Neron model of J_{1}(Np)_{/O}, and hence also on its cotangent space X. (For a proof that X[𝔪′] is at most onedimensional, which is readily adapted to this case, see Lemma 2.2 below. For similar versions in slightly simpler contexts see [Wi3, §6] or [Gro, §12]. Then the Cartier map induces an injection 9cf. Prop. 6.5 of [Wi3])

δ : \{Pic^{0}Σ_{1}^{μ} × Pic^{0}Σ_{1}^{ét}\}[p](\overline{F}_{p}) ⊗_{F_{p}} \overline{F}_{p} ↪ X.

The composite δ ∘ w_{ζ} 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)_{𝔪/O}(\overline{F}_{p})[𝔪] ≃ T/𝔪

as a T-module. This shows that if Ĥ is the Pontrjagin dual of H = J_{1}(Np)_{𝔪/O}(\overline{F}_{p}) then Ĥ ≃ T_{𝔪} since Ĥ/𝔪 ≃ T/𝔪. Thus

J_{1}(Np)_{𝔪/O}(\overline{F}_{p})[p] →^{∼} Hom(T_{𝔪}/p, Z/pZ).

Now our assumption that 𝔪 is D_{p}-distinguished enables us to identify

D^{0} = J_{1}(Np)_{𝔪/O}^{0}(\overline{Q}_{p}) , D^{E} = J_{1}(Np)_{𝔪/O}^{ét}(\overline{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 Hom( , Q_{p}/Z_{p}(1))). So

D^{0}[p] →^{∼} T_{𝔪}/p, D^{E}[p] →^{∼} Hom(T_{𝔪}/p, Z/pZ)

as T_{𝔪}-modules, the former following from the latter when we use duality under the pairing [ , ]. In particular as 𝔪 is D_{p}-distinguished,

(2.4)  D[p] ≃ T_{𝔪}/p ⊕ Hom(T_{𝔪}/p, Z/pZ).

We now use an argument of Tilouine [Ti1]. We pick a complex conjugation τ. This has distinct eigenvalues ±1 on ]ρ_{𝔪} so we may decompose D[p] into eigenspaces for τ:

D[p] = D[p]^{+} ⊕ D[p]^{−}.

Since T_{𝔪}/p and Hom(T_{𝔪}/p, Z/pZ) 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

D[𝔪] = D[𝔪]^{+} ⊕ D[𝔪]^{−}

one of the eigenspaces is isomorphic to T_{𝔪} and the other to (T_{𝔪}/p)[𝔪]. But since ρ_{𝔪} is irreducible it is easy to see by considering D[𝔪]⊕Hom(D[𝔪], det ρ_{𝔪}) that τ has the same number of eigenvalues equal to +1 as equal to −1 in D[𝔪], whence #(T_{𝔪}/p)[𝔪] = #(T/𝔪). This shows that D[𝔪]^{+} →^{∼} D[𝔪]^{−} ≃ T/𝔪 as required.

Now we consider the case where Δ_{(p)} is trivial mod 𝔪. This case was treated (but only for the group Γ_{0}(Np) and ρ_{𝔪} ‘new’ at p—–the crucial restriction being the last one) in [M Ri]. Let X_{1}(N, p)_{/Q} be the modular curve corresponding to Γ_{1}(N) ∩ Γ_{0}(p) and let J_{1}(N, p) be its Jacobian. Then since the composite of natural maps J_{1}(N, p) → J_{1}(Np) → J_{1}(N, p) is multiplication by an integer prime to p and since Δ_{(p)} is trivial mod 𝔪 we see that

J_{1}(N, p)_{𝔪}(\overline{Q}) ≃ J_{1}(Np)_{𝔪}(\overline{Q}).

It will be enough then to use J_{1}(N, p), and the corresponding ring T and ideal 𝔪.

The curve X_{1}(N, p) has a canonical model X_{1}(N, p)_{/Z_{p}} which over \overline{F}_{p} consists of two smooth curves Σ^{ét} and Σ^{μ} intersecting transversally at the supersingular points (again this is a theorem of Deligne and Rapoport; cf. [DR, Ch. 6, Th. 6.9], [KM] or [MW1] for more details). We will use the models described in [MW1, Ch. II] and in particular the cusp ∞ will lie on Σ^{μ}. Let Ω denote the sheaf of regular differentials on X_{1}(N, p)_{/F_{p}} (cf. [DR, Ch. 1 §2], [M Ri, §7]). Over \overline{F}_{p}, since X_{1}(N, p)_{/\overline{F}_{p}} has ordinary double point singularities, the differentials may be identified with the meromorphic differentials on the normalization X_{1}{ \atop (N, p)}_{/\overline{F}_{p}} = Σ^{ét} ∪ Σ^{μ} which have at most simple poles at the supersingular points (the intersection points of the two components) and satisfy res_{x_{1}} + res_{x_{2}} = 0 if x_{1} and x_{2} are the two points above such a supersingular point. We need the following lemma:

Lemma 2.2. dim_{T/𝔪} H^{0}(X_{1}(N, p)_{/F_{p}}, Ω)[𝔪] = 1.

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 explained below. We will first show that dim_{T/𝔪} H^{0}(X_{1}(N, p)_{/F_{p}}, Ω)[𝔪] ≤ 1, this being the essential step. If we embed T/𝔪 ↪ \overline{F}_{p} and then set 𝔪′ = ker : T ⊗ \overline{F}_{p} → \overline{F}_{p} (the map given by t ⊗ a ↦ at mod 𝔪) then it is enough to show that dim_{\overline{F}_{p}} H^{0}(X_{1}(N, p)_{/\overline{F}_{p}}, Ω)[𝔪′] ≤ 1. First we will suppose that there is no nonzero holomorphic differential in H^{0}(X_{1}(N, p)_{/\overline{F}_{p}}, Ω)[𝔪′], i.e., no differential form which pulls back to holomorphic differentials on Σ^{ét} and Σ^{μ}. Then if ω_{1} and ω_{2} are two differentials in H^{0}(X_{1}(N, p)_{/\overline{F}_{p}}, Ω)[𝔪′], the q-expansion principle shows that μω_{1} − λω_{2} has zero q-expansion at ∞ for some pair (μ, λ) ≠ (0, 0) in \overline{F}_{p}^{2} and thus is zero on Σ^{μ}. As μω_{1} − λω_{2} = 0 on Σ^{μ} it is holomorphic on Σ^{ét}. By our hypothesis it would then be zero which shows that ω_{1} and ω_{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 ω 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)_{/Z_{p}}, Ω), where Ω denotes the sheaf of regular differentials on X_{1}(N, p)_{/Z_{p}}, behaves well under the base changes Z_{p} → \overline{Z}_{p} and Z_{p} → \overline{Q}_{p}; cf. [Ma2, §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)_{/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)_{/Z_{p}}, Ω); see (2.5) below. (For a summary of the duality statements used in this context, see [Ma2, §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{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{F}_{p}}, Ω)[𝔪′].

We claim that the space of holomorphic differentials then has dimension 1 and that any such differential ω ≠ 0 is actually nonzero on Σ^{μ}. The dimension claim follows from the second assertion by using the q-expansion principle. To prove that ω ≠ 0 on Σ^{μ} we use the formula

U_{p∗}(x, y) = (F x, y′)

for (x, y) ∈ (Pic^{0}Σ^{ét} × Pic^{0}Σ^{μ})(\overline{F}_{p}), where F denotes the Frobenius endomorphism. The value of y′ will not be needed. This formula is a variant on the second part of Theorem 5.3 of [Wi3] where the corresponding result 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}(Σ^{μ}, Ω^{1}) ⊕ H^{0}(Σ^{ét}, Σ^{1}) viewed as a subspace of H^{0}(X_{1}(N, p)_{/\overline{F}_{p}}, Ω) is the same as the action on X_{0} viewed as the cotangent space of Pic^{0}Σ^{μ} × Pic^{0}Σ^{ét}. From this we see that if ω = 0 on Σ^{μ} then U_{p^{ω}} = 0 on Σ^{ét}. But U_{p} acts as a nonzero scalar which gives a contradiction if ω ≠ 0. We can thus assume that the space of 𝔪′-torsion holomorphic differentials has dimension 1 and is generated by ω. So if ω_{2} is now any differential in H^{0}(X_{1}(N, p)_{/\overline{F}_{p}}, Ω)[𝔪′] then ω_{2}−λω has zero q-expansion at ∞ for some choice of λ. Then ω_{2}−λω = 0 on Σ^{μ} whence ω_{2} − λω is holomorphic and so ω_{2} = λω. We have now shown in general that dim(H^{0}(X_{1}(N, p)_{/\overline{F}_{p}}, Ω)[𝔪′]) ≤ 1.

The singularities of X_{1}(N, p)_{/Z_{p}} at the supersingular points are formally isomorphic over Ẑ_{p}^{unr} to Ẑ_{p}^{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)_{/Z_{p}} then H^{0}(M_{1}(N, p)_{/F_{p}}, Ω) ≃ H^{0}(X_{1}(N, p)_{/F_{p}}, Ω) (see the argument in [Ma2, Prop. 3.4]), and a similar isomorphism holds for H^{0}(M_{1}(N, p)_{/Z_{p}}, Ω).

As M_{1}(N, p)_{/Z_{p}} is regular, a theorem of Raynaud [Ray2] says that the connected component of the Neron model of J_{1}(N, p)_{/Q_{p}} is J_{1}(N, p)_{/Z_{p}}^{0} ≃ Pic^{0}(M_{1}(N, p)_{/Z_{p}}). Taking tangent spaces at the origin, we obtain

(2.5)  Tan(J_{1}(N, p)_{/Z_{p}}^{0}) ≃ H^{1}(M_{1}(N, p)_{/Z_{p}}, O_{M_{1}(N,p)}).

Reducing both sides mod p and applying Grothendieck duality we get an isomorphism

(2.6)  Tan(J_{1}(N, p)_{/F_{p}}^{0}) →^{∼} Hom(H^{0}(X_{1}(N, p)_{/F_{p}}, Ω), F_{p}).

(To justify the reduction in detail see the arguments in [Ma2, §II. 3]). Since Tan(J_{1}(N, p)_{/Z_{p}}^{0}) is a faithful T ⊗ Z_{p}-module it follows that

H^{0}(X_{1}(N, p)_{/F_{p}}, Ω)[𝔪]

is nonzero. This completes the proof of the lemma. □

To complete the proof of the theorem we choose an abelian subvariety 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) → J_{1}(N) × J_{1}(N) under the natural map φ̂ described in Section 2 (see (2.10)). Then we have an exact sequence

0 → A → J_{1}(N, p) → B → 0

and J_{1}(N, p) has semistable reduction over Q_{p} and B has good reduction. By Proposition 1.3 of [Ma3] the corresponding sequence of connected group schemes

0 → A[p]_{/Z_{p}}^{0} → J_{1}(N, p)[p]_{/Z_{p}}^{0} → B[p]_{/Z_{p}}^{0} → 0

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

(2.7)  Tan(J_{1}(N, p)[p]_{/\overline{F}_{p}}^{t}) ⊗_{T_{p}} T_{𝔪} → Tan(J_{1}(N, p)_{/\overline{F}_{p}}) ⊗_{T_{p}} T_{𝔪}

is an isomorphism, where t denotes the maximal multiplicative-type subgroup scheme (cf. [Ma3, §1]). For it is enough to check such a relation on A and B separately and on B it is true because the 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

Tan(J_{1}(N, p))_{/Z_{p}} ⊗_{T_{p}} T_{𝔪} ≃ T_{𝔪}.

We claim that (2.7) together with this implies that as T_{𝔪}-modules

V := J_{1}(N, p)[p]^{t}(\overline{Q}_{p})_{𝔪} ≃ (T_{𝔪}/p).

To see this it is sufficient to exhibit an isomorphism of \overline{F}_{p}-vector spaces

(2.8)  Tan(G_{/\overline{F}_{p}}) ≃ G(\overline{Q}_{p}) ⊗_{F_{p}} \overline{F}_{p}

for any multiplicative-type group scheme (finite and flat) G_{/Z_{p}} which is killed by p and moreover to give such an isomorphism that respects the action of endomorphism of G_{/Z_{p}}. To obtain such an isomorphism observe that we have isomorphisms

(2.9)  Hom_{\overline{Q}_{p}}(μ_{p}, G) ⊗_{F_{p}} \overline{F}_{p} ≃ Hom_{\overline{F}_{p}}(μ_{p}, G) ⊗_{F_{p}} \overline{F}_{p}
≃ Hom(Tan(μ_{p}_{/\overline{F}_{p}}), Tan(G_{/\overline{F}_{p}}))

where Hom_{\overline{Q}_{p}} denotes homomorphisms of the group schemes viewed over \overline{Q}_{p} and similarly for Hom_{\overline{F}_{p}}. The second isomorphism can be checked by reducing to the case G = μ_{p}. Now picking a primitive p^{th} root of unity we can identify the left-hand term in (2.9) with G(\overline{Q}_{p}) ⊗_{F_{p}} \overline{F}_{p}. Picking an isomorphism of

Tan(μ_{p/\overline{F}_{p}}) with \overline{F}_{p} we can identify the last term in (2.9) with Tan(G_{/\overline{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 Gal(\overline{Q}_{p}/Q_{p}) on V is ramified on every subquotient, so V ⊆ D^{0}[p]. (Note that our assumption that Δ_{(p)} is trivial mod 𝔪 implies that the action on D^{0}[p] is ramified on every subquotient and on D^{E}[p] is unramified on every subquotient.) By again examining A and B separately we see that in fact V = D^{0}[p]. For A we note that A[p]/A[p]^{t} is unramified because it is dual to Â[p]^{t} where Â is the dual abelian variety. We can now proceed as we did in the case where Δ_{(p)} was nontrivial mod 𝔪. □

2. Congruences between Hecke rings

Suppose that q is a prime not dividing N. Let Γ_{1}(N, q) = Γ_{1}(N) ∩ Γ_{0}(q) and let X_{1}(N, q) = X_{1}(N, q)_{/Q} be the corresponding curve. The two natural maps X_{1}(N, q) → X_{1}(N) induced by the maps z → z and z → qz on the upper half plane permit us to define a map J_{1}(N) × J_{1}(N) → 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 φ by

(2.10)  0 → J_{1}(N) × J_{1}(N) ⟶^{φ} J_{1}(N, q).

Dualizing, we define B by

0 → B ⟶^{ψ} J_{1}(N, q) ⟶^{φ̂} J_{1}(N) × J_{1}(N) → 0.

Let T_{1}(N, q) be the ring of endomorphisms of J_{1}(N, q) generated by the standard Hecke operators {T_{l∗} for l ∤ Nq, U_{l∗} for l|Nq, ⟨a⟩ = ⟨a⟩_{∗} for (a, Nq) = 1}. One can check that U_{p} preserves B either by an explicit calculation 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) × 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 T_{H}(N) and T_{H}(N, q) for the associated Hecke rings. In this case the corresponding map φ may have a kernel. However since the kernel of J_{H}(N) → J_{1}(N) does not meet ker 𝔪 for any maximal ideal 𝔪 whose associated ρ_{𝔪} is irreducible, the above sequence remain exact if we restrict to 𝔪^{(q)}-divisible groups, 𝔪^{(q)} being the maximal ideal associated to 𝔪 of the ring T_{H}^{(q)}(N, q) generated by the standard Hecke operators but ommitting U_{q}. With this minor modification the proofs of the results below for H ≠ 1 follow from the cases of full level. We will use the same notation in the general case. Thus φ is the map J_{2} = J_{H}(N)^{2} → J_{H}(N, q) induced by z → z and z → qz on the two factors, and B = ker φ̂. (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} ≡ q(N) and n_{q} ≡ 1(q), and write ⟨q⟩ = ⟨n_{q}⟩ ∈ T_{H}(Nq).

Lemma 2.3 (Ribet). ψ(B) ∩ φ(J_{2})_{𝔪^{(q)}} = φ(J_{2})[U_{q}^{2} − ⟨q⟩]_{𝔪^{(q)}} for irreducible ρ_{𝔪}.

Proof. The left-hand side is (imφ ∩ ker φ̂), so we compute φ^{−1}(imφ ∩ ker φ̂) = ker(φ̂ ∘ φ).

An explicit calculation shows that

φ̂ ∘ φ = [ \begin{matrix}q + 1 & T_{q} \\ T_{q}^{∗} & q + 1 \end{matrix} ] on J_{2}

where T_{q}^{∗} = T_{q} · ⟨q⟩^{−1}. The matrix action here is on the left. We also find that on J_{2}

(2.11)  U_{q} ∘ φ = φ ∘ [ \begin{matrix}0 & −⟨q⟩ \\ q & T_{q} \end{matrix} ] ,

whence

(U_{q}^{2} − ⟨q⟩) ∘ φ = φ ∘ [ \begin{matrix}−⟨q⟩ & 0 \\ T_{q} & −⟨q⟩ \end{matrix} ] ∘ (φ̂ ∘ φ). □

Now suppose that 𝔪 is a maximal ideal of T_{H}(N), p ∈ 𝔪 and ρ_{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 ≠ p we will also strengthen but we will do this separately.) Assume the that p ∤ N and T_{p} ∉ 𝔪. Let a_{p} be the unit root of x^{2} − T_{p}x + p⟨p⟩ = 0 in T_{H}(N)_{𝔪}. We first define a maximal ideal 𝔪_{p} of T_{H}(N, p) with the same associated representation as 𝔪. To do this consider the ring

S_{1} = T_{H}(N)[U_{1}]/(U_{1}^{2} − T_{p}U_{1} + p⟨p⟩) ⊆ End(J_{H}(N)^{2})

where U_{1} is the endomorphism of J_{H}(N)^{2} given by the matrix

[ \begin{matrix}T_{p} & −⟨p⟩ \\ p & 0 \end{matrix} ] .

It is thus compatible with the action of U_{p} on J_{H}(N, p) when compared using φ̂. Now 𝔪_{1} = (𝔪, U_{1} − ã_{p}) is a maximal ideal of S_{1} where ã_{p} is any element of T_{H}(N) representing the class ā_{p} ∈ T_{H}(N)_{𝔪}/𝔪 ≃ T_{H}(N)/𝔪. Moreover S_{1,𝔪_{1}} ≃ T_{H}(N)_{𝔪} and we let 𝔪_{p} be the inverse image of 𝔪_{1} in T_{H}(N, p) under the natural map T_{H}(N, p) → S_{1}. One checks that 𝔪_{p} id D_{p}-distinguished. For any standard Hecke operator t except U_{p} (i.e., t = T_{l}, U_{q′} for q′ ≠ p or ⟨a⟩) the image of t is t. The image of U_{p} is U_{1}.

We need to check that the induced map

α : T_{H}(N, p)_{𝔪_{p}} ⟶ S_{1,𝔪_{1}} ≃ T_{H}(N)_{𝔪}

is surjective. The only problem is to show that T_{p} is in the image. In the present context one can prove this using the surjectivity of φ̂ in (2.12) and using the fact that the Tate-modules in the range and domain of φ̂ are free of rank 2 by Corollary 1 to Theorem 2.1. The result then follows from Nakayama’s lemma as one deduces easily that T_{H}(N)_{𝔪} is a cyclic T_{H}(N, p)_{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.

For the following lemma we let T^{M}, for an integer M, denote the subring of End(S_{2}(Γ_{1}(N))) generated by the Hecke operators T_{n} for positive integers n

relatively prime to M. Here S_{2}(Γ_{1}(N)) denotes the vector space of weight 2

cusp forms on Γ_{1}(N). Write T for T^{1}. It will be enough to show that T_{p} is a redundant operator in T^{1}, i.e., that T^{p} = T. The result for T_{H}(N)_{𝔪} then follows.

Lemma (Ribet). Suppose that (M, N) = 1. If M is odd then T^{M} = T. If M is even then T^{M} has finite index in T equal to a power of 2.

As the rings are finitely generated free Z-modules, it suffices to prove that T^{M} ⊗ F_{l} → T ⊗ F_{l} is surjective unless l and M are both even. The claim follows from

1. T^{M} ⊗ F_{l} → T^{M/p} ⊗ F_{l} is surjective if p|M and p ∤ lN.
2. T^{l} ⊗ F_{l} → T ⊗ F_{l} is surjective if l ∤ 2N.

Proof of 1. Let A denote the Tate module Ta_{l}(J_{1}(N)). Then R = T^{M/p} ⊗ Z_{l} acts faithfully on A. Let R′ = (R ⊗_{Z_{l}} Q_{l}) ∩ End_{Z_{l}}A and choose d so that

l^{d}R′ ⊂ lR. Consider the Gal(Q̄/Q)-module B = J_{1}(N)[l^{d}] × μ_{Nl^{d}}. By Čebotarev density, there is a prime q not dividing MNl so that Frobp = Frobq on B. Using the fact that T_{r} = Frobr + ⟨r⟩r(Frobr)^{−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}End_{Z_{l}}A and therefore in l^{d}R′ ⊂ lR.

Proof of 2. Let S be the set of cusp forms in S_{2}(Γ_{1}(N)) whose q-expansions at ∞ have coefficients in Z. Recall that S_{2}(Γ_{1}(N)) = S⊗C and that S is stable under the action of T (cf. [Sh1, Ch. 3] and [Hi4, §4]). The pairing T ⊗ S → Z defined by T ⊗ f ↦ a_{1}(T f) is easily checked to induce an isomorphism of T-modules

S ≅ Hom_{Z}(T, Z).

The surjectivity of T^{l}/lT^{l} → T/lT is equivalent to the injectivity of the dual map

Hom(T, F_{l}) → Hom(T^{l}, F_{l}).

Now use the isomorphism S/lS ≅ Hom(T, F_{l}) and note that if f is in the kernel of S → Hom(T^{l}, F_{l}), then a_{n}(f) = a_{1}(T_{n}f) 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 \frac{d}{dq}, and is therefore trivial if l is odd. (See Corollary 5 of the main theorem of [Ka].) Therefore f is in lS.

Remark. The argument does not prove that T^{Md} = T^{d} if (d, N) ≠ 1.

We now return to the assumptions that ρ_{𝔪} is irreducible, p ∤ N and T_{p} ∉ 𝔪. Next we define a principal ideal (Δ_{p}) of T_{H}(N)_{𝔪} as follows. Since T_{H}(N, p)_{𝔪_{p}} and T_{H}(N)_{𝔪} are both Gorenstein rings (by Corollary 2 of Theorem 2.1) we can define an adjoint α̂ to

α : T_{H}(N, p)_{𝔪_{p}} ⟶ S_{1,𝔪_{1}} ≃ T_{H}(N)_{𝔪}

in the manner described in the appendix and we set Δ_{p} = (α ∘ α̂)(1). Then (Δ_{p}) is independent of the choice of (Hecke-module) pairings on T_{H}(N, p)_{𝔪_{p}} and T_{H}(N)_{𝔪}. It is equal to the ideal generated by any composite map

T_{H}(N)_{𝔪} ⟶^{β} T_{H}(N, p)_{𝔪_{p}} ⟶^{α} T_{H}(N)_{𝔪}

provided that β is an injective map of T_{H}(N, p)_{𝔪_{p}}-modules with Z_{p} torsion-free cokernel. (The module structure on T_{H}(N)_{𝔪} is defined via α.)

Proposition 2.4. Assume that 𝔪 is D_{p}-distinguished and that ρ_{𝔪} is irreducible of level N with p ∤ N. Then

(Δ_{p}) = (T_{p}^{2} − ⟨p⟩(1 + p)^{2}) = (a_{p}^{2} − ⟨p⟩).

Proof. Consider the maps on p-adic Tate-modules induced by φ and φ̂:

Ta_{p}(J_{H}(N)^{2}) ⟶^{φ} Ta_{p}(J_{H}(N, p)) \underrightarrow{φ̂} Ta_{p}(J_{H}(N)^{2}).

