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\begin{document}
\title[Shafarevich-Tate Groups of Modular Motives]{Constructing Elements in Shafarevich-Tate Groups of Modular Motives}
\author{Neil Dummigan}
\author{William Stein}
\author{Mark Watkins}
\date{28 January 2003}
\subjclass{11F33, 11F67, 11G40.}
\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
Shafarevich-Tate group.}
\address{University of Sheffield\\ Department of Pure
Mathematics\\
Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
U.K.}
\address{Harvard University\\Department of Mathematics\\
One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
\address{Penn State Mathematics Department\\
University Park\\State College, PA 16802\\ U.S.A.}
\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
\email{watkins@math.psu.edu}
\begin{abstract}
We study Shafarevich-Tate groups of motives attached to modular
forms on $\Gamma_0(N)$ of weight bigger than~$2$. We deduce a
criterion for the existence of nontrivial elements of these
Shafarevich-Tate groups, and give $16$ examples in which a strong
form of the Beilinson-Bloch conjecture implies the existence of
such elements. We also use modular symbols and observations about
Tamagawa numbers to compute nontrivial conjectural lower bounds on
the orders of the Shafarevich-Tate groups of modular motives of
low level and weight at most $12$. Our methods build upon the
idea of visibility due to Cremona and Mazur, but in the context of
motives instead of abelian varieties.
\end{abstract}
\maketitle
\section{Introduction}
Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
be the associated $L$-function. The conjecture of Birch and
Swinnerton-Dyer \cite{BSD} predicts that the order of vanishing of $L(E,s)$
at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
also gives an interpretation of the leading term in the Taylor
expansion in terms of various quantities, including the order of
the Shafarevich-Tate group of~$E$.
Cremona and Mazur \cite{CM} look, among all strong Weil elliptic
curves over $\QQ$ of conductor $N\leq 5500$, at those with
nontrivial Shafarevich-Tate group (according to the Birch and
Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
group has predicted elements of prime order~$p$. In most cases
they find another elliptic curve, often of the same conductor,
whose $p$-torsion is Galois-isomorphic to that of the first one,
and which has positive rank. The rational points on the second elliptic
curve produce classes in the common $H^1(\QQ,E[p])$. They show
\cite{CM2} that these lie in the Shafarevich-Tate group of the
first curve, so rational points on one curve explain elements of
the Shafarevich-Tate group of the other curve.
The Bloch-Kato conjecture \cite{BK} is the generalisation to
arbitrary motives of the leading term part of the Birch and
Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
\cite{B, Be} generalises the part about the order of vanishing at the
central point, identifying it with the rank of a certain Chow
group.
This paper is a partial generalisation of \cite{CM} and \cite{AS}
from abelian varieties over $\QQ$ associated to modular forms of
weight~$2$ to the motives attached to modular forms of higher weight.
It also does for congruences between modular forms of equal weight
what \cite{Du2} did for congruences between modular forms of different
weights.
We consider the situation where two newforms~$f$ and~$g$, both of
even weight $k>2$ and level~$N$, are congruent modulo a maximal
ideal $\qq$ of odd residue characteristic, and $L(g,k/2)=0$ but
$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish
to order at least $2$ at $s=k/2$. In Section~\ref{sec:examples},
we give sixteen such examples (all with $k=4$ and $k=6$), and in
each example, we find that $\qq$ divides the numerator of the
algebraic number $L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$
is a certain canonical period.
In fact, we show how this divisibility may be deduced from the
vanishing of $L(g,k/2)$ using recent work of Vatsal \cite{V}. The
point is, the congruence between$f$ and~$g$ leads to a congruence
between suitable ``algebraic parts'' of the special values
$L(f,k/2)$ and $L(g,k/2)$. In slightly more detail, a multiplicity
one result of Faltings and Jordan shows that the congruence of
Fourier expansions leads to a congruence of certain associated
cohomology classes. These are then identified with the modular
symbols which give rise to the algebraic parts of special values.
If $L(g,k/2)$ vanishes then the congruence implies that
$L(f,k/2)/\vol_{\infty}$ must be divisible by $\qq$.
The Bloch-Kato conjecture sometimes then implies that the
Shafarevich-Tate group $\Sha$ attached to~$f$ has nonzero
$\qq$-torsion. Under certain hypotheses and assumptions, the most
substantial of which is the Beilinson-Bloch conjecture relating
the vanishing of $L(g,k/2)$ to the existence of algebraic cycles,
we are able to construct some of the predicted elements of~$\Sha$
using the Galois-theoretic interpretation of the congruence to
transfer elements from a Selmer group for~$g$ to a Selmer group
for~$f$. One might say that algebraic cycles for one motive
explain elements of~$\Sha$ for the other, or that we use the
congruence to link the Beilinson-Bloch conjecture for one motive
with the Bloch-Kato conjecture for the other.