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} = T_{H}(N)[U_{2}]/(U_{2}^{2} − T_{p}U_{2} + p⟨p⟩) ⊆ End(J_{H}(N)^{2})

where U_{2} is the endomorphism of J_{H}(N)^{2} defined by ({0 \atop p} {−⟨p⟩ \atop T_{p}}). It satisfies φU_{2} = U_{p}φ. Again 𝔪_{2} = (𝔪, U_{2} − ã_{p}) is a maximal ideal of S_{2} and we have, on restricting to the 𝔪_{1}, 𝔪_{p} and 𝔪_{2}-adic Tate-modules:

(2.12)

Ta_{𝔪_{2}}(J_{H}(N)^{2}) ⟶^{φ} Ta_{𝔪_{p}}(J_{H}(N, p)) \underrightarrow{φ̂} Ta_{𝔪_{1}}(J_{H}(N)^{2})
↑≀ v_{2}  ↑≀ v_{1}
Ta_{𝔪}(J_{H}(N))  Ta_{𝔪}(J_{H}(N)).

The vertical isomorphisms are defined by v_{2} : x → (−⟨p⟩x, a_{p}x) and v_{1} : x → (a_{p}x, px). (Here a_{p} ∈ T_{H}(N)_{𝔪} can be viewed as an element of T_{H}(N)_{p} ≃ ∏ T_{H}(N)_{𝔫} where the product is taken over the maximal ideals containing p. So v_{1} and v_{2} can be viewed as maps to Ta_{p}(J_{H}(N)^{2}) whose images are

respectively Ta_{𝔪_{1}}(J_{H}(N)^{2}) and Ta_{𝔪_{2}}(J_{H}(N)^{2}).)

Now φ̂ is surjective and φ is injective with torsion-free cokernel by the result of Ribet mentioned before. Also Ta_{𝔪}(J_{H}(N)) ≃ T_{H}(N)_{𝔪}^{2} and Ta_{𝔪_{p}}(J_{H}(N, p)) ≃ T_{H}(N, p)_{𝔪_{p}}^{2} by Corollary 1 to Theorem 2.1. So as φ, φ̂ are maps of T_{H}(N, p)_{𝔪_{p}}-modules we can use this diagram to compute Δ_{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,𝔪_{1}} and S_{2,𝔪_{2}} with T_{H}(N)_{𝔪}, we get U_{1} = U_{2} which is indeed the case.) We find that

v_{1}^{−1} ∘ φ̂ ∘ φ ⊗ v_{2}(z) = a_{p}^{−1}(a_{p}^{2} − ⟨p⟩)(z). □

We now apply to J_{1}(N, q^{2}) (but q ≠ p) the same analysis that we have just applied to J_{1}(N, q^{2}). Here X_{1}(A, B) is the curve corresponding to Γ_{1}(A)∩Γ_{0}(B) 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}) → X_{1}(Nq^{r−1}) induced by z → z and z → qz by π_{1,r} and π_{2,r} respectively. Similarly we denote the maps X_{1}(Nq^{r}, q^{r+1}) → X_{1}(Nq^{r}) induced by z → z and z → qz by π_{3,r} and π_{4,r} respectively. Also let π : X_{1}(Nq^{r}) → X_{1}(Nq^{r−1}, q^{r}) denote the natural map induced by z → z.

In the following lemma if 𝔪 is a maximal ideal of T_{1}(Nq^{r−1}) or T_{1}(Nq^{r}) we use 𝔪^{(q)} to denote the maximal ideal of T_{1}^{(q)}(Nq^{r}, q^{r+1}) compatible with 𝔪, the ring T_{1}^{(q)}(Nq^{r}, q^{r+1}) ⊂ T_{1}(Nq^{r}, q^{r+1}) being the subring obtained by omitting U_{q} from the list of generators.

Lemma 2.5. If q ≠ p is a prime and r ≥ 1 then the sequence of abelian varieties

0 → J_{1}(Nq^{r−1}) ⟶^{ξ_{1}} J_{1}(Nq^{r}) × J_{1}(Nq^{r}) ⟶^{ξ_{2}} J_{1}(Nq^{r}, q^{r+1})

where ξ_{1} = ((π_{1,r} ∘ π)^{∗}, −(π_{2,r} ∘ π)^{∗}) and ξ_{2} = (π_{4,r}^{∗}, π_{3,r}^{∗}) induces a corresponding sequence of p-divisible groups which becomes exact when localized at any 𝔪^{(q)} for which ρ_{𝔪} is irreducible.

Proof. Let Γ^{1}(Nq^{r}) denote the group \{(({a \atop c} {b \atop d})) ∈ Γ_{1}(N) : a ≡ d ≡ 1(q^{r}),

c ≡ 0(q^{r−1}), b ≡ 0(q)}. Let B_{1} and B^{1} be given by

B_{1} = Γ_{1}(Nq^{r})/Γ_{1}(Nq^{r}) ∩ Γ(q), B^{1} = Γ^{1}(Nq^{r})/Γ_{1}(Nq^{r}) ∩ Γ(q)

and let Δ_{q} = Γ_{1}(Nq^{r−1})/Γ_{1}(Nq^{r}) ∩ Γ(q). Thus Δ_{q} ≃ SL_{2}(Z/q) if r = 1 and is of order a power of q if r > 1.

The exact sequences of inflation-restriction give:

H_{1}(Γ_{1}(Nq^{r}), Q_{p}/Z_{p}) ⟶^{{λ_{1} \atop ∼}} H^{1}(Γ_{1}(Nq^{r}) ∩ Γ(q), Q_{p}/Z_{p})^{B_{1}},

together with a similar isomorphism with λ^{1} replacing λ_{1} and B^{1} replacing B_{1}. We also obtain

H^{1}(Γ_{1}(Nq^{r−1}), Q_{p}/Z_{p}) ⟶^{∼} H^{1}(Γ_{1}(Nq^{r}) ∩ Γ(q), Q_{p}/Z_{p})^{Δ_{q}}.

The vanishing of H^{2}(SL_{2}(Z/q), Q_{p}/Z_{p}) can be checked by restricting to the Sylow p-subgroup which is cyclic. Note that imλ_{1}∩imλ^{1} ⊆ H^{1}(Γ_{1}(Nq^{r})∩Γ(q), Q_{p}/Z_{p})^{Δ_{q}} since B_{1} and B^{1} together generate Δ_{q}. Now consider the sequence

(2.13)  0⟶H^{1}(Γ_{1}(Nq^{r−1}), Q_{p}/Z_{p})

⟶^{res_{1}⊕−res^{1}}H^{1}(Γ_{1}(Nq^{r}), Q_{p}/Z_{p}) ⊕ H^{1}(Γ^{1}(Nq^{r}), Q_{p}/Z_{p})

⟶^{λ_{1}⊕λ^{1}}H^{1}(Γ_{1}(Nq^{r}) ∩ Γ(q), Q_{p}/Z_{p}).

We claim it is exact. To check this, suppose that λ_{1}(x) = −λ^{1}(y). Then λ_{1}(x) ∈ H^{1}(Γ_{1}(Nq^{r}) ∩ Γ(q), Q_{p}/Z_{p})^{Δ_{q}}. So λ_{1}(x) is the restriction of an x′ ∈ H^{1}(Γ_{1}(Nq^{r−1}), Q_{p}/Z_{p}) whence x − res_{1}(x′) ∈ ker λ_{1} = 0. It follows

also that y = −res^{1}(x′).

Now conjugation by the matrix ({q \atop 0} {0 \atop 1}) induces isomorphisms

Γ^{1}(Nq^{r}) ≃ Γ_{1}(Nq^{r}), Γ_{1}(Nq^{r}) ∩ Γ(q) ≃ Γ_{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 ≡ 1modNq^{r+1} is nilpotent) so will vanish when we localize at the preimage of 𝔪^{(q)} in the abstract Hecke ring generated as a polynomial ring by all the standard Hecke operators excluding T_{q}. If M ≤ 3 then the group Γ_{1}(M) has torsion. For M = 1, 2, 3 we can restrict to Γ(3), Γ(4), Γ(3), respectively, where the cohomology 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. □

Let us denote the maps X_{1}(N, q) → X_{1}(N) induced by z → z and z → qz by π_{1} and π_{2} respectively. Similarly we denote the maps X_{1}(N, q^{2}) → X_{1}(N, q) induced by z → z and z → qz by π_{3} and π_{4} respectively.

¿From the lemma (with r = 1) and Ihara’s result (2.10) we deduce that there is a sequence

(2.14)  0 → J_{1}(N) × J_{1}(N) × J_{1}(N) ⟶^{ξ} J_{1}(N, q^{2})

where ξ = (π_{1} ∘ π_{3})^{∗} × (π_{2} ∘ π_{3})^{∗} × (π_{2} ∘ π_{4})^{∗} and that the induced map of pdivisible groups becomes injective after localization at 𝔪^{(q)}’s which correspond to irreducible ρ_{𝔪}’s. By duality we obtain a sequence

J_{1}(N, q^{2}) ⟶^{ξ̂} J_{1}(N)^{3} → 0

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 a kernel of order divisible by p in J_{H}(N) → J_{1}(N). However this kernel will not meet the 𝔪^{(q)}-divisible group for any maximal ideal 𝔪^{(q)} whose associated ρ_{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 𝔪^{(q)}. We use the same notation in the general case when H ≠ 1 so ξ is the map J_{H}(N)^{3} → J_{H}(N, q^{2}).

We suppose now that 𝔪 is a maximal ideal of T_{H}(N) (as always with p ∈ 𝔪) associated to an irreducible representation and that q is a prime, p ∤ Np. We now define a maximal ideal 𝔪_{q} of T_{H}(N, q^{2}) with the same associated representation as 𝔪. To do this consider the ring

S_{1} = T_{H}(N)[U_{1}]/U_{1}(U_{1}^{2} − T_{q}U_{1} + q⟨q⟩) ⊆ End(J_{H}(N)^{3})

where the action of U_{1} on J_{H}(N)^{3} is given by the matrix

[ \begin{matrix}T_{q} & −⟨q⟩ & 0 \\ q & 0 & 0 \\ 0 & q & 0 \end{matrix} ] .

Then U_{1} satisfies the compatibility

ξ̂∘ U_{q} = U_{1} ∘ ξ̂.

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 𝔪_{1} = (U_{1}, 𝔪) in S_{1} and S_{1,𝔪_{1}} ≃ T_{H}(N)_{𝔪}. We let 𝔪_{q} denote the reciprocal image of 𝔪_{1} in T_{H}(N, q^{2}) under the natural map T_{H}(N, q^{2}) → S_{1}.

Next we define a principal ideal (Δ′_{q}) of T_{H}(N)_{𝔪} using the fact that T_{H}(N, q^{2})_{𝔪_{q}} and T_{H}(N)_{𝔪} are both Gorenstein rings (cf. Corollary 2 to Theorem 2.1). Thus we set (Δ′_{q}) = (α̂ ∘ α′) where

α′ : T_{H}(N, q^{2})_{𝔪_{q}} → S_{1,𝔪_{1}} ≃ T_{H}(N)_{𝔪}

is the natural map and α′ is the adjoint with respect to selected Hecke-module pairings on T_{H}(N, q^{2})_{𝔪_{q}} and T_{H}(N)_{𝔪}. Note that α′ 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. §1) in which T_{q} = trace(Frob q) and apply the Čebotarev density theorem.

Proposition 2.6. Suppose that frakm is a maximal ideal of T_{H}(N) associated to an irreducible ρ_{𝔪}. Suppose also that q ∤ Np. Then

(Δ′_{p}) = (q − 1)(T_{q}^{2} − ⟨q⟩(1 + q)^{2}).

Proof. We prove this in the same manner as we proved Proposition 2.4. Consider the maps on p-adic Tate-modules induced by ξ and ξ̂:

(2.15)  Ta_{p}(J_{H}(N)^{3}) ⟶^{ξ} Ta_{p}(J_{H}(N, q^{2})) \underrightarrow{ξ̂} Ta_{p}(J_{H}(N)^{3}).

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} = T_{H}(N)[U_{2}]/U_{2}(U_{2}^{2} − T_{q}U_{2} + q⟨q⟩) ⊆ End(J_{H}(N)^{3})

where U_{2} is the endomorphism of J_{H}(N)^{3} given by the matrix

[ \begin{matrix}0 & 0 & 0 \\ q & 0 & −⟨q⟩ \\ 0 & q & T_{q} \end{matrix} ] .

Then U_{q}ξ = ξU_{2} as one can verify by checking the equality (ξ̂∘ξ)U^{2} = U_{1}(ξ̂∘ξ) because ξ̂ ∘ ξ is an isogeny. The formula for ξ̂ ∘ ξ is given below. Again 𝔪_{2} = (𝔪, U_{2}) is a maximal ideal of S_{2} and S_{2,𝔪_{2}} ≃ T_{H}(N)_{𝔪}. On restricting (2.15) to the 𝔪_{2}, 𝔪_{q} and 𝔪_{1}-adic Tate modules we get

(2.16)

Ta_{𝔪_{2}}(J_{H}(N)^{3}) ⟶^{ξ} Ta_{𝔪_{q}}(J_{H}(N, q^{2})) \underrightarrow{ξ̂} Ta_{𝔪_{1}}(J_{H}(N)^{3})
↑ ≀ u_{2}  ↑ ≀ u_{1}
Ta_{𝔪}(J_{H}(N))  Ta_{𝔪}(J_{H}(N)).

The vertical isomorphisms are induced by u_{2} : z → (⟨q⟩z, −T_{q}z, qz) and u_{1} : z → (0, 0, z). Now a calculation shows that on J_{H}(N)^{3}

ξ̂∘ ξ = [ \begin{matrix}q(q + 1) & T_{q} · q & T_{q}^{2} − ⟨q⟩(1 + q) \\ T_{q}^{∗} · q & q(q + 1) & T_{q} · q \\ T_{q}^{∗2} − ⟨q⟩^{−1}(1 + q) & T_{q}^{∗} · q & q(q + 1) \end{matrix} ]

where T_{q}^{∗} = ⟨q⟩^{−1}T_{q}.

We compute then that

(u_{1}^{−1} ∘ ξ̂∘ ξ ∘ u_{2}) = −⟨q^{−1}⟩(q − 1)(T_{q}^{2} − ⟨q⟩(1 + q)^{2}).

Now using the surjectivity of ξ̂ and that ξ has torsion-free cokernel in (2.16) (by Lemma 2.5) and that Ta_{𝔪}(J_{H}(N)) and Ta_{𝔪_{q}}(J_{H}(N, q^{2})) 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. □

There is one further (and completely elementary) generalization of this result. We let π : X_{H}(Nq, q^{2}) → X_{H}(N, q^{2}) be the map given by z → z. Then π^{∗} : J_{H}(N, q^{2}) → J_{H}(Nq, q^{2}) has kernel a cyclic group and as before this will vanish when we localize at 𝔪^{(q)} if 𝔪 is associated to an irreducible representation. (As before the superscript q denotes the omission of U_{q} from the list of generators of T_{H}(Nq, q^{2}) and 𝔪^{(q)} denotes the maximal ideal of T_{H}^{(q)}(Nq, q^{2}) compatible with 𝔪.)

We thus have a sequence (not necessarily exact)

0 → J_{H}(N)^{3} ⟶^{κ} J_{H}(Nq, q^{2}) → Z → 0

where κ = π^{∗} ∘ ξ which induces a corresponding sequence of p-divisible groups which becomes exact when localized at an 𝔪^{(q)} corresponding to an irreducible ρ_{𝔪}. Here Z is the quotient abelian variety J_{H}(Nq, q^{2})/imκ. As before there is a natural surjective homomorphism

α : T_{H}(Nq, q^{2})_{𝔪_{q}} → S_{1,𝔪_{1}} ≃ T_{H}(N)_{𝔪}

where 𝔪_{q} is the inverse image of 𝔪_{1} in T_{H}(Nq, q^{2}). (We note that one can replace T_{H}(Nq, q^{2}) by T_{H}(Nq^{2}) in the definition of α and Proposition 2.7 below would still hold unchanged.) Since both rings are again Gorenstein we can define an adjoint α̂ and a principal ideal

(Δ_{q}) = (α ∘ α̂).

Proposition 2.7. Suppose that 𝔪 is a maximal ideal of T = T_{H}(N) associated to an irreducible representation. Suppose that q ∤ Np. Then

(Δ_{q}) = (q − 1)^{2}(T_{q}^{2} − ⟨q⟩(1 + q)^{2})).

The proof is a trivial generalization of that of Proposition 2.6.

Remark 2.8. We have included the operator U_{q} in the definition of T_{𝔪_{q}} = T_{H}(Nq, q^{2})_{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 T_{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 ∤ N are known (cf. [Li]). They satisfy c(q, f)^{2} = ⟨q⟩ in O_{f} (the ring of integers generated by the Fourier coefficients of f) if f is on Γ_{1}(N, q), and |c(q, f)| = q^{1/2} if f is on Γ_{1}(Nq) but not on Γ_{1}(N, q). Also when f is a newform of level dividing N the roots of x^{2} − c(q, f)x + qχ_{f}(q) = 0 have absolute value q^{1/2} where c(q, f) is the eigenvalue of T_{q} and χ_{f}(q) of ⟨q⟩. Since for f on Γ_{1}(Nq, q^{2}), U_{q}f is a form on Γ_{1}(Nq) we see that

U_{q}(U_{q}^{2} − ⟨q⟩) ∏_{f∈S_{1}}(U_{q} − c(q, f)) ∏_{f∈S_{2}} (U_{q}^{2} − c(q, f)U_{q} + q⟨q⟩) = 0

in T_{H}(Nq, q^{2}) ⊗ C where S_{1} is the set of newforms on Γ_{1}(Nq) which are not on Γ_{1}(n, q) and S_{2} is the set of newforms of level dividing N. In particular as U_{q} is in 𝔪_{q} it must be zero in T_{𝔪_{q}}.

A slightly different situation arises if 𝔪 is a maximal ideal of T = T_{H}(N, q) (q ≠ p) which is not associated to any maximal ideal of level N (in the sense of having the same associated ρ_{𝔪}). In this case we may use the map ξ_{3} = (π_{4}^{∗}, π_{3}^{∗}) to give

(2.17)  J_{H}(N, q) × J_{H}(N, q) ⟶^{ξ_{3}} J_{H}(N, q^{2}) ⟶^{ξ̂_{3}} J_{H}(N, q) × J_{H}(N, q).

Then ξ̂_{3} ∘ ξ_{3} is given by the matrix

ξ̂_{3} ∘ ξ_{3} = [ \begin{matrix}q & U_{q}^{∗} \\ U_{q} & q \end{matrix} ]

on J_{H}(N, q)^{2}, where U_{q}^{∗} = U_{q}⟨q⟩^{−1} and U_{q}^{2} = ⟨q⟩ on the 𝔪-divisible group. The second of these formulae is standard as mentioned above; cf. for example [Li, Th. 3], since ρ_{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 inner product ⟨(U_{q}^{∗}U_{q} − 1)f(rz), f(mz)⟩ is zero for any r, m|(Nq/level f) by [Li, Th. 3]. This shows that U_{q}^{∗}U_{q}f(rz), a priori a linear combination of f(m_{i}z), is equal to f(rz). So U_{q}^{∗}U_{q} = 1 on the space of forms on Γ_{H}(N, q) which are new at q, i.e. the space spanned by forms {f(sz)} where f runs through newforms with q|level f. In particular U_{q}^{∗} preserves the 𝔪-divisible group and satisfies the same relation on it, again because ρ_{𝔪} is not associated to any maximal ideal of level N.

Remark 2.9. Assume that ρ_{𝔪} is of type (A) at q in the terminology of Chapter 1, §1 (which ensures that ρ_{𝔪} does not occur at level N). In this case T_{𝔪} = T_{H}(N, q)_{𝔪} is already generated by the standard Hecke operators with the omission of U_{q}. To see this, consider the GL_{2}(T_{𝔪}) representation of Gal(\overline{Q}/Q) associated to the 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 Z_{p}-subalgebra T_{𝔪}^{tr} of T_{𝔪} generated by the traces of Frobenius elements, i.e. by the T_{ℓ} for ℓ ∤ Nqp. In particular ⟨q⟩ ∈ T_{𝔪}^{tr}. Furthermore, as T_{m}^{tr} is local and complete, and as U_{q}^{2} = ⟨q⟩, it is enough to solve X^{2} = ⟨q⟩ in the residue field of T_{𝔪}^{tr}. But we can even do this in k_{0} (the minimal field of definition of ρ_{𝔪}) by letting X be the eigenvalue of Frob q on the unique unramified rank-one free quotient of k_{0}^{2} and invoking the π_{q} ≃ π(σ_{q}) theorem of Langlands (cf. [Ca1]). (It is to ensure that the unramified quotient is free of rank one that we assume ρ_{𝔪} to be of type (A).)

We assume now that ρ_{𝔪} is of type (A) at q. Define S_{1} this time by setting

S_{1} = T_{H}(N, q)[U_{1}]/U_{1}(U_{1} − U_{q}) ⊆ End(J_{H}(N, q)^{2})

where U_{1} is given by the matrix

(2.18)  U_{1} = [ \begin{matrix}0 & q \\ 0 & U_{q} \end{matrix} ]

on J_{H}(N, q)^{2}. The map ξ̂_{3} is not necessarily surjective and to remedy this we introduce 𝔪^{(q)} = 𝔪 ∩ T_{H}^{(q)}(N, q) where T_{H}^{(q)}(N, q) is the subring of T_{H}(N, q) generated by the standard Hecke operators but omitting U_{q}. We also write 𝔪^{(q)} for the corresponding maximal ideal of T_{H}^{(q)}(Nq, q^{2}). Then on 𝔪^{(q)}-divisible groups, ξ̂_{3} and ξ̂_{3} ∘ π_{∗} are surjective and we get a natural restriction map of localization T_{H}(Nq, q^{2})_{(𝔪^{(q)})} → S_{1(𝔪^{(q)})}. (Note that the image of U_{q} under this map is U_{1} and not U_{q}.) The ideal 𝔪_{1} = (𝔪, U_{1}) is maximal in S_{1} and so also in S_{1,(𝔪^{(q)})} and we let 𝔪_{q} denote the inverse image of 𝔪_{1} under this restriction map. The inverse image of 𝔪_{q} in T_{H}(Nq, q^{2}) is also a maximal ideal which we agin write 𝔪_{q}. Since the completions T_{H}(Nq, q^{2})_{𝔪_{q}} and S_{1,𝔪_{1}} ≃ T_{H}(N, q)_{𝔪} are both Gorenstein rings (by Corollary 2 of Theorem 2.1) we can define a principal ideal (Δ_{q}) of T_{H}(N, q)_{𝔪} by

(Δ_{q}) = (α ∘ α̂)

where α : T_{H}(Nq, q^{2})_{𝔪_{q}} ↠ S_{1,𝔪_{1}} ≃ T(N, q)_{𝔪} is the restriction map induced by the restriction map on 𝔪^{(q)}-localizations described above.

Proposition 2.10. Suppose that 𝔪 is a maximal ideal of T_{H}(N, q) associated to an irreducible 𝔪 of type (A). Then

(Δ_{q}) = (q − 1)^{2}(q + 1).

Proof. The method is a straightforward adaptation of that used for Propositions 2.4 and 2.6. We let S_{2} = 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

[ \begin{matrix}U_{q} & q \\ 0 & 0 \end{matrix} ] .

This satisfies the compatability ξ_{3}U_{2} = U_{q}ξ_{3}. We define 𝔪_{2} = (𝔪, U_{2}) in S_{2} and observe that S_{2}, 𝔪_{2} ≃ T_{H}(N, q)_{𝔪}.

Then we have maps

Ta_{𝔪_{2}}(J_{H}(N, q)^{2}) ↪^{π^{∗}∘ξ_{3}} Ta_{𝔪_{q}}(J_{H}(Nq, q^{2})) ↠^{ξ̂_{3}∘π_{∗}} Ta_{𝔪_{1}}(J_{H}(N, q)^{2})
↑ ≀ v_{2}  ↑ ≀ v_{1}
Ta_{𝔪}(J_{H}(N, q))  Ta_{𝔪}(J_{H}(N, q)).

The maps v_{1} and v_{2} are given by v_{2} : z → (−qz, a_{q}z) and v_{1} : z → (z, 0) where U_{q} = a_{q} in T_{H}(N, q)_{𝔪}. One checks then that v_{1}^{−1} ∘(ξ̂_{3} ∘π_{∗})∘(π^{∗} ∘ξ_{3})∘v_{2} is equal to −(q − 1)(q^{2} − 1) or −\frac{1}{2}(q − 1)(q^{2} − 1).

The surjectivity of ξ̂_{3}∘π_{∗} on the completions is equivalent to the statement that

J_{H}(Nq, q^{2})[p]_{𝔪_{q}} → J_{H}(N, q)^{2}[p]_{𝔪_{1}}

is surjective. We can replace this condition by a similar one with 𝔪^{(q)} substituted for 𝔪_{q} and for 𝔪_{1}, i.e., the surjectivity of

J_{H}(Nq, q^{2})[p]_{𝔪^{(q)}} → J_{H}(N, q)^{2}[p]_{𝔪^{(q)}}.

By our hypothesis that ρ_{𝔪} be of type (A) at q it is even sufficient to show that the cokernel of J_{H}(Nq, q^{2})[p] ⊗ \overline{F}_{p} → J_{H}(N, q)^{2}[p] ⊗ \overline{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 π^{∗} ∘ξ_{3} on the completions and the fact that it has torsion-free cokernel also follows from Lemma 2.5 and our hypothesis that ρ_{𝔪} be of type (A) at q. □

The case that corresponds to type (B) is similar. We assume in the analysis of type (B) (and also of type (C) below) that H decomposes as ΠH_{q} as described at the beginning of Section 1. We assume that 𝔪 is a maximal ideal of T_{H}(Nq^{r}) where H contains the Sylow p-subgroup S_{p} of (Z/q^{r}Z)^{∗} and that

(2.19)  ρ_{𝔪}|_{I_{q}} ≈ ( \begin{matrix}χ_{q} \\ & 1 \end{matrix} )

for a suitable choice of basis with χ_{q} ≠ 1 and condχ_{q} = q^{r}. Here q ∤ Np and we assume also that ρ_{𝔪} is irreducible. We use the sequence

J_{H}(Nq^{r}) × J_{H}(Nq^{r}) ⟶^{(π′)^{∗}∘ξ_{2}} J_{H′}(Nq^{r}, q^{r+1}) ⟶^{ξ̂_{2}∘π′_{∗}} J_{H}(Nq^{r}) × J_{H}(Nq^{r})

defined analogously to (2.17) where ξ_{2} was as defined in Lemma 2.5 and where H′ is defined as follows. Using the notation H = ΠH_{l} as at the beginning of Section 1 set H′_{l} = H_{l} for l ≠ q and H′_{q} × S_{p} = H_{q}. Then define H′ = ΠH′_{l} and let π′ : X_{H′}(Nq^{r}, q^{r+1}) → X_{H}(Nq^{r}, q^{r+1}) be the natural map z → z. Using Lemma 2.5 we check that ξ_{2} is injective on the 𝔪^{(q)}-divisible group. Again we set S_{1} = T_{H}(Nq^{r})[U_{1}]/U_{1}(U_{1} − U_{q}) ⊆ End(J_{H}(Nq^{r})^{2}) where U_{1} is given by the matrix in (2.18). We define 𝔪_{1} = (𝔪, U_{1}) and let 𝔪_{q} be the inverse image of 𝔪_{1} in T_{H′}(Nq^{r}, q^{r+1}). The natural map (in which U_{q} → U_{1})

α : T_{H′}(Nq^{r}, q^{r+1})_{𝔪_{q}} → S_{1,𝔪_{1}} ≃ T_{H}(Nq^{r})_{𝔪}

is surjective by the following remark.

Remark 2.11. When we assume that ρ_{𝔪} is of type (B) then the U_{q} operator is redundant in T_{𝔪} = T_{H}(Nq^{r})_{𝔪}. To see this, first assume that T_{𝔪} is reduced and consider the GL_{2}(T_{𝔪}) representation of Gal(\overline{Q}/Q) associated to the 𝔪adic Tate module. Pick a σ_{q} ∈ I_{q}, the inertia group in D_{q} in Gal(\overline{Q}/Q), such that χ_{q}(σ_{q}) ≠ 1. Then because the eigenvalues of σ_{q} are distinct mod 𝔪 we can diagonalize the representation with respect to σ_{q}. If Frobq is a Frobenius in D_{q}, then in the GL_{2}(T_{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 eigenvector for σ_{q}. This is by the π_{q} ≃ π(σ_{q}) theorem of Langlands as before (cf. [Ca1]) applied to each of the representations obtained from maps T_{𝔪} → O_{f,λ}. Since the representation is defined over the Z_{p}-algebra T_{𝔪}^{tr} generated by the traces, the same reasoning applied to T_{𝔪}^{tr} shows that U_{q} ∈ T_{𝔪}^{tr}.

If T_{𝔪} is not reduced the above argument shows only that there is an operator v_{q} ∈ T_{𝔪}^{tr} such that (U_{q} − v_{q}) is nilpotent. Now T_{H}(Nq^{r}) can be viewed as a ring of endomorphisms of S_{2}(Γ_{H}(Nq^{r})), the space of cusp forms of weight 2 on Γ_{H}(Nq^{r}). There is a restriction map T_{H}(Nq^{r}) → T_{H}(Nq^{r})^{new} where T_{H}(Nq^{r})^{new} is the image of T_{H}(Nq^{r}) in the ring of endomorphisms of S_{2}(Γ_{H}(Nq^{r}))/S_{2}(Γ_{H}(Nq^{r}))^{odd}, the old part being defined as the sum of two copies of S_{2}(Γ_{H}(Nq^{r−1})) mapped via z → z and z → qz. One sees that on 𝔪-completions T_{𝔪} ≃ (T_{H}(Nq^{r})^{new})_{𝔪} since the conductor of ρ_{𝔪} is divisible by q^{r}. It follows that U_{q} ∈ T_{𝔪} satisfies an equation of the form P (U_{q}) = 0 where P (x) is a polynomial with coefficients in W (k_{𝔪}) and with distinct roots. By extending scalars to O (the integers of a local field containing W (k_{𝔪})) we can assume that the roots lie in T ≃ T_{𝔪} ⊗_{W (k_{𝔪})} O.

Since (U_{q} − v_{q}) is nilpotent it follows that P (v_{q})^{r} = 0 for some r. Then since v_{q} ∈ T_{𝔪}^{tr} which is reduced, P (v_{q}) = 0. Now consider the map T → ΠT_{(𝔭)} where the product is taken over the localizations of T at the minimal primes 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_{(𝔭)}, U_{q} = α_{i} and v_{q} = α_{j} for roots α_{i}, α_{j} of P (x) = 0 because the roots are distinct. Since U_{q} − v_{q} ∈ 𝔭 for each 𝔭 it follows that α_{i} = α_{j} for each 𝔭 whence U_{q} = v_{q} in each T_{(𝔭)}. Hence U_{q} = v_{q} in T also and this finally shows that U_{q} ∈ T_{𝔪}^{tr} in general.

We can therefore define a principal ideal

(Δ_{q}) = (α ∘ α̂)

using,as previously, that the rings T_{H′}(Nq^{r}, q^{r+1})_{𝔪_{q}} and T_{H}(Nq^{r})_{𝔪} are Gorenstein. We compute (Δ_{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 Γ_{H}(Nq^{r}) which are new at q, i.e., the space spanned by forms {f(sz)} where f runs through newforms with q^{r}|level f. To see this let f be any newform of level divisible by q^{r} and observe that the Petersson inner product ⟨(U_{q}^{∗}U_{q} − q)f(rz), f(mz)⟩ = 0 for any m|(Nq^{r}/level f) by [Li, Th. 3(ii)]. This shows that (U_{q}^{∗}U_{q} − q)f(rz), a priori a linear combination of {f(m_{i}z)}, is zero. We obtain the following result.

Proposition 2.12. Suppose that 𝔪 is a maximal ideal of T_{H}(Nq^{r}) associated to an irreducible ρ_{𝔪} of type (B) at q, i.e., satisfying (2.19) including the hypothesis that H cantains S_{p}. (Again q ∤ Np.) Then

(Δ_{q}) = ((q − 1)^{2}).

Finally we have the case where ρ_{𝔪} is of type (C) at q. We assume then that 𝔪 is a maximal ideal of T_{H}(Nq^{r}) where H contains the Sylow p-subgroup S_{p} of (Z/q^{r}Z)^{∗} and that

(2.20)  H^{1}(Q_{q}, W_{λ}) = 0

where W_{λ} is defined as in (1.6) but with ρ_{𝔪} replacing ρ_{0}, i.e., W_{λ} = ad^{0}ρ_{𝔪}.

This time we let 𝔪_{q} be the inverse image of 𝔪 in T_{H′}(Nq^{r}) under the natural restriction map T_{H′}(Nq^{r}) ⟶ T_{H}(Nq^{r}) with H′ defined as in the case of type B. We set

(Δ_{q}) = (α ∘ α̂)

where α : T_{H′}(Nq^{r})_{𝔪_{q}} ↠ T_{H}(Nq^{r})_{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 ρ_{𝔪} satisfy (2.20) but this is the case in which we will use it.)

Proposition 2.13. Suppose that 𝔪 is a maximal of T_{H}(Nq^{r}) associated to an irreducible ρ_{𝔪} with H containing the Sylow p-subgroup of (Z/q^{r}Z)^{∗}. Then

(Δ_{q}) = (q − 1).

Finally, in this section we state Proposition 2.4 in the case q ≠ p as this will be used in Chapter 3. Let q be a prime, q ∤ Np and let S_{1} denote the ring

(2.21)  T_{H}(N)[U_{1}]/{U_{1}^{2} − T_{q}U_{1} + ⟨q⟩q} ⊆ End(J_{H}(N)^{2})

where φ̂ : J_{H}(N, q) → J_{H}(N)^{2} is the map defined after (2.10) and U_{1} is the matrix

[ \begin{matrix}T_{q} & −⟨q⟩ \\ q & 0 \end{matrix} ] .

Thus, φ̂U_{q} = U_{1}φ̂. Also ⟨q⟩ is defined as ⟨n_{q}⟩ where n_{q} ≡ q(N), n_{q} ≡ 1(q). Let 𝔪_{1} be a maximal ideal of S_{1} containing the image of 𝔪, where 𝔪 is a maximal ideal of T_{H}(N) with associated irreducible ρ_{𝔪}. We will also assume that ρ_{𝔪}(Frob q) has distinct eigenvalues. (We will only need this case and it simplifies the exposition.) Let 𝔪_{q} denote the corresponding maximal ideals of T_{H}(N, q) and T_{H}(Nq) under the natural restriction maps T_{H}(Nq) → T_{H}(N, q) → S_{1}. The corresponding maps on completions are

(2.22)  T_{H}(Nq)_{𝔪_{q}} ⟶^{β} T_{H}(N, q)_{𝔪_{q}}

⟶^{α} S_{1,𝔪_{1}} ≃ T_{H}(N)_{𝔪} ⊗_{W (k_{𝔪})} W (k^{+})

where k^{+} is the extension of k_{𝔪} generated by the eigenvalues of {ρ_{𝔪}(Frob q)}. That k^{+} is either equal to k_{𝔪} or its quadratic extension. The maps β, α are surjective, the latter because T_{q} is a trace in the 2-dimensional representation to GL_{2}(T_{H}(N)_{𝔪}) given after Theorem 2.1 and hence is ‘redundant’ by the Čebotarev density theorem. The completions are Gorenstein by Corollary 2 to Theorem 2.1 and so we define invariant ideals of S_{1,𝔪_{1}}

(2.23)  (Δ) = (α ∘ α̂), (Δ′) = (α ∘ β) ∘ ({ \atop α ∘ β}).

Let α_{q} be the image of U_{1} in T_{H}(N)_{𝔪} ⊗_{W (k_{𝔪})} W (k^{+}) under the last isomorphism

in (2.22). The proof of Proposition 2.4 yields

Proposition 2.4′. Suppose that ρ_{m} is irreducible where 𝔪 is a maximal ideal of T_{H}(N) and that ρ_{𝔪}(Frob q) has distinct eigenvalues. Then

\begin{aligned}(Δ) & = (α_{q}^{2} − ⟨q⟩), \\ (Δ′) & = (α_{q}^{2} − ⟨q⟩)(q − 1). \end{aligned}

Remark. Note that if we suppose also that q ≡ 1(p) then (Δ) is the unit ideal and α is an isomorphism in (2.22).

3. The main conjectures

As we suggested in Chapter 1, in order to study the deformation theory of ρ_{0} in detail we need to assume that it is modular. That this should always be so for det ρ_{0} odd was conjectured by Serre. Serre also made a conjecture (the ‘ε’-conjecture) making precise where one could find a lifting of ρ_{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 ρ_{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(ρ_{0}) be the prime to p part of the conductor of ρ_{0} as defined for example in [Se].

Theorem 2.14. Suppose that ρ_{0} is modular and satisfies (1.1) (so in particular is irreducible) and is of type D = (·, Σ, O, M) with · = Se, str or fl. Suppose that at least one of the following conditions holds (i) p > 3 or (ii) ρ_{0} is not induced from a character of Q(\sqrt{−3}). Then there exists a newform f of weight 2 and a prime λ of O_{f} such that ρ_{f,λ} is of type D′ = (·, Σ, O′, M) for some O′, and such that (ρ_{f,λ} mod λ) ≃ ρ_{0} over \overline{F}_{p}. Moreover we can assume that f has character χ_{f} of order prime to p and has level N(ρ_{0})p^{δ(ρ_{0})} where δ(ρ_{0}) = 0 if ρ_{0}|_{D_{p}} is associated to a finite flat group scheme over Z_{p}

and det ρ_{0}|_{I_{p}} = ω, and δ(ρ_{0}) = 1 otherwise. Furthermore in the Selmer case

we can assume that a_{p}(f) ≡ χ_{2}(Frob p) mod λ in the notation of (1.2) where a_{p}(f) is the eigenvalue of U_{p}.

For the rest of this chapter we will assume that ρ_{0} is modular and that if p = 3 then ρ_{0} is not induced from a character of 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 D = {·, Σ, O, M} we will now define a Hecke ring T_{D} except where · 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 ρ_{0} is not ordinary and that det ρ_{0}|_{I_{p}} = ω. Suppose that Σ = {q_{i}} and that N(ρ_{0}) = Πq_{i}^{s_{i}} with s_{i} ≥ 0. If U_{λ} ≃ k^{2} is the representation space of ρ_{0} we set n_{q} = dim_{k}(U_{λ})^{I_{q}} where I_{q} in the inertia group at q. Define M_{0} and M by

(2.24)  M_{0} = N(ρ_{0}) ∏_{{nq_{i} =1 \atop q_{i}∉M∪\{p\}}} q_{i} · ∏_{n_{q_{i}} =2} q_{i}^{2}, M = M_{0}p^{τ(ρ_{0})}

where τ(ρ_{0}) = 1 if ρ_{0} is ordinary and τ(ρ_{0}) = 0 otherwise. Let H be the subgroup of (Z/MZ)^{∗} generated by the Sylow p-subgroup of (Z/q_{i}Z)^{∗} for each q_{i} ∈ M as well as by all of (Z/q_{i}Z)^{∗} for each q_{i} ∈ M of type (A). Let T′_{H}(M) denote the ring generated by the standard Hecke operators {T_{l} for l ∤ Mp, ⟨a⟩ for (a, Mp) = 1}. Let 𝔪′ denote the maximal ideal of T′_{H}(M) associated to the f and λ given in the theorem and let k_{𝔪′} be the residue field T′_{H}(M)/𝔪. Note that 𝔪′ does not depend on the particular choice of pair (f, λ) in theorem 2.14. Then k_{𝔪′} ≃ k_{0} where k_{0} is the smallest possible field of definition for ρ_{0} because k_{𝔪′} is generated by the traces. Henceforth we will identify k_{0} with k_{𝔪′}. There is one exceptional case where ρ_{0} is ordinary and ρ_{0}|_{D_{p}} is isomorphic to a sum of two distinct unramified characters (χ_{1} and χ_{2} in the notation of Chapter 1, §1). If ρ_{0} is not exceptional we define

(2.25(a))  T_{D} = T′_{H}(M)_{𝔪′} ⊗_{W (k_{0})} O.

If ρ_{0} is exceptional we let T′′_{H}(M) denote the ring generated by the operators {T_{l} for l ∤ Mp, ⟨a⟩ for (a, Mp) = 1, U_{p}}. We choose 𝔪′′ to be a maximal ideal of T′′_{H}(M) lying above 𝔪′ for which there is an embedding k_{𝔪′′} ↪ k (over k_{0} = k_{𝔪′}) satisfying U_{p} → χ_{2}(Frob p). (Note that χ_{2} is specified by D.) Then in the exceptional case k_{𝔪′′} is either k_{0} or its quadratic extension and we define

(2.25(b))  T_{D} = T′′_{H}(M)_{𝔪′′} ⊗_{W (k_{𝔪′′})} O.

The omission of the Hecke operators U_{q} for q|M_{0} ensures that T_{D} is reduced.

We need to relate T_{D} to a Hecke ring with no missing operators in order to apply the results of Section 1.

Proposition 2.15. In the nonexceptional case there is a maximal ideal 𝔪 for T_{H}(M) with 𝔪 ∩′_{H} (M) = 𝔪′ and k_{0} = k_{𝔪}, and such that the natural map T′_{H}(M)_{𝔪′} → T_{H}(M)_{𝔪} is an isomorphism, thus given

T_{D} ≃ T_{H}(M)_{𝔪} ⊗_{W (k_{0})} O.

In the exceptional case the same statements hold with 𝔪′′ replacing 𝔪′, T′′_{H}(M) replacing T′_{H}(M) and k_{𝔪′′} replacing k_{0}.

Proof. For simplicity we describe the nonexceptional case indicating where appropriate the slight modifications needed in the exceptional case. To construct 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 ∈ Σ − {M ∪ p}. If ρ_{0} is ordinary and f has level prime to p we also remove the Euler factor (1 − β_{p} · p^{−s}) where β_{p} is the non-unit eigenvalue in O_{fλ}. (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 Γ_{H}(M) (this is ensured by the choice of f) with O_{f,λ} coefficients. We have a corresponding homomorphism π_{f_{0}} : T_{H}(M) → O_{f,λ} and we let 𝔪 = π_{f_{0}}^{−1}(λ).

Since the Hecke operators we have used to generate T′_{H}(M) are prime to the level these is an inclusion with finite index

T′_{H}(M) ↪ ∏ O_{g}

where g runs over representatives of the Galois conjugacy classes of newforms associated to Γ_{H}(M) and where we note that by multiplicity one O_{g} can also be described as the ring of integers generated by the eigenvalues of the operators in T′_{H}(M) acting on g. If we consider T_{H}(M) in place of T′_{H}(M) we get a similar map but we have to replace the ring O_{g} by the ring

S_{g} = O_{g}[X_{q_{1}}, …, X_{q_{r}}, X_{p}]/{Y_{i}, Z_{p}}_{i=1}^{r}

where {p, p_{1}, …, q_{r}} are the distinct primes dividing Mp. Here

(2.26)  Y_{i} = \{ \begin{matrix}X_{q_{i}}^{r_{i}−1}(X_{q_{i}} − α_{q_{i}}(g))(X_{q_{i}} − β_{q_{i}}(g)) & if q_{i} ∤ level(g) \\ X_{q_{i}}^{r_{i}}(X_{q_{i}} − a_{q_{i}}(g)) & if q_{i}| level(g), \end{matrix}

where the Euler factor of g at q_{i} (i.e., of its associated L-series) is (1−α_{q_{i}}(g)q_{i}^{−s})(1−β_{q_{i}}(g)q_{i}^{−s}) in the first cases and (1−a_{q_{i}}(g)q_{i}^{−s}) in the second case, and q_{i}^{r_{i}}||(M/level(g)). (We allow a_{q_{i}}(g) to be zero here.) Similarly Z_{p} is defined by

Z_{p} = \{ \begin{matrix}X_{p}^{2} − a_{p}(g)X_{p} + pχ_{g}(p) & if p|M, p ∤ level(g) \\ X_{p} − a_{p}(g) & if p ∤ M \\ X_{p} − a_{p}(g) & if p|level(g), \end{matrix}

where the Euler factor of g at p is (1 − a_{p}(g)p^{−s} + χ_{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

(2.27)  \begin{aligned}T′_{H}(M) & ⊂⟶ & ∏_{g} O_{g} \\ ∩ ↓ & & ∩ ↓ \\ T_{H}(M) & ⊂⟶ & ∏_{g} S_{g} = ∏_{g} O_{g}[X_{q_{1}}, …, X_{q_{r}}, X_{p}]/\{Y_{i}, Z_{p}\}_{i=1}^{r} \end{aligned}

where the lower map is given on {U_{q}, U_{p} or T_{p}} by U_{q_{i}} ⟶ X_{q_{i}}, U_{p} or T_{p} ⟶ X_{p} (according as p|M or p ∤ M). To verify the existence of such a homomorphism one considers the action of T_{H}(M) on the space of forms of weight 2 invariant under Γ_{H}(M) and uses that ∑_{j=1}^{r} g_{j}(m_{j}z) is a free generator as a T_{H}(M) ⊗ C-module where {g_{j}} runs over the set of newforms and m_{j} = M/level(g_{j}).

Now we tensor all the rings in (2.27) with Z_{p}. Then completing the top row of (2.27) with respect to 𝔪′ and the bottom row with respect to 𝔪 we get a commutative diagram

(2.28)  \begin{aligned}T′_{H}(M)_{𝔪′} & ⊂⟶ & ( ∏_{g} O_{g})_{𝔪′} & ≃ & ∏_{{g \atop 𝔪′→μ}} O_{g,μ} \\ ↓ & & ↓ & & ↓ \\ T_{H}(M)_{𝔪} & ⊂⟶ & ( ∏_{g} S_{g})_{𝔪} & ≃ & ∏(S_{g})_{𝔪}. \end{aligned}

Here μ runs through the primes above p in each O_{g} for which 𝔪′ → μ under T_{H′}(M) → O_{g}. Now (S_{g})_{𝔪} is given by

(2.29) (S_{g} ⊗ Z_{p})_{𝔪} ≃ ((O_{g} ⊗ Z_{p})[X_{q_{1}}, …, X_{q_{r}}, X_{p}]/{Y_{i}, Z_{p}}_{i=1}^{r})_{𝔪}

≃ ( ∏_{μ|p} O_{g,μ}[X_{q_{1}}, …, X_{q_{r}}, X_{p}]/{Y_{i}, Z_{p}}_{i=1}^{r})_{𝔪}

≃ ( ∏_{μ|p} A_{g,μ})_{m}

where A_{g,μ} denotes the product of the factors of the complete semi-local ring O_{g,μ}[X_{q_{1}}, …, X_{q_{r}}, X_{p}]/{Y_{i}, Z_{p}}_{i=1}^{r} in which X_{q_{i}} is topologically nilpotent for q_{i} ∉ 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}} ∈ 𝔪 if q_{i} ∉ M and U_{p} is a unit at 𝔪 in the ordinary case.

Now if 𝔪′ → μ then in (A_{g,μ})_{μ} we claim that Y_{i} is given up to a unit by X_{q_{i}} − b_{i} for some b_{i} ∈ O_{g,μ} with b_{i} = 0 if q_{i} ∉ M. Similarly Z_{p} is given up to a unit by X_{p} − α_{p}(g) where α_{p}(g) is the unit root of x^{2} − a_{p}(g)x + pχ_{g}(p) = 0 in O_{g,μ} if p ∤ level g and p|M and by X_{p} − a_{p}(g) if p|level g or p ∤ M. This will show that (A_{g,μ})_{𝔪} ≃ O_{g,μ} when 𝔪′ → μ and (A_{g,μ})_{𝔪} = 0 otherwise.