%In proving the local
%conditions at primes dividing the level, and also in examining the
%local Tamagawa factors at these primes, we make use of a higher weight
%level-lowering result due to Jordan and Livn\'e \cite{JL}.
We also compute data which, assuming the Bloch-Kato conjecture,
provides lower bounds for the orders of numerous Shafarevich-Tate
groups (see Section~\ref{sec:invis}). We thank the referee for
many constructive comments.
%Our data is consistent
%with the fact \cite{Fl2} that the part of $\#\Sha$ coprime to the
%congruence modulus is necessarily a perfect square (assuming that~$\Sha$
%is finite).
\section{Motives and Galois representations}
This section and the next provide definitions of some of the
quantities appearing later in the Bloch-Kato conjecture. Let
$f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
$\Gamma_0(N)$, with coefficients in an algebraic number field~$E$,
which is necessarily totally real. Let~$\lambda$ be any finite
prime of~$E$, and let~$\ell$ denote its residue characteristic. A
theorem of Deligne \cite{De1} implies the existence of a
two-dimensional vector space $V_{\lambda}$ over $E_{\lambda}$, and
a continuous representation
$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
such that
\begin{enumerate}
\item $\rho_{\lambda}$ is unramified at~$p$ for all primes~$p$
not dividing~$\ell N$, and
\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
characteristic polynomial of $\Frob_p^{-1}$ acting on
$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
\end{enumerate}
Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
both $2$-dimensional $E$-vector spaces. For details of the
construction see \cite{Sc}. The de Rham realisation has a Hodge
filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset
F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
cohomology, while $V_{\lambda}$ comes from \'etale $\ell$-adic
cohomology.
For each prime $\lambda$, there is a natural isomorphism
$V_B\otimes E_{\lambda}\simeq V_{\lambda}$. We may choose a
$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
which amounts to multiplying the action of $\Frob_p$ by $p^j$.
Following \cite{BK} (Section 3), for $p\neq \ell$ (including
$p=\infty$) let
$$
H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow
H^1(I_p,V_{\lambda}(j))).
$$
The subscript~$f$ stands for ``finite
part'', $D_p$ is a decomposition subgroup at a prime above~$p$,
$I_p$ is the inertia subgroup, and the cohomology is for
continuous cocycles and coboundaries. For $p=\ell$ let
$$
H^1_f(\QQ_{\ell},V_{\lambda}(j))=\ker
(H^1(D_{\ell},V_{\lambda}(j))\rightarrow
H^1(D_{\ell},V_{\lambda}(j)\otimes_{\QQ_{\ell}} B_{\cris}))
$$
(see Section 1 of
\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
$B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes~$p$.
There is a natural exact sequence
$$
\begin{CD}0@>>>T_{\lambda}(j)@>>>
V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.
$$
Let
$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
Define the $\lambda$-Selmer group
$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes~$p$. Note that the
condition at $p=\infty$ is superfluous unless $\ell=2$. Define the
Shafarevich-Tate group
$$
\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/
\pi_*H^1_f(\QQ,V_{\lambda}(j)).
$$
Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any
prime ideal~$\lambda$ is the length of the $\lambda$-component of
$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,
and write~$\Sha$ for~$\Sha(k/2)$. It depends on the choice of
$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
each $V_{\lambda}$. But if $A[\lambda]$ is irreducible then
$T_{\lambda}$ is unique up to scaling and the $\lambda$-part of
$\Sha$ is independent of choices.
In the case $k=2$ the motive comes from a (self-dual) isogeny class of
abelian varieties over $\QQ$, with endomorphism algebra
containing~$E$. Choose an abelian variety~$B$ in the isogeny class in
such a way that the endomorphism ring of~$B$ contains the full ring of
integers $O_E$. If one takes all the $T_{\lambda}(1)$ to be
$\lambda$-adic Tate modules, then what we have defined above coincides
with the usual Shafarevich-Tate group of~$B$ (assuming finiteness of
the latter, or just taking the quotient by its maximal divisible
subgroup). To see this one uses 3.11 of \cite{BK}, for $\ell=p$. For
$\ell\neq p$, $H^1_f(\QQ_p,V_{\ell})=0$. Considering the formal
group, every class in $B(\QQ_p)/\ell B(\QQ_p)$ is represented by an
$\ell$-power torsion point in $B(\QQ_p)$, so maps to zero in
$H^1(\QQ_p,A_{\ell})$.