For q_{i} ∈ M and for p, the claim is straightforward. For q_{i} ∉ M, it amounts to the following. Let U_{g,μ} denote the 2-dimensional K_{g,μ}-vector space with Galois action via ρ_{g,μ} and let n_{q_{i}}(g, μ) = dim(U_{g,μ})^{I_{q_{i}}}. We wish to check that Y_{i} = unit.X_{q_{i}} in (A_{g,μ})_{𝔪} and from the definition of Y_{i} in (2.26) this reduces to checking that r_{i} = n_{q_{i}}(g, μ) by the π_{q} ≃ π(σ_{q}) of theorem (cf. [Ca1]). We use here that α_{q_{i}}(g), β_{q_{i}}(g) and a_{q_{i}}(g) are p-adic units when they are nonzero since they are eigenvalues of Frob(q_{i}). Now by definition the power of q_{i} dividing M is the same as that dividing N(ρ_{0})q_{i}^{n_{q_{i}}} (cf. (2.21)). By an observation of Livné (cf. [Liv], [Ca2,§1]),

(2.30)  ord_{q_{i}}(level g) = ord_{q_{i}}(N(ρ_{0})q_{i}^{n_{q_{i}} −n_{q_{i}} (g,μ)}).

As by definition q_{i}^{r_{i}}||(M/level g) we deduce that r_{i} = n_{q_{i}}(g, μ) as reqired.

We have now shown that each A_{g,μ} ≃ O_{g,μ} (when 𝔪′ → μ) and it follows from (2.28) and (2.29) that we have homomorphisms

T′_{H}(M)_{𝔪′} ⊂⟶T_{H}(M)_{𝔪} ⊂⟶ ∏_{{g \atop 𝔪′→μ}} O_{g,μ}

where the inclusions are of finite index. Moreover we have seen that U_{q_{i}} = 0 in T_{H}(M)_{𝔪} for q_{i} ∉ M. We now consider the primes q_{i} ∈ M. We have to show that the operators U_{q} for q ∈ M are redundant in the sense that they lie in T′_{H}(M)_{𝔪′}, i.e., in the Z_{p}-subalgebra of T_{H}(M)_{𝔪} generated by the {T_{l} : l ∤ Mp, ⟨a⟩ : a ∈ (Z/MZ)^{∗}}. For q ∈ M of type (A), U_{q} ∈ T′_{H}(M)_{𝔪′} as explained in Remark 2.9 are for q ∈ M of type (B), U_{q} ∈ T′_{H}(M)_{𝔪′} as explained in Remark 2.11. For q ∈ M of type (C) but not of type (A), U_{q} = 0 by the π_{q} ≃ π(σ_{q}) theorem (cf. [Ca1]). For in this case n_{q} = 0 whence also n_{q}(g, μ) = 0 for each pair (g, μ) with 𝔪′ → μ. If ρ_{0} is strict or Selmer at p then U_{p} can be recovered from the two-dimensional representation ρ (described after the corollaries to Theorem 2.1) as the eigenvalue of Frobp on the (free, of rank one) unramified quotient (cf. Theorem 2.1.4 of [Wi4]). As this representation is defined over the 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 T′′_{H}(M)_{𝔪′′} by definition.

Finally we have to show that T_{p} is also redundant in the sense explained above when p ∤ M. A proof of this has already been given in Section 2 (Ribet’s lemma). Here we give an alternative argument using the Galois representations. We know that T_{p} ∈ 𝔪 and it will be enough to show that T_{p} ∈ (𝔪^{2}, p). Writing k_{𝔪} for the residue field T_{H}(M)_{𝔪}/𝔪 we reduce to the following situation. If T_{p} ∉ (𝔪^{2}, p) then there is a quotient

T_{H}(M)_{𝔪}/(𝔪^{2}, p) ↠ k_{𝔪}[ε] = T_{H}(M)_{𝔪}/𝔞

where k_{𝔪}[ε] is the ring of dual numbers (so ε^{2} = 0) with the property that T_{p} ↦ λε with λ ≠ 0 and such that the image of T′_{H}(M)_{𝔪′} lies in k_{𝔪}. Let G_{/Q} denote the four-dimensional k_{𝔪}-vector space associated to the representation

ρ_{ε} : Gal(\overline{Q}/Q) ⟶ GL_{2}(k_{𝔪}[ε])

induced from the representation in Theorem 2.1. It has the form

G_{/Q} ≃ G_{0}_{/Q} ⊕ G_{0}_{/Q}

where G_{0} is the corresponding space associated to ρ_{0} by our hypothesis that the traces lie in k_{𝔪}. The semisimplicity of G_{/Q} here is obtained from the main theorem of [BLR]. Now G_{/Q_{p}} extends to a finite flat group scheme G_{/Z_{p}}. Explicitly it is a quotient of the group scheme J_{H}(M)_{𝔪}[p]_{/Z_{p}}. Since extensions to Z_{p} are unique (cf. [Ray1]) we know

G_{/Z_{p}} ≃ G_{0}_{/Z_{p}} ⊕ G_{0}_{/Z_{p}}.

Now by the Eichler-Shimura relation we know that in J_{H}(M)_{/F_{p}}

T_{p} = F + ⟨p⟩F^{T} .

Since T_{p} ∈ 𝔪 it follows that F +⟨p⟩F^{T} = 0 on G_{0}_{/F_{p}} and hence the same holds on G_{/F_{p}}. But T_{p} is an endomorphism of G_{/Z_{p}} which is zero on the special fibre, so by [Ray1, Cor. 3.3.6], T_{p} = 0 on G_{/Z_{p}}. It follows that T_{p} = 0 in k_{𝔪}[ε] which contradicts our earlier hypothesis. So T_{p} ∈ (𝔪^{2}, p) as required. This completes the proof of the proposition. □

¿From the proof of the proposition it is also clear that 𝔪 is the unique maximal ideal of T_{H}(M) extending 𝔪′ and satisfying the conditions that U_{q} ∈ 𝔪 for q ∈ Σ − {M ∪ p} and U_{p} ∉ 𝔪 if ρ_{0} is ordinary. For the rest of this chapter we will always make this choice of 𝔪 (given ρ_{0}).

Next we define T_{D} in the case when D = (ord, Σ, O, M). If 𝔫 is any ordinary maximal ideal (i.e. U_{p} ∉ 𝔫) of T_{H}(Np) with N prime to p then Hida has constructed a 2-dimensional Noetherian local Hecke ring

T_{∞} = eT_{H}(Np^{∞})_{𝔫} := lim_{⟵} eT_{H}(Np^{r})_{𝔫_{r}}

which is a Λ = Z_{p}[[T]]-algebra satisfying T_{∞}/T ≃ T_{H}(Np)_{𝔫}. Here 𝔫_{r} is the inverse image of 𝔫 under the natural restriction map. Also T = lim_{⟵}⟨1+Np⟩−1 and e = lim ; r^{⟶} U_{p}^{r!}. For an irreducible ρ_{0} of type D we have defined T_{D′} in

(2.25(a)), where D′ = (Se, Σ, O, M) by

T_{D′} ≃ T_{H}(M_{0}p)_{𝔪} ⊗_{W (k_{𝔪})} O,

the isomorphism coming from Proposition 2.15. We will define T_{D} by

(2.31)  T_{D} = eT_{H}(M_{0}p^{∞})_{𝔪} ⊗_{W (k_{𝔪})} O.

In particular we see that

(2.32)  T_{D}/T ≃ T_{D′},

i.e., where D′ is the same as D but with ‘Selmer’ replacing ‘ord’. Moreover if 𝔮 is a height one prime ideal of T_{D} containing ((1 + T)^{p^{n}} − (1 + Np)^{p^{n}(k−2)}) for any integers n ≥ 0, k ≥ 2, then T_{D}/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 Λ-adic modular forms see for example [Wi1] or [Hi1].

For each n ≥ 1 let T_{n} = T_{H}(M_{0}p^{n})_{𝔪_{n}}. Then by the argument given after the statement ofTheorem 2.1 we can construct a Galois representation ρ_{n} unramified outside Mp with values in GL_{2}(T_{n}) satisfying traceρ_{n}(Frobl) = T_{l}, det ρ_{n}(Frob l) = l⟨l⟩ for (l, Mp) = 1. These representations can be patched together to give a continuous representation

(2.33)  ρ = lim_{⟵} ρ_{n} : Gal(Q_{Σ}/Q) ⟶ GL_{2}(T_{D})

where Σ is the set of primes dividing Mp. To see this we need to check the commutativity of the maps

\begin{matrix}R_{Σ} ⟶ T_{n} \\ ↘ ↓ \\ T_{n−1} \end{matrix}

where the horizontal maps are induced by ρ_{n} and ρ_{n−1} and the vertical map is the natural one. Now the commutativity is valid on elements of R_{Σ}, which are traces or determinants in the universal representation, since trace(Frobl) ↦ T_{l} under both horizontal maps and similarly for determinants. Here R_{Σ} is the universal deformation ring described in Chapter 1 with respect to ρ_{0} viewed with residue field k = k_{𝔪}. It suffices then to show that R_{Σ} is generated (topologically) by traces and this reduces to checking that there are no nonconstant deformations of ρ_{0} to k[ε] with traces lying in k (cf. [Ma1, §1.8]). For then if R_{Σ}^{tr} denotes the closed W (k)-subalgebra of R_{Σ} generated by the traces we see that R_{Σ}^{tr} → (R_{Σ}/m^{2}) is surjective, m being the maximal ideal of R_{Σ}, from which we easily conclude that R_{Σ}^{tr} = R_{Σ}. To see that the condition holds, assume that a basis is chosen so that ρ_{0}(c) = ({1 \atop 0} {0 \atop −1}) for a chosen complex cunjugation c and ρ_{0}(σ) = ({a_{σ} \atop c_{σ}} {b_{σ} \atop d_{σ}}) with b_{σ} = 1 and c_{σ} ≠ 0 for some σ. (This is possible because ρ_{0} is irreducible.) Then any deformation [p] to k[ε] can be represented by a representation ρ such that ρ(c) and ρ(σ) have the same properties. It follows easily that if the traces of ρ lie in k then ρ takes values in k whence it is equal to ρ_{0}. (Alternatively one sees that the universal representation can be defined over R_{Σ}^{tr} by diagonalizing complex conjugation as before. Since the two maps R_{Σ}^{tr} → T_{n−1} induced by the triangle are the same, so the associated 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 D = {·, Σ, O, M} where · is not unrestricted there is then a canonical surjective map

φ_{D} : R_{D} → T_{D}

which induces the representations described after the corollaries to Theorem 2.1 and in (2.33). It is enough to check this when O = W (k_{0}) (or W (k_{𝔪′′}) in the exceptional case). Then one just has to check that for every pair (g, μ) which appears in (2.28) the resulting representation is of type D. For then we claim that the image of the canonical map R_{D} → T̃_{D} = ΠO_{g,μ} is T_{D} where here ∼ denotes the normalization. (In the case where · is ord this needs to be checked instead for T_{n} ⊗_{W (k_{0})} O for each n.) For this we just need to see that R_{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 ∈ R_{D} where u is the eigenvalue of Frob p on the unique rank one unramified quotient of R_{D}^{2} with eigenvalue ≡ χ_{2}(Frobp) which is specified in the definition of D.) But we saw above that this was true for R_{Σ}. The same then holds for R_{D} as R_{Σ} → R_{D} is surjective because the map on reduced cotangent spaces is surjective (cf. (1.5)). To check the condition on the pairs (g, μ) observe first that for q ∈ M we have imposed the following conditions on the level and character of such g’s by our choice of M and H:

q of type (A): q|| level g, det ρ_{g,μ}|_{I_{q}} = 1,

q of type (B): cond χ_{q}|| level g, det ρ_{g,μ}|_{I_{q}} = χ_{q},

q of type (C): det ρ_{g,μ}|_{I_{q}} is the Teichmüller lifting of det ρ_{0}|_{I_{q}}.

In the first two cases the desired form of ρ_{q,μ}|_{D_{q}} then follows from the

π_{q} ≃ π(σ_{q}) theorem of Langlands (cf. [Ca1]). The third case is already of 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 D = (ord, Σ, W (k_{0}), ϕ); cf. [MT].

Conjecture 2.16. φ_{D} is an isomorphism.

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 𝔪. We remind the reader that throughout this section we are assuming that if p = 3 then ρ_{0} is not induced from a character of Q(\sqrt{−3} ).

Remark. The case of most interest to us is when p = 3 and ρ_{0} is a representation with values in GL_{2}(F_{3}). In this case it is a theorem of Tunnell, extending results of Langlands, that ρ_{0} is always modular. For GL_{2}(F_{3}) is a double cover of S_{4} and can be embedded in GL_{2}(Z[\sqrt{−2} ]) whence in GL_{2}(C); cf. [Se] and [Tu]. The conjecture will be proved with a mild restriction on ρ_{0} at the end of Chapter 3.

Remark. Our original restriction to the types (A), (B), (C) for ρ_{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 Q to an extension unramified at Σ, ρ_{0} ⊗ ψ has smaller ‘conductor’ for some character ψ. The typical example of this is where ρ_{0}|_{D_{q}} = Ind_{K}^{Q_{q}}(χ) with q ≡ −1(p) and

χ is a ramified character over K, the unramified quadratic extension of Q_{q}. What makes this difficult for us is that there are then nontrivial ramified local deformations (Ind_{K}^{Q_{p}}χξ for ξ a ramified character of order p of K) which we cannot detect by a change of level.

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 D = (·, Σ, O, M) is a standard deformation problem (associated to ρ_{0}) with · = Se, str or fl and suppose that H, M_{0}, M and 𝔪 are defined as in (2.24) and Proposition 2.15. We choose a finite set of primes Q = {q_{1}, …, q_{r}} with q_{i} ∤ Mp. Furthermore we assume that each q_{i} ≡ 1(p) and that the eigenvalues {α_{i}, β_{i}} of ρ_{0}(Frob q_{i}) are distinct for each q_{i} ∈ Q. This last condition ensures that ρ_{0} does not occur as the residual representation of the λ-adic representation associated to any newform on Γ_{H}(M, q_{1} … q_{r}) where any q_{i} divides the level of the form. This can be seen directly by looking at (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 O contains α_{i}, β_{i} for each q_{i}.

Pick α_{i} for each i. We let D_{Q} be the deformation problem associated to representations ρ of Gal(Q_{Σ∪Q}/Q) which are of type D and which in addition satisfy the property that at each q_{i} ∈ Q

(2.34)  ρ|_{D_{q_{i}}} ∼ ( \begin{matrix}χ_{1,q_{i}} \\ & χ_{2,q_{i}} \end{matrix} )

with χ_{2,q_{i}} unramified and χ_{2,q_{i}}(Frob q_{i}) ≡ α_{i} mod 𝔪 for a suitable choice of basis. One checks as in Chapter 1 that associated to 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(ρ_{0})p^{δ(ρ_{0})} where δ(ρ_{0}) in as defined in Theorem 2.14. Let 𝔪_{0} denote a maximal ideal of T_{H}(N) given by Theorem 2.14 with the property that ρ_{𝔪_{0}} ≃ ρ_{0} over \overline{F}_{p} relative to a suitable embedding of k_{𝔪_{0}} → k over k_{0}. (In the exceptional case we also impose the same condition on 𝔪_{0} about the reduction of U_{p} as in the definition of T_{D} in the exceptional case before (2.25)(b).) Thus ρ_{𝔪_{0}} ≃ ρ_{f,λ} mod λ over the residue field of O_{f,λ} for some choice of f and λ 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 𝔪_{Q} of T_{H}(Nq_{1} … q_{r}) with the property that ρ_{𝔪_{Q}} ≃ ρ_{0} over \overline{F}_{p} relative to a suitable embedding k_{𝔪_{Q}} → k over k_{𝔪_{0}}. The field k_{𝔪_{Q}} is the extension of k_{0} (or k_{𝔪′′} in the exceptional case) generated by the α_{i}, β_{i}. We set

(2.35)  T_{Q} = T_{H}(Nq_{1} … q_{r})_{𝔪_{Q}} ⊗_{W (k_{𝔪_{Q}})} O.

It is easy to see directly (or by the arguments of Proposition 2.15) that T_{Q} is reduced and that there is an inclusion with finite index

(2.36)  Q_{Q} ↪ T̃_{Q} = ∏ O_{g,μ}

where the product is taken over representatives of the Galois conjugacy classes of eigenforms g of level Nq_{1} … q_{r} with 𝔪_{Q} → μ. Now define D_{Q} using the choices α_{i} for which U_{q_{i}} → α_{i} under the chosen embedding k_{𝔪_{Q}} → k. Then each of the 2-dimensional representations associated to each factor O_{g,μ} is of type D_{Q}. We can check this for each q ∈ Q using either the π_{q} ≃ π(σ_{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 O-algebras R_{Q} → T̃_{Q} and hence also an O-algebra map

(2.37)  φ_{Q} : R_{Q} → T_{Q}

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(ρ_{0})p, we check that U_{q} is in the image of φ_{Q} using the arguments in the second half of the proof of Proposition 2.15. For q ∈ Q we use the fact that U_{q} is the image of the value of χ_{2,q}(Frob q) in the universal representation;cf. (2.34). For q|M, but not of the previous types, T_{q} is a trace in ρ_{T_{Q}} and we can apply the Čebotarev density theorem to show that it is in the image of φ_{Q}.

Finally, if there is a section π : T_{Q} → O, then set 𝔭_{Q} = ker π and let ρ_{𝔭} denote the 2-dimensional representation to GL_{2}(O) obtained from ρ_{T_{Q}} mod 𝔭_{Q}. Let V = Adρ_{𝔭} ⊗_{O} K/O where K is the field of fractions of O. We pick a basis

for ρ_{𝔭} satisfying (2.34) and then let

(2.38)  \begin{aligned}V^{(q_{i})} & = \{ ( \begin{matrix}a & 0 \\ 0 & 0 \end{matrix} ) \} \\ ⊆ Adρ_{𝔪} ⊗_{O} K/O & = \{ ( \begin{matrix}a & b \\ c & d \end{matrix} ) : a, b, c, d ∈ O\} ⊗_{O} K/O \end{aligned}

and let V_{(q_{i})} = V/V^{(q_{i})}. Then as in Proposition 1.2 we have an isomorphism

(2.39)  Hom_{O}(𝔭_{R_{Q}}/𝔭_{R_{Q}}^{2}, K/O) ≃ H_{D_{Q}}^{1}(Q_{Σ∪Q}/Q, V)

where 𝔭_{R_{Q}} = ker(π ∘ φ_{Q}) and the second term is defined by

(2.40) H_{D_{Q}}^{1}(Q_{Σ∪Q}/Q, V) = ker : H_{D}^{1}(Q_{Σ∪Q}/Q, V) → ∏_{i=1}^{r} H^{1}(Q_{q_{i}}^{unr}, V_{(q_{i})}).

We return now to our discussion of Conjecture 2.16. We will call a deformation theory D minimal if Σ = M ∪ {p} and · is Selmer, strict or flat. 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 ρ_{0} has an associated minimal D even when ρ_{0} is flat or ordinary at p as explained in the remarks after Conjecture 2.16. However this could be achieved if one replaced Q by a suitable finite extension depending on ρ_{0}.

Suppose now that f is a (normalized) newform, λ is a prime of O_{f} above p and ρf, λ a deformation of ρ_{0} of type D where D = (·, Σ, O_{f,λ}, M) with · = Se, str or fl. (Strictly speaking we may be changing ρ_{0} as we wish to choose its field of definition to be k = O_{f,λ}/λ.) Suppose further that level(f)|M where M is defined by (2.24).

Now let us set O = O_{f,λ} for the rest of this section. There is a homomorphism

(2.41)  π = π_{D,f} : T_{D} → O

whose kernel is the prime ideal 𝔭_{T,f} associated to f and λ. Similary there is a homomorphism

R_{D} → O

whose kernel is the prime ideal 𝔭_{R,f} associated to f and λ and which factors through π_{f}. Pick perfect pairings of O-modules, the second one T_{D}-bilinear,

(2.42)  O × O → O, ⟨ , ⟩ : T_{D} × T_{D} → O.

In each case we use the term perfect pairing to signify that the pairs of induced maps O → Hom_{O}(O, O) and T_{D} → Hom_{O}(T_{D}, O) are isomorphisms. In addition the second one is required to be T_{D}-linear. The existence of the second 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 T_{D}-module Hom_{O}(T_{D}, O) we set ⟨t_{1}, t_{2}⟩ = h(t_{1}t_{2}).

A priori T_{H}(M)_{𝔪} (occurring in the description of T_{D} in Proposition 2.15) is only Gorenstein as a Z_{p}-algebra but it follows immediately that it is also a Gorenstein W (k_{𝔪})-algebra. (The notion of Gorenstein O-algebra is explained in the appendix.) Indeed the map

Hom_{W (k_{𝔪})}(T_{H}(M)_{𝔪}, W (k_{𝔪})) → Hom_{Z_{p}}(T_{H}(M)_{,}Z_{p})

given by φ ↦ trace ∘ φ is easily seen to be an isomorphism, as the reduction mod p is injective and the ranks are equal. Thus T_{D} is a Gorenstein O-algebra.

Now let π̂ : O → T_{D} be the adjoint of π with respect to these pairings. Then define a principal ideal (η) of T_{D} by

(η) = (η_{D,f}) = (π̂(1)).

This is well-defined independently of the pairings and moreover one sees that T_{D}/η is torsion-free (see the appendix). From its description (η) is invariant under extensions of O to O′ in an obvious way. Since T_{D} is reduced π(η) ≠ 0.

One can also verify that

(2.43)  π(η) = ⟨η, η⟩

up to a unit in O.

We will say that D_{1} ⊃ D if we obtain D_{1} by relaxing certain of the hypotheses on D, i.e., if D = (·, Σ, O, M) and D_{1} = (·, Σ_{1}, O_{1}, M_{1}) we allow that Σ_{1} ⊃ Σ, any O_{1}, M ⊃ M_{1} (but of the same type) and if · is Se or str in D it can be Se, str, ord or unrestricted in D_{1}, if · is fl in D_{1} it can be fl or unrestricted in D_{1}. We use the term restricted to signify that · is Se, str, fl or ord. The following theorem reduces conjecture 2.16 to a ‘class number’ criterion. For an interpretation of the right-hand side of the inequality in 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.

Theorem 2.17. Assume, as above, that ρ_{f,λ} is a deformation of ρ_{0} of type D = (·, Σ, O = O_{f,λ}, M) with · = Se, str or fl. Suppose that

#O/π(η_{D,f}) ≥ #𝔭_{R,f}/𝔭_{R,f}^{2}.

Then

(i) φ_{D_{1}} : R_{D_{1}} ≃ T_{D_{1}} is an isomorphism for all (restricted) D_{1} ⊃ D.

(ii) T_{D_{1}} is a complete intersection (over O_{1} if · is Se, str or fl) for all restricted D_{1} ⊃ D.

Proof. Let us write T for T_{D}, 𝔭_{T} for 𝔭_{T,f}, 𝔭_{R} for 𝔭_{R,f} and η for η_{η,f}. Then we always have

(2.44)  #O/η ≤ #𝔭_{T}/𝔭_{T}^{2}.

(Here and in what follows we sometimes write η for π(η) if the context makes this reasonable.) This is proved as follows. T/η acts faithfully on 𝔭_{T}. Hence the Fitting ideal of 𝔭_{T} as a T/η-module is zero. The same is then true of 𝔭_{T}/𝔭_{T}^{2} as an O/η = (T/η)/𝔭_{T}-module. So the Fitting ideal of 𝔭_{T}/𝔭_{T}^{2} as an O-module is contained in (η) and the conclusion is then easy. So together with the hypothesis of the theorem we get inequality (and hence equalities)

#O/π(η) ≥ #𝔭_{R}/𝔭_{R}^{2} ≥ #𝔭_{T}/𝔭_{T}^{2} ≥ #O/π(η).

By Proposition 2 of the appendix T is a complete intersection over O. Part (ii) of the theorem then follows for D. Part (i) follows for D from Proposition 1 of the appendox.

We now prove inductively that we can deduce the same inequality

(2.45)  #O_{1}/η_{D_{1},f} ≥ #𝔭_{R_{1},f}/𝔭_{R_{1},f}^{2}

for D_{1} ⊃ D and R_{1} = R_{D_{1}}. The above argument will then prove the theorem for D_{1}. We explain this first in the case D_{1} = D_{q} where D_{q} differs from D only in replacing Σ by Σ∪{q}. Let us write T_{q} for T_{D_{q}}, 𝔭_{R,q} for 𝔭_{R,f} with R = R_{D_{q}} and η_{q} for η_{D_{q},f}. We recall that U_{q} = 0 in T_{q}.

We choose isomorphisms

(2.46)  T ≃ Hom_{O}(T, O), T_{q} ≃ Hom_{O}(T_{q}, O)

coming from the fact that each of the rings is a Gorenstein O-algebra. If α_{q} : T_{q} → T is the natural map we may consider the element Δ_{q} = α_{q} ∘α̂_{q} ∈ T where the adjoint is with respect to the above isomorphisms. Then it is clear that

(2.47)  (α_{q}(η_{q})) = (ηΔ_{q})

as principal ideals of T. In particular π(η_{q}) = π(ηΔ_{q}) in O.

Now it follows from Proposition 2.7 that the principal ideal (Δ_{q}) is given by

(2.48)  (Δ_{q}) = ((q − 1)^{2}(T_{q}^{2} − ⟨q⟩(1 + q)^{2})).

In the statement of Proposition 2.7 we used Z_{p}-pairings

T ≃ Hom_{Z_{p}}(T, Z_{p}), T_{q} ≃ Hom_{Z_{p}}(T_{q}, Z_{p})

to define (Δ_{q}) = (α_{q} ∘ α̂_{q}). However, using the description of the pairings as W (k_{𝔪})-algebras derived from these Z_{p}-pairings in the paragraph following (2.42) we see that the ideal (Δ_{q}) is unchanged when we use W (k_{𝔪})-algebra pairings, and hence also when we extend scalars to O as in (2.42).

On the other hand

#𝔭_{R,q}/𝔭_{R,q}^{2} ≤ #𝔭_{R}/𝔭_{R}^{2} · #{O/(q − 1)^{2}(T_{q}^{2} − ⟨q⟩(1 + q)^{2})}

by Propositions 1.2 and 1.7. Combining this with (2.47) and (2.48) gives (2.45).

If M ≠ ϕ we use a similar argument to pass from D to D_{q} where this time D_{q} signifies that D is unchanged except for dropping q from M. In each of types (A), (B), and (C) one checks from Propositions 1.2 and 1.8 that

#𝔭_{R,q}/𝔭_{R,q}^{2} ≤ #𝔭_{R}/𝔭_{R}^{2} · #H^{0}(Q_{q}, V^{∗}).

This is in agreement with Propositions 2.10, 2.12 and 2.13 which give the corresponding change in η by the method described above.

To change from an O-algebra to an O_{1}-algebra is straightforward (the complete intersection property can be checked using [Ku1, Cor. 2.8 on p. 209]), 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 ∈ 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. □

Remark 2.18. If we suppose in the Selmer case that f has level N with p ∤ N we can also consider the ring T_{H}(M_{0})_{𝔪_{0}} (with M_{0} as in (2.24) and 𝔪_{0} defined in the same way as for T_{H}(M)). This time set

T_{0} = T_{H}(M_{0})_{𝔪_{0}} ⊗_{W (k_{𝔪_{0}} )} O, T = T_{H}(M)_{𝔪} ⊗_{W (k_{𝔪})} O.

Define η_{0}, η, 𝔭_{0} and 𝔭 with respect to these rings, and let (Δ_{p}) = α_{p} ∘ α̂_{p} where α_{p} : T → T_{0} and the adjoint is taken with respect to O-pairings on T and T_{0}. We then have by Proposition 2.4

(2.49)  (η_{p}) = (η · Δ_{p}) = (η · (T_{p}^{2} − ⟨p⟩(1 + p)^{2})) = (η · (a_{p}^{2} − ⟨p⟩))

as principal ideals of T, where a_{p} is the unit root of x^{2} − T_{p}x + p⟨p⟩ = 0.

Remark. For some earlier work on how deformation rings change with Σ see [Bo].

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.

Estimates for the Selmer group

Let ρ_{0} : Gal(Q_{Σ}/Q) → GL_{2}(k) be an odd irreducible representation which we will assume is modular. Let D be a deformation theory of type (·, Σ, O, M) such that ρ_{0} is type D, where · is Selmer, strict or flat. We remind the reader that k is assumed to be the residue field of O. Then as explained in Theorem 2.14, we can pick a modular lifting ρ_{f,λ} of ρ_{0} of type D (altering k if necessary and replacing O by a ring containing O_{f,λ}) provided that ρ_{0} is not induced from a character of Q(\sqrt{−3} ) if p = 3. For the rest of this chapter, we will make the assumption that ρ_{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 ρ_{0}|_{D_{p}} is associated to a finite flat group scheme over Z_{p} and det ρ_{0}|_{I_{p}} = ω.

In Chapter 2, Section 3, we defined a ring T_{D} associated to D. Here we make a slight modification of this ring. In the case where · is Selmer and ρ_{0}|_{D_{p}} is associated to a finite flat group scheme and det ρ_{0}|_{I_{p}} = ω we set

(3.1)  T_{D_{0}} = T′_{H}(M_{0})_{𝔪′_{0}} ⊗_{W (k_{0})} O

with M_{0} as in (2.24), H defined following (2.24) (it is actually a subgroup of (Z/M_{0}Z)^{∗}) and 𝔪′_{0} the maximal ideal of T′_{H}(M_{0}) associated to ρ_{0}. The same proof as in Proposition 2.15 ensures that there is a maximal ideal 𝔪_{0} of T_{H}(M_{0}) with 𝔪_{0} ∩ T′_{H}(M_{0}) = 𝔪′_{0} and such that the natural map

(3.2)  T_{D_{0}} = T′_{H}(M_{0})_{𝔪′_{0}} ⊗_{W (k_{0})} O → T_{H}(M_{0})_{𝔪_{0}} ⊗_{W (k_{0})} O

is an isomorphism. The maximal ideal 𝔪_{0} which we choose is characterized by the properties that ρ_{𝔪_{0}} = ρ_{0} and U_{q} ∈ 𝔪_{0} for q ∈ Σ − M ∪ {p}. (The value of T_{p} or of U_{q} for q ∈ M is determined by the other operators; see the proof of Proposition 2.15.) We now define T_{D_{0}} in general by the following:

T_{D_{0}} is given by (3.1) if · is Se and ρ_{0}|_{D_{p}} is associated

to a finite flat group scheme over Z_{p} and det ρ_{0}|_{I_{p}} = ω;

(3.3)

T_{D_{0}} = T_{D} if · is str or fl, or ρ_{0}|D_{p} is not associated

to a finite flat group scheme over Z_{p}, or det ρ_{0}|_{I_{p}} ≠ ω.

We choose a pair (f, λ) of minimum level and character as given by Theorem 2.14 and this gives a homomorphism of O-algebras

π_{f} : T_{CalD_{0}} → O ⊇ O_{f,λ}.

We set 𝔭_{T,f} = ker π_{f} and similarly we let 𝔭_{R,f} denote the inverse image of 𝔭_{T,f} in R_{D}. We define a principal ideal (η_{T,f}) of T_{D_{0}} by taking an adjoint π̂_{f} of π_{f} with respect to parings as in (2.42) and write

η_{T,f} = (π̂_{f}(1)).

Note that 𝔭_{T,f}/𝔭_{T,f}^{2} is finite and π_{f}(η_{T,f}) ≠ 0 because T_{D_{0}} is reduced. We also write η_{T,f} for π_{f}(η_{𝔗,f}) if the context makes this usage reasonable. We let V_{f} = Ad ρ_{𝔭} ⊗_{O} K/O where ρ_{𝔭} is the extension of scalars of ρ_{f,λ} to O.

Theorem 3.1. Assume that D is minimal, i.e., ∑ = M ∪ {p}, and that

ρ_{0} is absolutely irreducible when restricted to Q(\sqrt{(−1)^{\frac{p−1}{2}} p} ). Then

(i) #H_{D}^{1}(Q_{Σ}/Q, V_{f}) ≤ #(𝔭_{T,f}/𝔭_{T,f}^{2})^{2} · c_{p}/#(O/η_{T,f})

where c_{p} = #(O/U_{p}^{2} − ⟨p⟩) < ∞ when ρ_{0} is Selmer and ρ_{0}|_{D_{p}} is associated to a finite flat group scheme over Z_{p} and det ρ_{0}|_{I_{p}} = ω, and c_{p} = 1 otherwise;

(ii) if T_{D_{0}} is a complete intersection over O then (i) is an equality, R_{D} ≃ T_{D} and T_{D} is a complete intersection.

In general, for any (not necessarily minimal) D of Selmer, strict or flat type, and any ρ_{f,λ} of type D, #H_{D}^{1}(Q_{Σ}/Q, V_{f}) < ∞ if ρ_{0} is as above.

Remarks. The finiteness was proved by Flach in [Fl] under some restrictions on f, p and 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_{D}^{1}(Q_{Σ}/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 D’s associated to ρ_{0} under the condition that a minimal D exists associated to ρ_{0}. This is stated in Theorem 3.3.

The case where the projective representation associated to ρ_{0} is dihedral does not always have the property that a twist of it has an associated minimal D. In the case where the associated quadratic field is imaginary we will give a different argument in Chapter 4.

Proof. We will assume throughout the proof that 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 Σ satisfying q ≡ 1(p) and ρ_{0}(Frobq) having distinct eigenvalues for each q ∈ Q. For the minimal deformation problem D = (·, Σ, O, M), let D_{Q} be the deformation problem described before (2.34); i.e., it is the refinement of (·, Σ ∪ Q, O, M) obtained by imposing the additional restriction (2.34) at each q ∈ Q. (We will assume for the proof that O is chosen so O/λ = k contains the eigenvalues of ρ_{0}(Frob q) for each q ∈ Q.) We set

T = T_{D_{0}}, R = R_{D}

and recall the definition of T_{Q} and R_{Q} from Chapter 2, §3 (cf. (2.35)). We write V for V_{f} and recall the definition of V_{(q)} following (2.38). Also remember that 𝔪_{Q} is a maximal ideal of T_{H}(Nq_{1} … q_{r}) as in (2.35) for which ρ_{𝔪_{Q}} ≃ ρ_{0} over F̄_{p} (recall that this uses the same choice of embedding k_{𝔪_{Q}} ⟶ k as in the definition of T_{Q}). We use 𝔪_{Q} also to denote the maximal ideal of T_{Q} if the context makes this reasonable.

Consider the exact and commutative diagram

0 → H_{D}^{1}(Q_{Σ}/Q, V) → H_{D_{Q}}^{1}(Q_{Σ∪Q}/Q, V) →^{δ_{Q}} ∏_{q∈Q} H^{1}(Q_{q}^{unr}, V^{(q)})^{Gal(Q_{q}^{unr}/Q_{q})}
|≀  |≀
0 →  (𝔭_{R}/𝔭_{R}^{2})^{∗}  →  (𝔭_{R_{Q}}/𝔭_{R_{Q}}^{2})^{∗}  ↑ι_{Q}
↑  ↑
0 →  (𝔭_{T}/𝔭_{T}^{2})^{∗}  →  (𝔭_{T_{Q}}/𝔭_{T_{Q}}^{2})^{∗}  →^{u_{Q}} K_{Q} → 0

where K_{Q} is by definition the cokernel in the horizontal sequence and ∗ denotes Hom_{O}( , K/O) for K the field of fractions of O. The key result is:

Lemma 3.2. The map ι_{Q} is injective for any finite set of primes Q satisfying

q ≡ 1(p), T_{q}^{2} ≢ ⟨q⟩(1 + q)^{2}mod 𝔪 f or all q ∈ Q.

Proof. Note that the hypotheses of the lemma ensure that ρ_{0}(Frob q) has distinct eigenvaluesw for each q ∈ Q. First, consider the ideal 𝔞_{Q} of R_{Q} defined by

(3.4) 𝔞_{Q} = \{a_{i} −1, b_{i}, c_{i}, d_{i} −1 : ( \begin{matrix}a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix} ) = ρ_{D_{Q}}(σ_{i}) with σ_{i} ∈ I_{q_{i}}, q_{i} ∈ Q\}.

Then the universal property of R_{Q} shows that R_{Q}/𝔞_{Q} ≃ R. This permits us to identify (𝔭_{R}/𝔭_{R}^{2})^{∗} as

(𝔭_{R}/𝔭_{R}^{2})^{∗} = {f ∈ (𝔭_{R_{Q}}/𝔭_{R_{Q}}^{2})^{∗} : f(𝔞_{Q}) = 0}.

If we prove the same relation for the Hecke rings, i.e., with T and T_{Q} replacing R and R_{Q} then we will have the injectivity of ι_{Q}. We will write 𝔞̄_{Q} for the image of 𝔞_{Q} in T_{Q} under the map φ_{Q} of (2.37).

It will be enough to check that for any q ∈ Q′, Q′ a subset of Q, T_{Q′}/𝔞̄_{q} ≃ T_{Q′−{q}} where 𝔞_{q} is defined as in (3.4) but with Q replaced by q. Let N′ = N(ρ_{0})p^{δ(ρ_{0})} · ∏_{q_{i}∈Q′−{q}} q_{i} where δ(ρ_{0}) is as defined in Theorem 2.14. Then take an element σ ∈ I_{q} ⊆ Gal(Q̄_{q}/Q_{q}) which restricts to a generator of Gal(Q(ζ_{N′q}/Q(ζ_{N′})). Then det(σ) = ⟨t_{q}⟩ ∈ T_{Q′} in the representation to GL_{2}(T_{Q′}) defined after Theorem 2.1. (Thus t_{q} ≡ 1(N′) and t_{q} is a primitive root mod q.) It is easily checked that

(3.5)  J_{H}(N′.q)_{𝔪_{Q′}} (Q̄) ≃ J_{H}(N′q)_{𝔪_{Q′}} (Q̄)[⟨t_{q}⟩ − 1].

Here H is still a subgroup of (Z/M_{0}Z)^{∗}. (We use here that ρ_{0} is not reducible for the injectivity and also that ρ_{0} is not induced from a character of Q(\sqrt{−3} ) for the surjectivity when p = 3. The latter is to avoid the ramification points of the covering X_{H}(N′q) → 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

(3.6)  (T_{H}(N′q)_{𝔪_{Q′}})^{2}/(⟨t_{q}⟩ − 1) ≃ (T_{H}(N′, q)_{𝔪_{Q′}})^{2}.

The hypotheses of the lemma imply the condition that ρ_{0}(Frob q) has distinct eigenvalues. So applying Proposition 2.4’ (at the end of §2) and the remark following it (or using the fact remarked in Chapter 2, §3 that this condition implies that ρ_{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_{𝔪_{Q′}}) with O, the right-hand side of (3.6) can be replaced by T_{Q′−{q}}^{2} thus giving

T_{Q′/𝔞̄_{q}}^{2} ≃ T_{Q′−{q}}^{2},

since ⟨t_{q}⟩ − 1 ∈ 𝔞̄_{q}. Repeated inductively this gives the desired relation T_{Q}/𝔞̄_{Q} ≃ T, and completes the proof of the lemma. □

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 §2)

η_{T_{Q,f}} /η_{T,f} = ∏_{q∈Q}(q − 1),

the factors (α_{q}^{2} − ⟨q⟩) being units by our hypotheses on q ∈ 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))

\begin{aligned}#(𝔭_{T}/𝔭_{T}^{2}) & ≥ #(O/η_{T_{f}}) \\ #(𝔭_{T_{Q}}/𝔭_{T_{Q}}^{2} & ≥ #(O/η_{T_{Q,f}}). \end{aligned}

We dedeuce that

#K_{Q} ≥ #(O/ ∏_{q∈Q}(q − 1)) · t^{−1}

where t = #(𝔭_{T}/𝔭_{T}^{2})/#(O/η_{T,f}). Since the range of ι_{Q} has order given by

#{O/ ∏_{q∈Q}(q − 1)},

we compute that the index of the image of ι_{Q} is ≤ t as ι_{Q} is injective.

Keeping our assumption on Q from Lemma 3.2, consider the kernel of λ^{M} applied to the diagram at the beginning of the proof of the theorem. Then with M chosen large enough so that λ^{M} annihilates 𝔭_{T}/𝔭_{T}^{2} (which is finite because T is reduced) we get:

0 → H_{D}^{1}(Q_{Σ}/Q, V [λ^{M}]) → H_{D_{Q}}^{1}(Q_{Σ∪Q}/Q, V [λ^{M}]) →^{δ_{Q}} ∏_{q∈Q}H^{1}(Q_{q}^{unr}, V^{(q)}[λ^{M}])^{Gal(Q_{q}^{unr}/Q_{q})}

↑  ↑ ψ_{Q}  ↑ ι_{Q}
0 → (𝔭_{T}/𝔭_{T}^{2})^{∗}  → (𝔭_{T_{Q}}/𝔭_{T_{Q}}^{2})^{∗}[λ^{M}]  →  K_{Q}[λ^{M}] → (𝔭_{T}/𝔭_{T}^{2})^{∗}.

See (1.7) for the justification that λ^{M} can be taken inside the parentheses in the first two terms. Let X_{Q} = ψ_{Q}((𝔭_{T_{Q}}/𝔭_{T_{Q}}^{2})^{∗}[λ^{M}]). Then we can estimate the order of δ_{Q}(X_{Q}) using the fact that the image if ι_{Q} has index at most t. We get

(3.7)  #δ_{Q}(X_{Q}) ≥ ( ∏_{q∈Q} #O/(λ^{M}, q − 1)) · (1/t) · (1/#(𝔭_{T}/𝔭_{T}^{2})).

Now we choose Q to be a set of primes with the property that

(3.8)  ε_{Q} : H_{D^{∗}}^{1}(Q_{Σ}/Q, V_{λ^{M}}^{∗}) → ∏_{q∈Q} H^{1}(Q_{q}, V_{λ^{M}}^{∗})

is injective. We also keep the condition that ι_{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 ≡ 1(p^{M}).

To see that this can be done, suppose that x ∈ ker ε_{Q} and λx = 0 but x ≠ 0. We have a commutative diagram

H^{1}(Q_{Σ}/Q, V_{λ^{M}}^{∗} [λ] →^{ε_{Q}}  ∏_{q∈Q} H^{1}(Q_{q}, V_{λ^{M}}^{∗})[λ]
|≀  |≀
H^{1}(Q_{Σ}/Q, V_{λ}^{∗}) →^{ε̄_{Q}}  ∏_{q∈Q} H^{1}(Q_{q}, V_{λ}^{∗})

the left-hand isomorphisms coming from our particular choices of q’s and the left-hand isomorphism from our hypothesis on ρ_{0}. The same diagram will hold if we replace Q by Q_{0} = Q ∪ {q_{0}} and we now need to show that we can choose q_{0} so that ε̄_{Q_{0}}(x) ≠ 0.

The restriction map

H^{1}(Q_{Σ}/Q, V_{λ}^{∗}) → Hom(Gal(Q̄/K_{0}(ζ_{p})), V_{λ}^{∗})^{Gal(K_{0}(ζ_{p})/Q)}

has kernel H^{1}(K_{0}(ζ_{p})/Q, k(1)) by Proposition 1.11 where here K_{0} is the splitting field of ρ_{0}. Now if x ∈ H^{1}(K_{0}(ζ_{p})/Q, k(1)) and x ≠ 0 then p = 3 and x factors through an abelian extension L of Q(ζ_{3}) of exponent 3 which is nonabelian over Q. In this exeptional case, L must ramify at some prime 𝔮 of Q(ζ_{3}), and if 𝔮 lies over the rational prime q ≠ 3 then the composite map

H^{1}(K_{0}(ζ_{3})/Q, k(1)) → H^{1}(Q_{q}^{unr}, k(1)) → H^{1}(Q_{q}^{unr}, (O/λ^{M})(1))

is nonzero on x. But then x is not of type D^{∗} which gives a contradiction. This only leaves the possibility that L = Q(ζ_{3}, \sqrt[3]{3}) but again this means that x is not of type D^{∗} as locally at the prime above 3, L is not generated by the cube root of a unit over Q_{3}(ζ_{3}). This argument holds whether or not D is minimal.

So x, which we view in ker ε̄_{Q}, gives a nontrivial Galois-equivalent homomorphism f_{x} ∈ Hom(Gal(Q̄/K_{0}(ζ_{p})), V_{λ}^{∗}) which factors through an abelian extension M_{x} of K_{0}(ζ_{p}) of exponent p. Specifically we choose M_{x} to be the minimal such extension. Assume first that the projective representation ρ̃_{0} associated to ρ_{0} is not dihedral so that Sym^{2}ρ_{0} is absolutely irreducible. Pick a σ ∈ Gal(M_{x}(ζ_{p^{M}})/Q) satisfying

(3.9) (i) ρ_{0}(σ) has order m ≥ 3 with (m, p) = 1,

(ii) σ fixes Q(det ρ_{0})(ζ_{p^{M}}),
(iii) f_{x}(σ^{m}) ≠ 0.

To show that this is possible, observe first that the first two conditions can be achieved by Lemma 1.10(i) and the subsequent remark. Let σ_{1} be an element satisfying (i) and (ii) and let σ̄_{1} denote its image in Gal(K_{0}(ζ_{p})/Q). Then ⟨σ̄_{1}⟩ acts on G = Gal(M_{x}/K_{0}(ζ_{p})) and under this action G decomposes as G ≃ G_{1} ⊕ G′_{1} where σ_{1} acts trivially on G_{1} and without fixed points on G′_{1}. If X is any irreducible Galois stable k̄-subspace of f_{x}(G) ⊗_{F_{p}} k̄ then ker(σ_{1} − 1)|_{X} ≠ 0 since Sym^{2}ρ_{0} is assumed absolutely irreducible. So also ker(σ_{1} − 1)|_{f_{x}(G)} ≠ 0 and thus we can find τ ∈ G_{1} such that f_{x}(τ) ≠ 0. Viewing τ as an element of G we then take

τ_{1} = τ × 1 ∈ Gal(M_{x}(ζ_{p^{M}})/K_{0}(ζ_{p})) ≃ G × Gal(K_{0}(ζ_{p^{M}})/K_{0}(ζ_{p}))

(This decomposition holds because M_{x} is minimal and because Sym^{2}ρ_{0} and μ_{p} are distinct from the trivial representation.) Now τ_{1} commutes with σ_{1} and either f_{x}((τ_{1}σ_{1})^{m}) ≠ 0 or f_{x}(σ_{1}^{m}) ≠ 0. Since ρ_{0}(τ_{1}σ_{1}) = ρ_{0}(σ_{1}) this gives (3.9) with at least one of σ = τ_{1}σ_{1} or σ = σ_{1}. We may then choose q_{0} so that Frobq_{0} = σ and we will then have ε̄_{Q_{0}}(x) ≠ 0. Note that conditions (i) and (ii) imply that q_{0} ≡ 1(p) and also that ρ_{0}(σ) has distinct eigenvalues, thus giving both the hypothses of Lemma 3.2.

If on the other hand ρ̃_{0} is dihedral then we pick σ’s satisfying

(i) ρ̃_{0}(σ) ≠ 1,

(ii) σ fixes Q(ζ_{p^{M}}),

(iii) f_{x}(σ^{m}) ≠ 0,

with m the order of ρ_{0}(σ) (and p ∤ m since ρ̃_{0} is dihedral). The first two conditions can be achieved using Lemma 1.12 and, in addition, we can assume that σ takes the eigenvalue 1 on any given irreducible Galois stable subspace X of W_{λ} ⊗ k̄. Arguing as above, we find a τ ∈ G_{1} such that f_{x}(τ) ≠ 0 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 ε_{Q} is injective.

We have thus shown that we can choose Q = {q_{1}, …, q_{r}} to be a finite set of primes q_{i} ≡ 1(p^{M}) satisfying the hypotheses of Lemma 3.2 as well as the injectivity of ε_{Q} in (3.8). By Proposition 1.6, the injectivity of ε_{Q} implies that

(3.10)  #H_{D}^{1}(Q_{Σ∪Q}/Q, V_{g}[λ^{M}]) = h_{∞} · ∏_{q∈Σ∪Q} h_{q}.

Here we are using the convention explained after Proposition 1.6 to define H_{D}^{1}. Now, as D was chosen to be minimal, h_{q} = 1 for q ∈ ∑ −{p} by Proposition 1.8. Also, h_{q} = #(O/λ^{M})^{2} for q ∈ Q. If · is str or fl then h_{∞}h_{p} = 1 by Proposition 1.9 (iv) and (v). If · is Se, h_{∞}h_{p} ≤ c_{p} by Proposition 1.9 (iii). (To compute this we can assume that I_{p} acts on W_{λ}^{0} via ω, as otherwise we get h_{∞}h_{p} ≤ 1. Then with this hypothesis, (W_{λ^{n}}^{0})^{∗} is easily verified to be unramified with Frob p acting as U_{p}^{2}⟨p⟩^{−1} by the description of ρ_{f,λ}|_{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 D_{Q}. We also have classes in

Hom(Gal(Q_{Σ∪Q}/Q), O/λ^{M}) = H^{1}(Q_{Σ∪Q}/Q, O/λ^{M}) ↪ H^{1}(Q_{Σ∪Q}/Q, V_{λ^{M}})

coming from the cyclotomic extension Q(ζ_{q_{1}} … ζ_{q_{r}}). These are of type D and disjoint from the classes obtained from (3.7). Combining these with (3.10) gives

#H_{D}^{1}(Q_{Σ}/Q, V_{f}[λ^{M}]) ≤ t · #(𝔭_{T}/𝔭_{T}^{2}) · c_{p}

as required. This proves part (i) of Theorem 3.1.

Now if we assume that 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_{D} ≃ T_{D} by Proposition 1 of the appendix together with Proposition 1.2. In the Selmer case we get

(3.11)  #(𝔭_{T}/𝔭_{T}^{2}) · c_{p} = #(O/η_{T,f})c_{p} = #(O/η_{T_{D,f}}) ≤ #(𝔭_{T_{D}}/𝔭_{T_{D}}^{2})

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, T_{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 ρ_{f,λ} is multiplicative type at q then ρ_{f,λ|_{Dq}} does not split.) □

Remark. The ring T_{D_{0}} defined in (3.1) and used in this chapter should be the deformation ring associated to the following deformation prblem D_{0}. One alters 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 ρ of ρ_{0} to GL_{2}(A) has the property that for any quotient A/𝔞 of finite order, ρ|_{D_{p}}mod 𝔞 is the Galois representation associated to the Q̄_{p}-points of a finite flat group scheme over Z_{p}. (Of course, ρ_{0} is ordinary here in contrast to our usual assumption for flat deformations.)

¿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 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 Selmer case) and Proposition 1.2. Strictly speaking, Theorem 1 of [TW] refers to a slightly smaller class of D’s than those covered by Theorem 3.1 but up to a twist every such D is covered. It is straightforward to see that it is enough to check Theorem 3.3 for ρ_{0} up to a suitable twist.

Theorem 3.3. Assume that ρ_{0} is modular and absolutely irreducible

when restricted to Q(\sqrt{(−1)^{\frac{p−1}{2}} p} ). Assume also that ρ_{0} is of type (A), (B) or (C) at each q ≠ p in Σ. Then the map φ_{D} : R_{D} → T_{D} of Conjecture 2.16 is an isomorphism for all D associated to ρ_{0}, i.e., where D = (·, Σ, O, M) with · = Se, str, fl or ord. In particular if · = Se, str or fl and f is any newform for which ρ_{f,λ} is a deformation of ρ_{0} of type D then

#H_{D}^{1}(Q_{Σ}/Q, V_{f}) = #(O/η_{D,f}) < ∞

where η_{D,f} is the invariant defined in Chapter 2 prior to (2.43).

The condition at q ≠ p in Σ ensures that there is a minimal D associated to ρ_{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 ρ_{0} as in Theorem 0.2 satisfies the hypotheses of Theorem 3.3.

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 the η-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.

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 η-invariant.

We assume that

(4.1)  ρ = Ind_{L}^{Q}κ : Gal(Q̄/Q) → GL_{2}(O)

is the p-adic representation associated to a character κ : Gal(\overline{L}/L) → O^{×} of an imaginary quadratic field L. We assume that p is unramified in L and that κ factors through an extension of L whose Galois group has the form A ≃ Z_{p} ⊕T where T is a finite group of order prime to p. The ring O is assumed to be the ring of integers of a local field with maximal ideal λ and we also assume that ρ is a Selmer deformation of ρ_{0} = ρ mod λ which is supposed irreducible with det ρ_{0}|_{I_{p}} = ω. In particular it follows that p splits in L, p = 𝔭𝔭̄ say, and that precisely one of κ, κ^{∗} is ramified at 𝔭 (κ^{∗} being the character τ → κ(στσ^{−1}) for any σ representing the nontrivial coset in Gal(Q̄/Q)/Gal(Q̄/L)). We can suppose without loss of generality that κ is ramified at 𝔭.

We consider the representation module V ≃ (K/O)^{4} (where K is the field of fractions of O) and the representation is Ad ρ. In this case V splits as

V ≃ Y ⊕ (K/O)(ψ) ⊕ K/O

where ψ is the quadratic character of Gal(Q̄/Q) associated to L. We let Σ denote a finite set of primes including all those which ramify in ρ (and in particular p). Our aim is to compute H_{Se}^{1}(Q_{Σ}/Q, V). The decomposition of V gives a corresponding decomposition of H^{1}(Q_{Σ}/Q, V) and we can use it to define H_{Se}^{1}(Q_{Σ}/Q, Y). Since W^{0} ⊂ Y (see Chapter 1 for the definition of W^{0}) we can define H_{Se}^{1}(Q_{Σ}/Q, Y) by

H_{Se}^{1}(Q_{Σ}/Q, Y) = ker{H^{1}(Q_{Σ}/Q, Y) → H^{1}(Q_{p}^{unr}, Y/W^{0})}.

Let Y^{∗} be the arithmetic dual of Y, i.e., Hom(Y, μ_{p^{∞}}) ⊗ Q_{p}/Z_{p}. Where ν for κε/κ^{∗} and let L(ν) be the splitting field of ν. Then we claim that Gal(L(ν)/L) ≃ Z_{p} ⊕ T′ with T′ a finite group of order prime to p. For this it is enough to show that χ = κκ^{∗}/ε factors through a group of order prime to p since ν = κ^{2}χ^{−1}. Suppose that χ has order m = m_{0}p^{r} with (m_{0}, p) = 1. Then χ^{m_{0}} extends to a character of Q which is then unramified at p since the same is true of χ. Also it factors through an abelian extension of L with Galois group isomorphic to Z_{p}^{2} since χ factors through such an extension with Galois group isomorphic to Z_{p}^{2} ⊕ T_{1} with T_{1} of order prime to p (the composite of the splitting fields of κ and κ^{∗}). It follows that χ^{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^{∗} ≃ (K/O)(ν) ⊕ (K/O)(ν^{−1}ε^{2}).

In analogy to the above we define H_{Se}^{1}(Q_{Σ}/Q, Y^{∗}) by

H_{Se}^{1}(Q_{Σ}/Q, Y^{∗}) = ker{H^{1}(Q_{Σ}/Q, Y^{∗}) → H^{1}(Q_{p}^{unr}, (W^{0})^{∗})}.

Analogous definitions apply if Y^{∗} is replaced by Y_{λ^{n}}^{∗}. Also we say informally that a cohomology class is Selmer at p if it vanishes in H^{1}(Q_{p}^{unr}, (W^{0})^{∗}) (resp.

H^{1}(Q_{p}^{unr}, (W_{λ^{n}}^{0})^{∗})). Let M_{∞} be the maximal abelian p-extension of L(ν) unramified outside 𝔭. The following proposition generalizes [CS, Prop. 5.9].

Proposition 4.1. There is an isomorphism

H_{unr}^{1}(Q_{Σ}/Q, Y^{∗}) →^{∼} Hom(Gal(M_{∞}/L(ν)), (K/O)(ν))^{Gal(L(ν)/L)}

where H_{unr}^{1} denotes the subgroup of classes which are Selmer at p and unramified everywhere else.

Proof. The sequence is obtained from the inflation-restriction sequence as follows. First we can replace H^{1}(Q_{Σ}/Q, Y^{∗}) by

{H^{1}(Q_{Σ}/L, (K/O)(ν)) ⊕ H^{1}(Q_{Σ}/L, (K/O)(ν^{−1}ε^{2}))}^{Δ}

where Δ = Gal(L/Q). The unramified condition then translates into the requirement that the cohomology class should lie in

{H_{unr in Σ−𝔭}^{1}(Q_{Σ}/L, (K/O)(ν)) ⊕ H_{unr in Σ−𝔭^{∗}}^{1}(Q_{Σ}/L, (K/O)(ν^{−1}ε^{2}))}^{Δ}.

Since Δ interchanges the two groups inside the parentheses it is enough to compute the first of them, i.e.,

(4.2)  H_{unr in Σ−𝔭}^{1}(Q_{Σ}/L, K/O(ν)).

The inflation-restriction sequence applied to this gives an exact sequence

(4.3)  0 → H_{unr in Σ−𝔭}^{1}(L(ν)/L, (K/O)(ν))
→ H_{unr in Σ−𝔭}^{1}(Q_{Σ}/L, (K/O)(ν))
→ Hom(Gal(M_{∞}/L(ν)), (K/O)(ν))^{Gal(L(ν)/L)}.

The first term is zero as one easily check using the divisibility of (K/O)(ν). Next note that H^{2}(L(ν)/L, (K/O)(ν)) is trivial. If ν ≢ 1(λ) this is straightforward (cf. Lemma 2.2 of [Ru1]). If ν ≡ 1(λ) then Gal(L(ν)/L) ≃ 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}(Q_{Σ}/L, (K/O)(ν)). Let L_{0} be the splitting field of Y_{λ}^{∗}. Then M_{∞}L_{0}/L_{0} is unramified outside 𝔭 and L_{0}/L has degree prime to p. It follows that c is unramified outside 𝔭. □

Now write H_{str}^{1}(Q_{Σ}/Q, Y_{n}^{∗}) (where Y_{n}^{∗} = Y_{λ^{n}}^{∗} and similarly for Y_{n}) for the supgroup of H_{unr}^{1}(Q_{Σ}/Q, Y_{n}^{∗}) given by

H_{str}^{1}(Q_{Σ}/Q, Y_{n}^{∗}) = {α ∈ H_{unr}^{1}(Q_{Σ}/Q, Y_{n}^{∗}) : α_{p} = 0 in H^{1}(Q_{p}, Y_{n}^{∗}/(Y_{n}^{∗})^{0})}

where (Y_{n}^{∗})^{0} is the first step in the filtration under D_{p}, thus equal to (Y_{n}/Y_{n}^{0})^{∗} or equivalently to (Y^{∗})_{λ^{n}}^{0} where (Y^{∗})^{0} is the divisible submodule of Y^{∗} on which the action of I_{p} is via ε^{2}. (If p ≠ 3 one can characterize (Y_{n}^{∗})^{0} as the maximal submodule on which I_{p} acts via ε^{2}.) A similar definition applies with Y_{n} replacing Y_{n}^{∗}. It follows from an examination of the action of I_{p} on Y_{λ} that

(4.4)  H_{str}^{1}(Q_{Σ}/Q, Y_{n}) = H_{unr}^{1}(Q_{Σ}/Q, Y_{n}).

In the case of Y^{∗} we will use the inequality

(4.5)  #H_{str}^{1}(Q_{Σ}/Q, Y^{∗}) ≤ #H_{unr}^{1}(Q_{Σ}/Q, Y^{∗}).

We also need the fact that for n sufficiently large the map

(4.6)  H_{str}^{1}(Q_{Σ}/Q, Y_{n}^{∗}) → H_{str}^{1}(Q_{Σ}/Q, Y^{∗})

is injective. One can check this by replacing these groups by the subgroups of H^{1}(L, (K/O)(ν)_{λ^{n}}) and H^{1}(L, (K/O)(ν)) which are unramified outside 𝔭 and trivial at 𝔭^{∗}, in a manner similar to the beginning of the proof of Proposition 4.1. the above map is then injective whenever the connecting homomorphism

H^{0}(L_{𝔭^{∗}}, (K/O)(ν)) → H^{1}(L_{𝔭^{∗}}, (K/O)(ν)_{λ^{n}})

is injective, which holds for sufficiently large n.

Now, by Propsition 1.6,

(4.7)  \frac{#H_{str}^{1}(Q_{Σ}/Q, Y_{n})}{#H_{str}^{1}(Q_{Σ}/Q, Y_{n}^{∗})} = #H^{0}(Q_{p}, (Y_{n}^{0})^{∗})\frac{#H^{0}(Q, Y_{n})}{#H^{0}(Q, Y_{n}^{∗})}.

Also, H^{0}(Q, Y_{n}) = 0 and a simple calculation shows that

#H^{0}(Q, Y_{n}^{∗}) = \{ \begin{matrix}inf_{𝔮} #(O/1 − ν(𝔮)) & if ν = 1 mod λ \\ 1 & otherwise \end{matrix}

where 𝔮 runs through a set of primes of O_{L} prime to p cond(ν) of density one. This can be checked since Y^{∗} = Ind_{L}^{Q}(ν) ⊗_{O} K/O. So, setting

(4.8)  t = \{ \begin{matrix}inf_{𝔮}#(O/(1 − ν(𝔮))) & if ν mod λ = 1 \\ 1 & if ν mod λ ≠ 1 \end{matrix}

we get (4.9)

#H_{Se}^{1}(Q_{Σ}/Q, Y) ≤ \frac{1}{t} · ∏_{∈Σ} ℓ_{q} · #Hom(Gal(M_{∞}/L(ν)), (K/O)(ν))^{Gal(L(ν)/L)}

where ℓ_{q} = #H^{0}(Q_{q}, Y^{∗}) for q ≠ p, ℓ_{p} = lim_{n→∞} #H^{0}(Q_{p}, (Y_{n}^{0})^{∗}). This follows from Proposition 4.1, (4.4)-(4.7) and the elementary estimate

(4.10)  #(H_{Se}^{1}(Q_{Σ}/Q, Y)/H_{unr}^{1}(Q_{Σ}/Q, Y)) ≤ ∏_{q∈Σ−{p}} ℓ_{q},

which follows from the fact that #H^{1}(Q_{q}^{unr}, Y)^{Gal(Q_{q}^{unr}/Q_{q})} = ℓ_{q}.

Our objective is to compute H_{Se}^{1}(Q_{Σ}/Q, V) and the main problem is to estimate H_{Se}^{1}(Q_{Σ}/Q, Y). By (4.5) this in turn reduces to the problem of estimating #Hom(Gal(M_{∞}/L(ν)), (K/O)(ν))^{Gal(L(ν)/L)}). This order can be computed using the ‘main conjecture’ established by Rubin using ideas of Kolyvagin. (cf.

[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_{𝔣} denote the number of roots of unity ζ of L such that ζ ≡ 1 mod 𝔣 (𝔣 an integral ideal of O_{L}). We choose an 𝔣 prime to p such that w_{𝔣} = 1. Then there is a grossencharacter φ of L satisfying φ((α)) = α for α ≡ 1 mod 𝔣 (cf. [de Sh, II.1.4]). According to Weil, after fixing an embedding Q̄ ↪ Q̄_{p} we can asssociate a p-adic character φ_{p} to φ (cf. [de Sh, II.1.1 (5)]). We choose an embedding corresponding to a prime above 𝔭 and then we find φ_{p} = κ · χ for some χ of finite order and conductor prime to p. Indeed φ_{p} and κ are both unramified at 𝔭^{∗} and satisfy φ_{p}|_{I_{𝔭}} = κ|_{I_{𝔭}} = ε where ε is the cyclotomic character and I_{𝔭} is an inertia group at 𝔭. Without altering 𝔣 we can even choose φ so that the order of χ is prime to p. This is by our hyppothesis that κ factored through an extension of the form Z_{p} ⊕ T with T of order prime to p. To see this pick an abelian splitting field of φ_{p} and κ whose Galois group has the form G ⊕ G′ with G a pro-p-group and G′ of order prime to p. Then we see that φ|_{G} has conductor dividing 𝔣𝔭^{∞}. Also the only primes which ramify in a Z_{p}- extension lie above p so our hypothesis on κ ensures that κ|_{G} has conductor dividing 𝔣𝔭^{∞}. The same is then true of the p-part of χ which therefore has conductor dividing 𝔣. We can therefore adjust φ so that χ has order prime to p as claimed. We will not however choose φ so that χ is 1 as this would require 𝔣𝔭^{∞} to be divisible by cond χ. However we will make the assumption, by altering 𝔣 if necessary, but still keeping 𝔣 prime to p, that both ν and φ_{p} have conductor dividing 𝔣𝔭^{∞}. Thus we replace 𝔣𝔭^{∞} by l.c.m.{f, cond ν}.

The grossencharacter φ (or more precisely φ ∘ N_{F/L}) is associated to a (unique) elliptic curve E defined over F = L(𝔣), the ray class field of conductor 𝔣, with complex multiplication by O_{L} and isomorphic over C to C/O_{L} (cf. [de Sh, II. Lemma 1.4]). We may even fix a Weierstrass model of E over O_{F} which has good reduction at all primes above 𝔭. For each prime 𝔓 of F above 𝔭 we have a formal group Ê_{𝔓}, and this is a relative Lubin-Tate group with respect to F_{𝔓} over L_{𝔭} (cf. [de Sh, Ch. II, §1.10]). We let λ = λÊ_{𝔓} be the logarithm of this formal group.

Let U_{∞} be the product of the principal local units at the primes above 𝔭 of L(𝔣𝔭^{∞}); i.e.,

U_{∞} = ∏_{𝔓|𝔭} U_{∞,𝔓} where U_{∞,𝔓} = lim_{⟵} U_{n}, 𝔓,

each U_{n,𝔓} being the principal local units in L(𝔣𝔭^{n})_{𝔓}. (Note that the primes of L(𝔣) above 𝔭 are totally ramified in L(𝔣𝔭^{∞}) so we still call them {𝔓}.) We wish to define certain homomorphisms δ_{k} on U_{∞}. These were first introduced in [CW] in the case where the local field F_{𝔓} is Q_{p}.

Assume for the moment that F_{𝔓} is Q_{p}. In this case Ê_{𝔓} is isomorphic to the Lubin-Tate group associated to πx + x^{p} where π = φ(𝔭). Then letting ω_{n} be nontrivial roots of [π^{n}](x) = 0 chosen so that [π](ω_{n}) = ω_{n−1}, it was shown in [CW] that to each element u = lim_{⟵} u_{n} ∈ U_{∞,𝔓} there corresponded a unique

power series f_{u}(T) ∈ Z_{p}[[T]]^{×} such that f_{u}(ω_{n}) = u_{n} for n ≥ 1. The definition of δ_{k,𝔓}(k ≥ 1) in this case was then

δ_{k,𝔓}(u) = (\frac{1}{λ′(T)} \frac{d}{dT} )^{k} log f_{n}(T)|_{T =0}.

It is easy to see that δ_{k,𝔓} gives a homomorphism: U_{∞} → U_{∞,𝔓} → O_{𝔭} satisfying δ_{k,𝔓}(ε^{σ}) = θ(σ)^{k}δ_{k,𝔓}(ε) where θ : Gal(\overline{F} /F ) → O_{𝔭}^{×} is the character giving the action on E[𝔭^{∞}].

The construction of the power series in [CW] does not extend to the case where the formal group has height > 1 or to the case where it is defined over an extension of Q_{p}. A more natural approach was developed bt Coleman [Co] which works in general. (See also [Iw1].) The corresponding generalizations of δ_{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_{𝔓} is not assumed to be Q_{p}.

To an element u = lim_{⟵} u_{n} ∈ U_{∞} we can associate a power series f_{u,𝔓}(T) ∈

O_{𝔓}[[T]]^{×} where O_{𝔓} is the ring of integers of F_{𝔓}; see [de Sh, Ch. II §4.5]. (More precisely f_{u,𝔓}(T) is the 𝔓-component of the power series described there.) For 𝔓 we will choose the prime above 𝔭 corresponding to our chosen embedding \overline{Q} ↪ \overline{Q}_{p}. This power series satisfies u_{n,𝔓} = (f_{u,𝔓})(ω_{n}) for all n > 0, n ≡ 0(d) where d = [F_{𝔓} : L_{𝔭}] and {ω_{n}} is chosen as before as an inverse system of π^{n} division points of Ê_{𝔓}. We define a homomorphism δ_{k} : U_{∞} → O_{𝔓} by

(4.11)  δ_{k}(u) := δ_{k,𝔓}(u) = (\frac{1}{λ′ ; Ê_{𝔓}(T)} \frac{d}{dT} )^{k} log f_{u,𝔓}(T)|_{T =0}.

Then

(4.12)  δ_{k}(u^{τ}) = θ(τ)^{k}δ_{k}(u) for τ ∈ Gal(F̄/F)

where θ again denotes the action on E[𝔭^{∞}]. Now θ = φ_{p} on Gal(F̄/F). We actually want a homomorphism on u_{∞} with a transformation property corresponding to ν on all of Gal(L̄/L). Observe that ν = φ_{p}^{2} on Gal(F̄/F). Let S be a set of coset representatives for Gal(L̄/L)/Gal(L̄/F) and define

(4.13)  Φ_{2}(u) = ∑_{σ∈S} ν^{−1}(σ)δ_{2}(u^{σ}) ∈ O_{𝔓}[ν].

Each term is independent of the choice of coset representative by (4.8) and it is easily checked that

Φ_{2}(u^{σ}) = ν(σ)Φ_{2}(u).

It takes integral values in O_{𝔓}[ν]. Let U_{∞}(ν) denote the product of the groups of local principal units at the primes above 𝔭 of the field L(ν) (by which we mean projective limis of local principal units as before). Then Φ_{2} factors through U_{∞}(ν) and thus defines a continuous homomorphism

Φ_{2} : U_{∞}(ν) → C_{p}.

Let C_{∞} be the group of projective limits of elliptic units in L(ν) as defined in [Ru4]. Then we have a crucial theorem of Rubin (cf. [Ru4], [Ru2]), proved using the ideas of Kolyvagin:

Theorem 4.2. There is an equality of characteristic ideals as Λ = Z_{p}[[Gal(L(ν)/L)]]-modules:

char_{∧}(Gal(M_{∞}/L(ν))) = char_{∧}(U_{∞}(ν)/\overline{C}_{∞}).

Let ν_{0} = ν mod λ. For any Z_{p}[Gal(L(ν_{0})/L)]-module X we write X^{(ν_{0})} for the maximal quotient of X ⊗_{Z_{p}} O on which the action of Gal(L(ν_{0})/L) is via

the Teichmüller lift of ν_{0}. Since Gal(L(ν)/L) decomposes into a direct product of a pro-p group and a group of order prime to p,

Gal(L(ν)/L) ≃ Gal(L(ν)/L(ν_{0})) × Gal(L(ν_{0})/L),

we can also consider any Z_{p}[[Gal(L(ν)/L)]]-module also as a Z_{p}[Gal(L(ν_{0})/L)]- module. In particular X^{(ν_{0})} is a module over Z_{p}[Gal(L(ν_{0})/L)]^{(ν_{0})} ≃ O. Also Λ^{ν_{0})} ≃ O[[T]].

Now according to results of Iwasawa ([Iw2, §12], [Ru2, Theorem 5.1]), U_{∞}(ν)^{(ν_{0})} is a free Λ^{(ν_{0})}-module of rank one. We extend Φ_{2} O-linearly to U_{∞}(ν) ⊗_{Z_{p}} O and it then factors through U_{∞}(ν)^{(ν_{0})}. Suppose that u is a generator of U_{∞}(ν)^{(ν_{0})} and β an element of C̄_{∞}^{(ν_{0})}. Then f(γ −1)u = β for some f(T) ∈ O[[T]] and γ a topological generator of Gal(L(ν)/L(ν_{0})). Computing Φ_{2} on both u and β gives

(4.14)  f(ν(γ) − 1) = ϕ_{2}(β)/Φ_{2}(u).

Next we let e(𝔞) be the projective limit of elliptic units in lim_{⟵} L_{𝔣𝔭^{n}}^{×} for 𝔞 some ideal prime to 6𝔣𝔭 described in [de Sh, Ch. II,§4.9]. Then by the proposition of Chapter II, §2.7 of [de Sh] this is a 12^{th} power in lim_{⟵} L_{𝔣𝔭^{n}}^{×}. We let β_{1} = β(𝔞)^{1/12} be the projection of e(𝔞)^{1/12} to U_{∞} and take β = Norm β_{1} where the norm is from L_{𝔣𝔭^{∞}} to L(ν). A generalization of the calculation in [CW] which may be found in [de Sh, Ch. II, §4.10] shows that

(4.15)  Φ_{2}(β) = (root of unity)Ω^{−2}(N𝔞 − ν(𝔞))L_{𝔣}(2, ν̄) ∈ O_{𝔓}[ν]

where Ω is a basis for the O_{L}-module of periods of our chosen Weierstrass model of E_{/F}. (Recall that this was chosen to have good reduction at primes above 𝔭. The periods are those of the standard Neron differential.) Also ν here should be interpreted as the grossencharacter whose associated p-adic character, via the chosen embedding \overline{Q} ↪ \overline{Q}_{p}, is ν, and \overline{ν} is the complex conjugate of ν.

The only restrictions we have placed on 𝔣 are that (i) 𝔣 is prime to 𝔭; (ii) w_{𝔣} = 1; and (iii) cond ν|𝔣𝔭^{∞}. Now let 𝔣_{0}𝔭^{∞} be the conductor of ν with 𝔣_{0} prime to p. We show now that we can choose 𝔣 such that L_{𝔣}(2, \overline{ν})/L_{𝔣_{0}}(2, \overline{ν}) is a p-adic unit unless ν_{0} = 1 in which case we can choose it to be t as defined in (4.4). We can clearly choose L_{𝔣}(2, \overline{ν})/L_{𝔣_{0}}(2, \overline{ν}) to be a unit if ν_{0} ≠ 1, as \overline{ν}(𝔮)ν(𝔮) = Norm 𝔮^{2} for any ideal 𝔮 prime to 𝔣_{0}𝔭. Note that if ν_{0} = 1 then also p = 3. Also if ν_{0} = 1 then we see that

inf_{𝔮} #\{O/\{L_{𝔣_{0}𝔮}(2, \overline{ν})/L_{𝔣_{0}}(2, \overline{ν})\}\} = t

since \overline{ν}ε^{−2} = ν^{−1}.

We can compute Φ_{2}(u) by choosing a special local unit and showing that Φ_{2}(u) is a p-adic unit, but it is sufficient for us to know that it is integral. Then since Gal(M_{∞}/L(ν)) has no finite Λ-submodule (by a result of Greenberg; see [Gre2, end of §4]) we deduce from Theorem 4.2, (4.14) and (4.15) that

#Hom(Gal(M_{∞}/L(ν)), (K/O)(ν))^{Gal(L(ν)/L)}

≤ \{ \begin{matrix}#O/Ω^{−2}L_{𝔣_{0}}(2, ν̄) & if ν_{0} ≠ 1 \\ (#O/Ω^{−2}L_{𝔣_{0}}(2, ν̄)) · t & if ν_{0} = 1. \end{matrix}

Combining this with (4.9) gives:

#H_{Se}^{1}(Q_{Σ}/Q, Y) ≤ #(O/Ω^{−2}L_{𝔣_{0}}(2, ν̄)) · ∏_{q∈Σ} ℓ_{q}

where ℓ_{q} = #H^{0}(Q_{q}, Y^{∗}) (for q ≠ p), ℓ_{p} = #H^{0}(Q_{p}, (Y^{0})^{∗}).

Since V ≃ Y ⊕ (K/O)(ψ) ⊕ K/O we need also a formula for

# ker {H^{1}(Q_{Σ}/Q, (K/O)(ψ) ⊕ K/O) → H^{1}(Q_{p}^{unr}, (K/O)(ψ) ⊕ K/O)}.

This is easily computed to be

(4.16)  #(O/h_{L}) · ∏_{q∈Σ−{p}} ℓ_{q}

where ℓ_{q} = #H^{0}(Q_{q}, ((K/O)(ψ) ⊕ K/O)^{∗}) and h_{L} is the class number of O_{L}. Combining these gives:

Proposition 4.3.

#H_{Se}^{1}(Q_{Σ}/Q, V) ≤ #(O/Ω^{−2}L_{𝔣_{0}}(2, \overline{ν})) · #(O/h_{L}) · ∏_{q∈Σ} ℓ_{q}

where ℓ_{q} = #H^{0}(Q_{q}, V^{∗}) (for q ≠ p), ℓ_{p} = #H^{0}(Q_{p}, (Y^{0})^{∗}).

2. Calculation of η

We need to calculate explicitly the invariants η_{D,f} introduced in Chapter 2, §3 in a special case. Let ρ_{0} be an irreducible representation as in (1.1). Suppose that f is a newform of weight 2 and level N, λ a prime of O_{f} above p and ρ_{f,λ} a deformation of ρ_{0}. Let 𝔪 be the kernel of the homomorphism T_{1}(N) → O_{f}/λ arising from f. We write T for T_{1}(N)_{𝔪} ⊗_{W (k_{𝔪})} O, where O = O_{f,λ} and k_{𝔪} is

the residue field of 𝔪. Assume that p ∤ N. We assume here that k is the residue field of O and that it is chosen to contain k_{𝔪}. Then by Corollary 1 of Theorem 2.1, T_{1}(N)_{𝔪} is Gorenstein andit follows that T is also a Gorenstein O-algebra (see the discussion following (2.42)). So we can use perfect pairings (the second one T -bilinear)

O × O → O, ⟨ , ⟩ : T × T → O

to define an invariant η of T. If π : T → O is the natural map, we set (η) = (π̂(1)) where π̂ is the adjoint of π with respect to the pairings. It is well-defined as an ideal of T, depending only on π. Furthermore, as we noted in Chapter 2, §3, π(η) = ⟨η, η⟩ up to a unit in O and as noted in the appendix η = Ann 𝔭 = T [𝔭] where 𝔭 = ker π. We now give an explicit formula for η developed by Hida (cf. [Hi2] for a survey of his earlier results) by interpreting ⟨ , ⟩ 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

(4.17)  ( , ) : H^{1}(X_{1}(N), O_{f}) × H^{1}(X_{1}(N), O_{f}) → O_{f}

be the cup product pairing with O_{f} as coefficients. (We sometimes drop the C from X_{1}(N)_{/C} or J_{1}(N)_{/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), O_{f}) given by x ↦ t^{∗}x and simply write tx for t^{∗}x. This is the same as the action induced by t_{∗} ∈ T_{1}(N) on H^{1}(J_{1}(N), O_{f}) ≃ H^{1}(X_{1}(N), O_{f}). Let 𝔭_{f} be the minimal prime of T_{1}(N) ⊗ O_{f} associated to f (i.e., the kernel of T_{1}(N) ⊗ O_{f} → O_{f} given by t_{l} ⊗ β ↦ βc_{t}(f) where tf = c_{t}(f)f), and let

L_{f} = H^{1}(X_{1}(N), O_{f})[𝔭_{f}].

If f = Σa_{n}q^{n} let f^{ρ} = Σā_{n}q^{n}. Then f^{ρ} is again a newform and we define L_{f^{ρ}} by replacing f by f^{ρ} in the definition of L_{f}. (Note here that O_{f} = O_{f^{ρ}} 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 O_{f} by any ring of integers containing it.) Then the pairing ( , ) induces another by restriction

(4.18)  ( , ) : L_{f} × L_{f^{ρ}} → O_{f}.

Replacing O (and the O_{f}-modules) by the localization of O_{f} at p (if necessary) we can assume that L_{f} and L_{f^{ρ}} are free of rank 2 and direct summands as O_{f}-modules of the respective cohomology groups. Let δ_{1}, δ_{2} be a basis of L_{f}. Then also δ̄_{1}, δ̄_{2} is a basis of L_{f^{ρ}} = \overline{L_{f}}. Here complex conjugation acts on H^{1}(X_{1}(N), O_{f}) via its action on O_{f}. We can then verify that

(δ, δ̄) := det(δ_{i}, δ̄_{j})

is an element of O_{f} (or its localization at p) whose image in O_{f,λ} is given by π(η^{2}) (unit). To see this, consider a modified pairing ⟨ , ⟩ defined by

(4.19)  ⟨x, y⟩ = (x, w_{ζ}y)

where w_{ζ} is defined as in (2.4). Then ⟨tx, y⟩ = ⟨x, ty⟩ for all x, y and Hecke operators t. Furthermore

det⟨δ_{i}, δ_{j}⟩ = det(δ_{i}, w_{ζ}δ_{j}) = c det(δ_{i}\overline{δ}_{j})

for some p-adic unit c (in O_{f}). This is because w_{ζ}(L_{f^{ρ}}) = L_{f} and w_{ζ}(L_{f}) = L_{f^{ρ}}. (One can check this, foe example, using the explicit bases described below.) Moreover, by Theorem 2.1,

\begin{aligned}H^{1}(X_{1}(N), Z) ⊗_{T_{1}(N)} T_{1}(N)_{𝔪} & ≃ T_{1}(N)_{𝔪}^{2}, \\ H^{1}(X_{1}(N), O_{f}) ⊗_{T_{1}(N)⊗O_{f}} T & ≃ T^{2}. \end{aligned}

Thus (4.18) can be viewed (after tensoring with O_{f,λ} and modifying it as in (4.19)) as a perfect pairing of T -modules and so this serves to compute π(η^{2}) as explained earlier (the square coming from the fact that we have a rank 2 module).

To give a more useful expression for (δ, δ̄) we observe that f and \overline{f^{ρ}} can be viewed as elements of H^{1}(X_{1}(N), C) ≃ H_{DR}^{1}(X_{1}(N), C) via f ↦ f(z)dz, \overline{f^{ρ}} ↦ \overline{f^{ρ}}d\overline{z}. Then \{f, \overline{f^{ρ}}\} form a basis for L_{f} ⊗_{O_{f}} C. Similarly \{f̄, f^{ρ}\} form a basis for L_{f^{ρ}} ⊗_{O_{f}} C. Define the vectors ω_{1} = (f, \overline{f^{ρ}}), ω_{2} = (f̄, f^{ρ}) and write ω_{1} = Cδ and ω_{2} = C̄δ̄ with C ∈ M_{2}(C). Then writing f_{1} = f, f_{2} = \overline{f^{ρ}} we set

(ω, ω̄) := det((f_{i}, \overline{f_{j}})) = (δ, δ̄) det(CC̄).

Now (ω, ω̄) is given explicitly in terms of the (non-normalized) Petersson inner product ⟨ , ⟩:

(ω, ω̄) = −4⟨f, f⟩^{2}

where ⟨f, f⟩ = ∫_{ℌ/Γ_{1}(N)} ff̄dx dy.

To compute det(C) we consider integrals over classes in H_{1}(X_{1}(N), O_{f}). By Poincar’e duality there exist classes c_{1}, c_{2} in H_{1}(X_{1}(N), O_{f}) such that det(∫_{c_{j}} δ_{i}) is a unit in O_{f}. Hence det C generates the same O_{f}-module as

is generated by { det ( ∫_{c_{j}} f_{i})} for all such choices of classes (c_{1}, c_{2}) and with

\{f_{1}, f_{2}\} = \{f, \overline{f^{ρ}}\}. Letting u_{f} be a generator of the O_{f}-module \{ det ( ∫_{c_{j}} f_{i})\} we have the following formula of Hida:

Proposition 4.4. π(η^{2}) = ⟨f, f⟩^{2}/u_{f} ū_{f} × ( unit in O_{f,λ}).

Now we restrict to the case where ρ_{0} = Ind_{L}^{Q}κ_{0} for some imaginary quadratic field L which is unramified at p and some k^{×}-valued character κ_{0} of Gal(L̄/L). We assume that ρ_{0} is irreducible, i.e., that κ_{0} ≠ κ_{0,σ} where κ_{0,σ}(δ) = κ_{0}(σ^{−1}δσ) for any σ representing the nontrivial coset of Gal(L̄/Q)/Gal(L̄/L). In addition we wish to assume that ρ_{0} is ordinary and det ρ_{0}|_{I_{p}} = ω. In particular p splits in L. These conditions imply that, if 𝔭 is a prime of L above pκ_{0}(α) ≡ α^{−1} mod 𝔭 on U_{𝔭} after possible replacement of κ_{0} by κ_{0,σ}. Here the U_{𝔭} are the units of L_{𝔭} and since κ_{0} is a character, the restriction of κ_{0} to an inertia group I_{𝔭} induces a homomorphism on U_{𝔭}. We assume now that 𝔭 is fixed and κ_{0} chosen to satisfy this congruence. Our choice of κ_{0} will imply that the grossencharacter introduced below has conductor prime to p.

We choose a (primitive) grossencharacter φ on L together with an embedding \overline{Q} ↪ \overline{Q}_{p} corresponding to the prime 𝔭 above p such that the induced p-adic character φ_{p} has the properties:

(i) φ_{p} mod \overline{p} = κ_{0} (\overline{p} = maximal ideal of \overline{Q}_{p}).

(ii) φ_{p} factors through an abelian extension isomorphic to Z_{p} ⊕ T with T of finite order prime to p.

(iii) φ((α)) = α for α ≡ 1(𝔣) for some integral ideal 𝔣 prime to p.

To obtain φ it is necessary first to define φ_{p}. Let M_{∞} denote the maximal abelian extension of L which is unramified outside 𝔭. Let θ : Gal(M_{∞}/L) → \overline{Q_{p}}^{×} be any character which factors through a Z_{p}-extension and induces the homomorphism α ↦ α^{−1} on U_{𝔭,1} ↦ Gal(M_{∞}/L) where U_{𝔭,1} = {u ∈ U_{𝔭} : u ≡ 1(𝔭)}. Then set φ_{p} = κ_{0}θ, and pick a grossencharacter φ such that (φ)_{p} = φ_{p}. Note that our choice of φ here is not necessarily intended to be the same as the choice of grossencharacter in Section 1.

Now let 𝔣_{φ} be the conductor of φ and let F be the ray class field of conductor 𝔣_{φ}𝔣̄_{φ}. Then over F there is an elliptic curve, unique up to isomorphism, with complex multiplication by O_{L} and period lattice free, of rank one over O_{L} and with associated grossencharacter φ∘N_{F/L}. The curve E_{/F} is the extension of scalars of a unique elliptic curve E_{/F^{+}} where F^{+} is the subfield of F of index 2. (See [Sh1, (5.4.3)].) Over F^{+} this elliptic curve has only the p-power isogenies of the form ±p^{m} for m ∈ Z. To see this observe that F is unramified at p and ρ_{0} is ordinary so that the only isogenies of degree p over F are the ones that correspond to division by ker 𝔭 and ker 𝔭′ where 𝔭𝔭′ = (p) in L. Over F^{+} these two subgroups are interchanged by complex conjugation, which gives the assertion. We let E/_{O_{F^{+},(p)}} denote a Weierstrass model over O_{F^{+},(p)}, the localization of O_{F^{+}} at p, with good reduction at the primes above p. Let ω_{E} be a Neron differential of E_{/O_{F^{+},(p)}}. Let Ω be a basis for the O_{L}-module of periods of ω_{E}. Then \overline{Ω} = u · Ω for some p-adic unit in F^{×}.

According to a theorem of Hecke, φ is associated to a cusp form f_{φ} in such a way that the L-series L(s, φ) and L(s, f_{φ}) are equal (cf. [Sh4, Lemma 3]). Moreover since φ was assumed primitive, f = f_{φ} is a newform. Thus the integer N = cond f = |Δ_{L/Q}|Norm_{L/Q}(cond φ) is prime to p and there is a homomorphism

ψ_{f} : T_{1}(N) ↠ R_{f} ⊂ O_{f} ⊂ O_{φ}

satisfying ψ_{f}(T_{l}) = φ(𝔠)+φ(𝔠̂) if l = 𝔠𝔠̂ in L, (l ∤ N) and ψ_{f}(T_{l}) = 0 if l is inert in L (l ∤ N). Also ψ_{f}(l⟨l⟩) = φ((l))ψ(l) where ψ is the quadratic character associated to L. Using the embedding of Q̄ in Q̄_{p} chosen above we get a prime λ of O_{f} above p, a maximal ideal 𝔪 of T_{1}(N) and a homomorphism T_{1}(N)_{𝔪} → O_{f,λ} such that the associated representation ρ_{f,λ} reduces to ρ_{0} mod λ.

Let 𝔭_{0} = ker ψ_{f} : T_{1}(N) → O_{f} and let

A_{f} = J_{1}(N)/𝔭_{0}J_{1}(N)

be the abelian variety associated to f by Shimura. Over F^{+} there is an isogeny

A_{f/F^{+}} ∼ (E_{/F^{+}})^{d}

where d = [O_{f} : Z] (see [Sh4, Th. 1]). To see this one checks that the p-adic Galois representation associated to the Tate modules on each side are equivalent to (Ind_{F}^{F^{+}}φ_{o})⊗_{Z_{p}} K_{f,p} where K_{f,p} = O_{f} ⊗Q_{p} and where φ_{p} : Gal(\overline{F} /F) → Z_{p}^{×} is the p-adic character associated to φ and restricted to F. (one compares trace(Frob ℓ) in the two representations for ℓ ∤ Np and ℓ split completely in F^{+}; cf. the discussion after Theorem 2.1 for the representation on A_{f}.)

Now pick a nonconstant map

π : X_{1}(N)_{/F^{+}} → E_{/F^{+}}

which factors through A_{f/F^{+}}. Let M be the composite of F^{+} and the normal closure of K_{f} viewed in C. Let ω_{E} be a Neron differential of E_{/O_{F^{+},(p)}}. Extending scalars to M we can write

π^{∗}ω_{E} = ∑_{σ∈Hom(K_{f} ,C)} a_{σ}ω_{f^{σ}}, a_{σ} ∈ M

where ω_{f^{σ}} = ∑_{n=1}^{∞} a_{n}(f^{σ})q^{n} \frac{dq}{q} for each σ. By suitably choosing π we can assume

that a_{id} ≠ 0. Then there exist λ_{i} ∈ O_{M} and t_{i} ∈ T_{1}(N) such that

∑ λ_{i}t_{i}π^{∗}ω_{E} = c_{1}ω_{f} for some c_{1} ∈ M.

We consider the map

(4.20)  π′ : H_{1}(X_{1}(N)_{/C}, Z) ⊗ O_{M,(p)} → H_{1}(E_{/C}, Z) ⊗ O_{M,(p)}

given by π′ = ∑ λ_{i}(π ∘ t_{i}). Even if π′ is not surjective we claim that the image of π′ always has the form H_{1}(E_{/C}, Z) ⊗ aO_{M,(p)} for some a ∈ O_{M}. This is because tensored with Z_{p} π′ can be viewed as a Gal(\overline{Q}/F^{+})-equivariant map of p-adic Tate-modules, and the omly p-power isogenies on E_{/F^{+}} have the form ±p^{m} for some m ∈ Z. It follows that we can factor π′ as (1 ⊗ a) ∘ α for some other surjective α

α : H_{1}(X_{1}(N)_{/C}, Z) ⊗ O_{M} → H^{1}(E_{/C}, Z) ⊗ O_{M},

now allowing a to be in O_{M,(p)}. Now define α^{∗} on Ω_{E/C}^{1} by α^{∗} = ∑ a^{−1}λ_{i}t_{i}∘π^{∗} where π^{∗} : Ω_{E/C}^{1} → Ω_{J_{1}(N)/C}^{1} is the map induced by π and t_{i} has the usual action on Ω_{J_{1}(N)/C}^{1}. Then α^{∗}(ω_{E}) = cω_{f} for some c ∈ M and

(4.21)  ∫_{γ} α^{∗}(ω_{E}) = ∫_{α(γ)} ω_{E}

for any class γ ∈ H_{1}(X_{1}(N)_{/C}, O_{M}). We note that α (on homology as in (4.20)) also comes from a map of abelian varieties α : J_{1}(N)_{/F^{+}} ⊗_{Z} O_{M} → E_{/F^{+}} ⊗_{Z} O_{M} although we have not used this to define α^{∗}.

We claim now that c ∈ O_{M,(p)}. We can compute α^{∗}(ω_{E}) by considering α^{∗}(ω_{E} ⊗ 1) = ∑ t_{i}π^{∗} ⊗ a^{−1}λ_{i} on Ω_{E/F^{+}}^{1} ⊗ O_{M} and then mapping the image in Ω_{J_{1}(N)/F^{+}}^{1} ⊗ O_{M} to Ω_{J_{1}(N)/F^{+}}^{1} ⊗_{O_{F^{+}}} O_{M} = Ω_{J_{1}(N)/M}^{1}. Now let us write O_{1} for O_{F^{+},(p)}. Then there are isomorphisms

Ω_{J_{1}(N)_{/O_{1}} ⊗O_{2}}^{1} ⟶^{{s_{1} \atop ∼}} Hom(O_{M}, Ω_{J_{1}(N)_{/O_{1}}}^{1} ) ⟶^{{s_{2} \atop ∼}} Ω_{J_{1}(N)_{/O_{1}}}^{1} ⊗ δ^{−1}

where δ is the different of M/Q. The first isomorphism can be described as follows. Let e(γ) : J_{1}(N) → J_{1}(N) ⊗ O_{M} for γ ∈ O_{M} be the map x ↦ x ⊗ γ. Then t_{1}(ω)(δ) = e(γ)^{∗}ω. Similar identifications occur for E in place of J_{1}(N). So to check that α^{∗}(ω_{E} ⊗ 1) ∈ Ω_{J_{1}(N)_{/O_{1}}}^{1} ⊗ O_{M} it is enough to observe that by its construction α comes from a homomorphism J_{1}(N)_{/O_{1}} ⊗O_{M} → E_{/O_{1}} ⊗O_{M}. It follows that we can compare the periods of f and of ω_{E}.

For f^{ρ} we use the fact that \overline{∫_{γ} f^{ρ} dz} = ∫_{γ^{c}} f dz where c is the O_{M}-linear map on homology coming from complex conjugation on the curve. We deduce:

Proposition 4.5. u_{f} = \frac{1}{4π^{2}} Ω^{2}.(1/p-adic integer)).

We now give an expression for ⟨f_{φ}, f_{φ}⟩ in terms of the L-function of φ. This was first observed by Shimura [Sh2] although the precise form we want was given by Hida.

Proposition 4.6.

⟨f_{φ}, f_{φ}⟩ = \frac{1}{16π^{3}} N^{2}\{ ∏_{{q|N \atop q∉Sφ}} (1 − \frac{1}{q})\}L_{N}(2, φ^{2}¯χ̂)L_{N}(1, ψ)

where χ is the character of f_{φ} and χ̂ its restriction to L; ψ is the quadratic character associated to L; L_{N}( ) denotes that the Euler factors for primes dividing N have been

removed;

S_{φ} is the set of primes q|N such that q = 𝔮𝔮′ with 𝔮 ∤ cond φ and 𝔮, 𝔮′ primes of L, not necessarily distinct.

Proof. One begins with a formula of Petterson that for an eigenform of weight 2 on Γ_{1}(N) says

⟨f, f⟩ = (4π)^{−2}Γ(2)(\frac{1}{3})π[SL_{2}(Z) : Γ_{1}(N) · (±1)] · Res_{s=2}D(s, f, f^{ρ})

where D(s, f, f^{ρ}) = ∑_{n=1}^{∞} |a_{n}|^{2}n^{−s} if f = ∑_{n=1}^{∞} a_{n}q^{n} (cf. [Hi3, (5.13)]). One checks

that, removing the Euler factors at primes dividing N,

D_{N}(s, f, f^{ρ}) = L_{N}(s, φ^{2}¯χ̂)L_{N}(s − 1, ψ)ζ_{Q,N}(s − 1)/ζ_{Q,N}(2s − 2)

by using Lemma 1 of [Sh3]. For each Euler factor of f at a q|N of the form (1−α_{q}q^{−s}) we get also an Euler factor in D(s, f, f^{ρ}) of the form (1−α_{q}ᾱ_{q}q^{−s}). When f = f_{φ} this can only happen for a split prime 𝔮 where 𝔮′ divides the conductor of φ but 𝔮 does not, or for a ramified prime 𝔮 which does not divide the conductor of φ. In this case we get a term (1 − q^{1−s}) since |φ(𝔮)|^{2} = q.

Putting together the propositions of this section we now have a formula for π(η) as defined at the beginning of this section. Actually it is more convenient to give a formula for π(η_{M}), an invariant defined in the same way but with T_{1}(M)_{𝔪_{1}} ⊗_{W (k_{𝔪_{1}} )} O replacing T_{1}(N)_{𝔪} ⊗_{W (k_{𝔪})} O where M = pM_{0} with p ∤ M_{0} and M/N is of the form

∏_{q∈S_{φ}} q · ∏_{{q∤N \atop q|M_{0}}} q^{2}.

Here 𝔪_{1} is defined by the requirements that ρ_{𝔪_{1}} = ρ_{0}, U_{q} ∈ 𝔪 if q|M(q ≠ p) and there is an embedding (which we fix) k_{𝔪_{1}} ↪ k over k_{0} taking U_{p} → α_{p} where α_{p} is the unit eigenvalue of Frob p in ρ_{f,λ}. So if f′ is the eigenform obtained from f by ‘removing the Euler factors’ at q|(M/N)(q ≠ p) and removing the non-unit Euler factor at p we have η_{M} = π̂(1) where π : T_{1} = T_{1}(M)_{𝔪_{1}} ⊗_{W (k_{𝔪_{1})}} O → O corresponds to f′ and the adjoint is taken with respect

to perfect pairings of T_{1} and O with themselves as O-modules, the first one assumed T_{1}-bilinear.

Property (ii) of φ_{p} ensures that M is as in (2.24) with D = (Se, Σ, O, ϕ) where Σ is the set of primes dividing M. (Note that S_{φ} is precisely the set of primes q for which n_{q} = 1 in the notation of Chapter 2, §3.) As in Chapter 2, §3 there is a canonical map

R_{D} → T_{D} ≃ T_{1}(M)_{𝔪_{1}} ⊗_{W (k_{𝔪_{1})}} 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, §3 as we are allowing ρ_{0} to be induced from a character of Q(\sqrt{−3}). In this special case we define T_{D} to be T_{1}(M)_{𝔪_{1}} ⊗_{W (k_{𝔪_{1})}} O. The existence of the map

in (4.22) is proved as in Chapter 2, §3. For the surjectivity, note that for each q|M (with q ≠ p) U_{q} is zero in T_{D} as U_{q} ∈ 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_{D} we use that it is the eigenvalue of Frob p on the unique unramified quotient which is free of rank one in the representation ρ described after the corollaries to Theorem 2.1 (cf. Theorem 2.1.4 of [Wi1]). To verify this one checks that T_{D} is reduced or alternatively one can apply the method of Remark 2.11. We deduce that U_{p} ∈ T_{D}^{tr}, the W (k_{𝔪_{1}})-subalgebra of T_{1}(M)_{𝔪_{1}} generated by the traces, and it follows then that it is in the image of R_{D}. We also need to give a definition of T_{D} where D = (ord, Σ, O, ϕ) and ρ_{0} is induced from a character of Q(\sqrt{−3}). For this we use (2.31).

Now we take

M = Np ∏_{q∈S_{φ}} q.

The arguments in the proof of Theorem 2.17 show that

π(η_{M}) is divisible by π(η)(α_{p}^{2} − ⟨p⟩) · ∏_{q∈S_{φ}}(q − 1)

where α_{p} is the unit eigenvalue of Frob p in ρ_{f,λ}. 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

(4.23) π(η_{M}) is divisible by Ω^{−2}L_{N}(2, φ^{2}¯χ̂)\frac{L_{N}(1, ϕ)}{π}(α_{p}^{2} − ⟨p⟩) ∏_{q|N}(q − 1).

We deduce:

Theorem 4.7. #(O/π(η_{M})) = #H_{Se}^{1}(Q_{Σ}/Q, V).

Proof. As explained in Chapter 2, §3 it is sufficient to prove the inequality #(O/π(η_{M})) ≥ #H_{Se}^{1}(Q_{Σ}/Q, V) as the opposite one is immediate. For this it suffices to compare (4.23) with Proposition 4.3. Since

L_{N}(2, ν̄) = L_{N}(2, ν) = L_{N}(2, φ^{2}¯χ̂)

(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_{Se}^{1}(Q_{Σ}/Q, V). □

We now deduce the main theorem in the CM case using the method of Theorem 2.17.

Theorem 4.8. Suppose that ρ_{0} as in (1.1) is an irreducible representation of odd determinant such that ρ_{0} = Ind_{L}^{Q}κ_{0} for a character κ_{0} of an imaginary quadratic extension L of Q which is unramified at p. Assume also that:

(i) det ρ_{0}|_{I_{p}} = ω;

(ii) ρ_{0} is ordinary.

Then for every D = (·, Σ, O, ϕ) uch that ρ_{0} os of type D with · = Se or ord,

R_{D} ≃ T_{D}

and T_{D} is a complete intersection.

Corollary. For any ρ_{0} as in the theorem suppose that

ρ : Gal(Q̄/Q) → GL_{2}(O)

is a continuous representation with values in the ring of integers of a local field, unramified outside a finite set of primes, satisfying ρ̄ ≃ ρ_{0} when viewed as representations to GL_{2}(F̄_{p}). Suppose further that:

(i) ρ|_{D_{p}} is ordinary;

(ii) det ρ|_{I_{p}} = χε^{k−1} with χ of finite order, k ≥ 2.

Then ρ is associated to a modular form of weight k.

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.

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.

Theorem 5.1 (Langlands-Tunnell). Suppose that ρ : Gal(Q̄/Q) → GL_{2}(C) is a continuous irreducible representation whose image is finite and solvable. Suppose further that det ρ is odd. Then there exists a weight one newform f such that L(s, f) = L(s, ρ) 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 Q). However in the crucial case where the image in PGL_{2}(C) is S_{4}, the result was only obtained with an additional hypothesis. This was subsequently removed by Tunnell in [Tu].

Suppose then that

ρ_{0} : Gal(Q̄/Q) → GL_{2}(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 F̄_{3}, ρ_{0} ≈ ρ_{g,μ} mod μ for some pair (g, μ) with g some newform of weight 2 (cf. [Se, §5.3]). There exists a representation

i : GL_{2}(F_{3}) ↪ GL_{2}(Z[\sqrt{−2}]) ⊂ GL_{2}(C).

By composing i with an automorphism of GL_{2}(F_{3}) if necessary we can assume that i induces the identity on reduction mod (1 + \sqrt{−2}). So if we consider i ∘ ρ_{0} : Gal(Q̄/Q) → GL_{2}(C) we obtain an irreducible representation which is easily seen to be odd and whose image is solvable. Applying the theorem we find a newform f of weight one associated to this representation. Its eigenvalues lie in Z[\sqrt{−2}]. Now pick a modular form E of weight one such that E ≡ 1(3). For example, we can take E = 6E_{1,χ} where E_{1,χ} is the Eisenstein series with Mellin transform given by ζ(s)ζ(s, χ) for χ the quadratic character associated to Q(\sqrt{−3}). Then fE ≡ f mod 3 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 μ′ 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′, μ′) by (g, μ) for some prime μ above (1 + \sqrt{−2}). Then the pair (g, μ) satisfies our requirements for a suitable choice of μ (compatible with μ′).

We can apply this to an elliptic curve E defined over 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.

Theorem 5.2. All semistable elliptic curves over Q are modular.

Proof. Suppose that E is a semistable elliptic curve over Q. Assume first that the representation ρ̄_{E,3} on E[3] is irreducible. Then if ρ_{0} = ρ̄_{E,3} restricted to Gal(Q̄/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 ρ_{0} have order dividing 3 this means that the splitting field of ρ_{0} is unramified outside 3. However, Q(\sqrt{−3}) has no nontrivial abelian extensions unramified outside 3 and of order prime to 3. So ρ_{0} itself would factor through an abelian extension of Q and this is a contradiction as ρ_{0} is assumed odd and irreducible. So ρ_{0} restricted to Gal(Q̄/Q(\sqrt{−3})) is absolutely irreducible and ρ_{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 ρ̄_{E,3} is reducible. Then we claim that the representation ρ̄_{E,5} on the 5-division points is irreducible. This is because X_{0}(15)(Q) has only four rational points besides the cusps and these correspond to nonsemistable curves which in any case are modular; cf. [BiKu, pp. 79-80]. If we knew that ρ̄_{E,5} was modular we could now prove the theorem in the same way we did knowing that ρ̄_{E,3} was modular once we observe that ρ̄_{E,5} restricted to Gal(Q̄/Q(\sqrt{5})) is absolutely irreducible. This irreducibility follows a similar argument to the one for ρ̄_{E,3} since the only nontrivial abelian extension of Q(\sqrt{5}) unramified outside 5 and of order prime to 5 is Q(ζ_{5}) which is abelian over Q. Alternatively, it is enough to check that there are no elliptic curves E for which ρ̄_{E,5} is an induced representation over Q(\sqrt{5}) and E is semistable at 5. This can be checked in the supersingular case using the description of ρ̄_{E,5}|_{D_{5}} (in particular it is induced from a character of the unramified quadratic extension of Q_{5} whose restriction to inertia is the fundamental character of level 2) and in the ordinary case it is straightforward.

Consider the twisted form X(ρ)_{/Q} of X(5)_{/Q} defined as follows. Let X(5)_{/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}(Gal(L/Q), Aut !X(5)_{/L})

given by ρ̄_{E,5} : Gal(L/Q) ⟶ GL_{2}(Z/5Z) ⊆ Aut X(5)_{/L} where L denotes the splitting field of ρ̄_{E,5}. Then E defines a rational point on X(ρ)_{/Q} and hence also of an irreducible component of it which we denote C. This curve C is smooth as X(ρ)_{/Q̄} = X(5)_{/Q̄} is smooth. It has genus zero since the same is true of the irreducible components of X(5)_{/Q̄}.

A rational point on C (necessarily non-cuspidal) corresponds to an elliptic curve E′ over Q with an isomorphism E′[5] ≃ E[5] as Galois modules (cf. [DR, VI, Prop. 3.2]). We claim that we can choose such a point with the two properties that (i) the Galois representation ρ̄_{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 ≠ 5 we just use the fact that E′ is semistable at q ⟺ #ρ̄_{E′,5}(I_{q})|5.) So E′ will be modular and hence so too will ρ̄_{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 ρ̄_{E′,3} is reducible. But this corresponds to E′ being the image of a rational point on an irreducible covering of C of degree 4. Let Q(t) be the function field of C. We have therefore an irreducible polynomial f(x, t) ∈ Q(t)[x] of degree > 1 and we need to ensure that for many values t_{0} in Q, f(x, t_{0}) has no rational solution. Hilbert’s theorem ensures that there exists a t_{1} such that f(x, t_{1}) is irreducible. Then we pick a prime p_{1} ≠ 5 such that f(x, t_{1}) has no root mod p_{1}. (This is easily achieved using the Čebotarev density theorem; cf. [CF, ex. 6.2, p. 362].) So finally we pick any t_{0} ∈ 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}) or a quadratic twist has semistable reduction at 5. To see this, observe that since j_{E} ≠ 0, 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 E(j_{0}) equal to j_{0} whenever the g_{i}(j_{0}) are finite. Then E is given by a quadratic twist of E(j_{E}) and so after a change of functions of the form g_{2}(j) ↦ u^{2}g_{2}(j), g_{3}(j) ↦ u^{3}g_{3}(j) with u ∈ Q^{×} we can assume that E(j_{E}) = E and that the equation E(j_{E}) is minimal at 5. Then for j′ ∈ Q close enough 5-adically to j_{E} the equation E(j′) is still minimal and semistable at 5, since a criterion for this, for an integral model, is that either ord_{5}(△(E(j′))) = 0 or ord_{5}(c_{4}(E(j′))) = 0. So up to a quadratic twist E′ is also semistable. □

This kind of argument can be applied more generally.

Theorem 5.3. Suppose that E is an elliptic curve defined over Q with the following properties:

(i) E has good or multiplicative reduction at 3, 5,

(ii) For p = 3, 5 and for any prime q ≡ −1 mod p either ρ̄_{E,p}|_{D_{q}} is reducible over F̄_{p} or ρ̄_{E,p}|I_{q} is irreducible over F̄_{p}.

Then E is modular.

Proof. the main point to be checked is that one can carry over condition (ii) to the new curve E′. For this we use that for any odd prime p ≠ q,

ρ̄_{E,p}|_{D_{q}} is absolutely irreducible and ρ̄_{E,p}|_{I_{q}} is absolutely reducible

and 3 ∤ #ρ̄_{E,p}(I_{q})
⇕

E acquires good reduction over an abelian 2-power extension of Q_{q}^{unr} but not over an abelian extension of Q_{q}.

Suppose then that q ≡ −1(3) and that E′ does not satisfy condition (ii) at q (for p = 3). Then we claim that also 3 ∤ #ρ̄_{E′,3}(I_{q}). For otherwise ρ̄_{E′,3}(I_{q}) has its normalizer in GL_{2}(F_{3}) contained in a Borel, whence ρ̄_{E′,3}(D_{q}) would be reducible which contradicts our hypothesis. So using the above equivalence we deduce, by passing via ρ̄_{E′,5} ≃ ρ̄_{E,5}, that E also does not satisfy hypothesis (ii) at p = 3.

We also need to ensure that ρ̄_{E′,3} is absolutely irreducible over Q(\sqrt{−3} ). This we can do by observing that the property that the image of ρ̄_{E′,3} lies in the Sylow 2-subgroup of GL_{2}(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 ρ̄_{E′,3} was reducible, this time using two separate coverings to ensure that the image of ρ̄_{E′,3} is neither reducible nor contained in a Sylow 2-subgroup.

Finally one also has to show that if both ρ̄_{E,5} is irreducible and ρ̄_{E,3} is induced from a character of Q(\sqrt{−3} ) then E is modular. (The case where both were reducible has already been considered.) Taylor has pointed out that curves satisfying both these conditions are classified by the non-cuspidal 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 Q. The non-cuspidal rational points correspond to modular elliptic curves of conductor 338. □

Appendix

Gorenstein rings and local complete intersections

Proposition 1. Suppose that O is a complete discrete valuation ring and that φ : S → T is a surjective local O-algebra homomorphism between complete local Noetherian O-algebras. Suppose further that 𝔭_{T} is a prime ideal of T such that T/𝔭_{T} ⟶^{∼} O and let 𝔭_{S} = φ^{−1}(𝔭_{T}). Assume that

(i) T ≃ O[[x_{1}, …, x_{r}]]/(f_{1}, …, f_{r−u}) where r is the size of a minimal set of O-generators of 𝔭_{T} /𝔭_{T}^{2} ,

(ii) φ induces an isomorphism 𝔭_{S}/𝔭_{S}^{2} ⟶^{∼} 𝔭_{T} /𝔭_{T}^{2} and that these are finitely generated O-modules whose free part has rank u.

Then φ is an isomorphism.

Proof. First we consider the case where u = 0. We may assume that the generators x_{1}, …, x_{r} lie in 𝔭_{T} by subtracting their residues in T/𝔭_{T} ⟶^{∼} O. By (ii) we may also write

S ≃ O[[x_{1}, …, x_{r}]]/(g_{1}, …, g_{s})

with s ≥ r (by allowing repetitions if necessary) and 𝔭_{S} generated by the images of {x_{1} …, x_{r}}. Let 𝔭 = (x_{1}, …, x_{r}) in [[x_{1}, …, x_{r}]]. Writing f_{i} ≡ Σa_{ij}x_{j} mod 𝔭^{2} with a_{ij} ∈ O, we see that the Fitting ideal as an O-module of 𝔭_{T} /𝔭_{T}^{2} is given by

F_{O}(𝔭_{T} /𝔭_{T}^{2} ) = det(a_{ij}) ∈ O

and that this is nonzero by the hypothesis that u = 0. Similarly, if each g_{i} ≡ Σb_{ij}x_{j} mod 𝔭^{2}, then

F_{O}(𝔭_{S}/𝔭_{S}^{2}) = {det(b_{ij}) : i ∈ I, #I = r, I ⊆ {1, …, s}}.

By (ii) again we see that det(a_{ij}) = det(b_{ij}) as ideals of O for some choice I_{0} of I. After renumbering we may assume that I_{0} = {1, …, r}. Then each g_{i} (i = 1, …, r) can be written g_{i} = Σr_{ij}f_{i} for some r_{ij} ∈ [[x_{1}, …, x_{r}]] and we have

det(b_{ij}) ≡ det(r_{ij}) · det(a_{ij}) mod 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 ≃ T.

We can extend this to the case u ≠ 0 by picking x_{1}, …, x_{r−u} so that they generate (𝔭_{T} /𝔭_{T}^{2} )^{tors}. Then we can write each f_{i} ≡ ∑_{i=1}^{r−u} a_{ij}x_{j} mod 𝔭^{2} and likewise for the g_{i}’s. The argument is now just as before but applied to the Fitting ideals of (𝔭_{T} /𝔭_{T}^{2} )^{tors}. □

For the next proposition we continue to assume that O is a complete discrete valuation ring. Let T be a local O-algebra which as a module is finite and free over O. In addition, we assume the existence of an isomorphism of T -modules T ⟶^{∼} Hom_{O}(T, O). We call a local O-algebra which is finite and free and satisfies this extra condition a Gorenstein O-algebra (cf. §5 of [Ti1]). Now suppose that 𝔭 is a prime ideal of T such that T/𝔭 ≃ O.

Let β : T → T/𝔭 ≃ O be the natural map and define a principal ideal of T by

(η_{T}) = (β̂(1))

where β̂ : O → T is the adjoint of β with respect to perfect O-pairings on O and T, and where the pairing of T with itself is T -bilinear. (By a perfect pairing on a free O-module M of finite rank we mean a pairing M × M → O such that both the induced maps M →Hom_{O}(M, O) are isomorphisms. When M = T we are thus requiring that this be an isomorphism of T -modules also.) The ideal (η_{T}) is independent of the pairing. Also T/η_{T} is torsion-free as an O-module, as can be seen by applying Hom( , O) to the sequence

0 → 𝔭 → T → O → 0,

to obtain a homomorphism T/η_{T} ↪ Hom(𝔭, O). This also shows that (η_{T}) = Ann𝔭.

If we let l(M) denote the length of an O-module M, then

l(𝔭/𝔭^{2}) ≥ l(O/\overline{η_{T}})

(where we write \overline{η_{T}} for β(η_{T})) because 𝔭 is a faithful T/η_{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/η_{T}} (𝔭) = 0 ⇒ F_{T} (𝔭) ⊂ (η_{T}) ⇒ F_{T/𝔭}(𝔭/𝔭^{2}) ⊂ (\overline{η_{T}})

and we then use the fact that the length of an O-module M is equal to the length of O/F_{O}(M) as O is a discrete valuation ring. In particular when 𝔭/𝔭^{2} is a torsion O-module then \overline{η}_{T} ≠ 0.

We need a criterion for a Gorenstein O-algebra to be a complete intersection. We will say that a local O-algebra S which is finite and free over O is a complete intersection over O if there is an O-algebra isomorphism S ≃ O[[x_{1}, …, x_{r}]]/(f_{1}, …, f_{r}) for some r. Such a ring is necessarily a Gorenstein O-algebra and {f_{1}, …, f_{r}} is necessarily a regular sequence. That (i) ⇒ (ii) in the following proposition is due to Tate (see A.3, conclusion 4, in the appendix in [M Ro].)

Proposition 2. Assume that O is a complete discrete valuation ring and that T is a local Gorenstein O-algebra which is finite and free over O and that 𝔭_{T} is a prime ideal of T such that T/𝔭_{T} ≅ O and 𝔭_{T} /𝔭_{T}^{2} is a torsion O-module. Then the following two conditions are equivalent:

(i) T is a complete intersection over O.

(ii) l(𝔭_{T} /𝔭_{T}^{2} ) = l(O/\overline{η_{T}}) as O-modules.

Proof. To prove that (ii) ⇒ (i), pick a complete intersection S over O (so assumed finite and flat over O) such that α:S↠T and such that 𝔭_{S}/𝔭_{S}^{2} ≃ 𝔭_{T} /𝔭_{T}^{2} where 𝔭_{S} = α^{−1}(𝔭_{T}). The existence of such an S seems to be well known (cf. [Ti2, §6]) but here is an argument suggested by N. Katz and H. Lenstra (independently).

Write T = O[x_{1}, …, x_{r}]/(f_{1}, …, f_{s}) with 𝔭_{T} the image in T of 𝔭 = (x_{1}, …, x_{r}). Since T is local and finite and free over O, it follows that also T ≃ O[[x_{1}, …, x_{r}]]/(f_{1}, …, f_{s}). We can pick g_{1}, …, g_{r} such that g_{i} = Σa_{ij}f_{j} with a_{ij} ∈ O and such that

(f_{1}, …, f_{s}, 𝔭^{2}) = (g_{1}, …, g_{r}, 𝔭^{2}).

We then modify g_{1}, …, g_{r} by the addition of elements {α_{i}} of (f_{1}, …, f_{s})^{2} and set (g′_{1} = g_{1} + α_{1}, …, g′_{r} = g_{r} + α_{r}). Since T is finite over 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}, …, x_{r}) of 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 α_{i} = (x_{i}^{N} − h_{i}(x_{1}, …, x_{r}))^{2}. Then set S = O[[x_{1}, …, x_{r}]]/(g′_{1}, …, g′_{r}). Then S is finite over O by construction and also dim(S) ≤ 1 since dim(S/λ) = 0 where (λ) is the maximal ideal of O. It follows that {g′_{1}, …, g′_{r}} is a regular sequence and hence that depth(S) = dim(S) = 1. In particular the maximal O-torsion submodule of S is zero since it is also a finite length S-submodule of S.

Now O/(η̄_{S}) ≃ O/(η̄_{T}), since l(O/(η̄_{S})) = l(𝔭_{S}/𝔭_{S}^{2}) by (i) ⇒ (ii) and l(O/(η̂_{T})) = l(𝔭_{T} /𝔭_{T}^{2} ) by hypothesis. Pick isomorphisms

T ≃ Hom_{O}(T, O), S ≃ Hom_{O}(S, O)

as T -modules and S-modules, respectively. The existence of the latter for complete intersections over O is well known; cf. conclusion 1 of Theorem A.3 of [M Ro]. Then we have a sequence of maps, in which α̂ and β̂ denote the adjoints with respect to these isomorphisms:

O ⟶^{β̂} T ⟶^{α̂} S ⟶^{α} T ⟶^{β} O.

One checks that α̂ is a map of S-modules (T being given an S-action via α) and in particular that α ∘ α̂ is multiplication by an element t of T. Now (β ∘β̂) = (η̄_{T}) in O and (β ∘α)∘( { \atop β ∘ α} ) = (η̄_{S}) in O. As (η̄_{S}) = (η̄_{T}) in O, we have that t is a unit mod 𝔭_{T} and hence that α∘α̂ is an isomorphism. It follows that S ≃ T, as otherwise S ≃ ker α ⊕ imα̂ is a nontrivial decomposition as S-modules, which contradicts S being local. □

Remark. Lenstra has made an important improvement to this proposition by showing that replacing η̄_{T} by β(ann 𝔭) gives a criterion valid for all local O-algebra which are finite and free over O, thus without the Gorenstein hypothesis.

Princeton University, Princeton, NJ

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(Received October 14, 1994)