Define the group of global torsion points
$$
\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).
$$
This is analogous to the group of rational torsion points on an
elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in
which the exponent of any prime ideal~$\lambda$ is the length of
the $\lambda$-component of $\Gamma_{\QQ}$.
\section{Canonical periods}
We assume from now on for convenience that $N\geq 3$. We need to
choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
and de Rham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
in a way such that $T_B$ and $T_{\dR}\otimes_{O_E}O_E[1/Nk!]$
agree with (respectively) the $O_E$-lattice $\mathfrak{M}_{f,B}$
and the $O_E[1/Nk!]$-lattice $\mathfrak{M}_{f,\dR}$ defined in
\cite{DFG} using cohomology, with non-constant coefficients, of
modular curves. (In \cite{DFG}, see especially Sections 2.2 and
5.4, and the paragraph preceding Lemma 2.3.)
For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes
E_{\lambda}\simeq V_{\lambda}$. Then the $O_{\lambda}$-module
$T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable.
Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
generalised elliptic curves with full level-$N$ structure. Let
$\mathfrak{E}$ be the universal generalised elliptic curve over
$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product
of $\mathfrak{E}$ over $M(N)$. (The motive $M_f$ is constructed
using a projector on the cohomology of a desingularisation of
$\mathfrak{E}^{k-2}$). Realising $M(N)(\CC)$ as the disjoint union
of $\phi(N)$ copies of the quotient
$\Gamma(N)\backslash\mathfrak{H}^*$ (where $\mathfrak{H}^*$ is the
completed upper half plane), and letting $\tau$ be a variable on
$\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is isomorphic to
the elliptic curve with period lattice generated by $1$ and
$\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a variable on
the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the fibre product.
Then $2\pi i f(\tau)\,d\tau\wedge dz_1\wedge\ldots\wedge dz_{k-2}$
is a well-defined differential form on (a desingularisation of)
$\mathfrak{E}^{k-2}$ and naturally represents a generating element
of $F^{k-1}T_{\dR}$. (At least we can make our choices locally at
primes dividing $Nk!$ so that this is the case.) We shall call
this element $e(f)$.
Under the de Rham isomorphism between $V_{\dR}\otimes\CC$ and
$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
a natural action of complex conjugation on $V_B$, breaking it up
into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
are rank one $O_E$-modules, but not necessarily free, since the
class number of $O_E$ may be greater than one. Choose nonzero
elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
$\Omega_f^{\pm}$ by $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$.
\section{The Bloch-Kato conjecture}\label{sec:bkconj}
In this section we extract from the Bloch-Kato conjecture for
$L(f,k/2)$ a prediction about the order of the Shafarevich-Tate
group, by analysing the other terms in the formula.
Let $L(f,s)$ be the $L$-function attached to~$f$. For
$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with
Euler product
$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but
there is an analytic continuation given by an integral, as
described in the next section. Suppose that $L(f,k/2)\neq 0$. The
Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the
following equality of fractional ideals of~$E$:
$$
\frac{L(f,k/2)}{\vol_{\infty}}=
\left(\prod_pc_p(k/2)\right)
\frac{\#\Sha}{\aaa^{\pm}(\#\Gamma_{\QQ})^2}.
$$
Here, {\bf and from this point onwards, }$\pm$ represents the
parity of $(k/2)-1$. The quantity
$\vol_{\infty}$ is equal to $(2\pi i)^{k/2}$
multiplied by the determinant of the isomorphism
$V_B^{\pm}\otimes\CC\simeq (V_{\dR}/F^{k/2})\otimes\CC$,
calculated with respect to the lattices $O_E\delta_f^{\pm}$ and
the image of $T_{\dR}$. For $l\neq p$, $\ord_{\lambda}(c_p(j))$ is
defined to be
\begin{align*}
\length&\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-
\ord_{\lambda}(P_p(p^{-j}))\\
=&\length\>\> \left(H^0(\QQ_p,A_{\lambda}(j))/H^0\left(\QQ_p, V_{\lambda}(j)^{I_p}/T_{\lambda}(j)^{I_p}\right)\right).
\end{align*}
We omit the definition of $\ord_{\lambda}(c_p(j))$ for
$\lambda\mid p$, which requires one to assume Fontaine's de Rham
conjecture (\cite{Fo}, Appendix A6), and depends on the choices of
$T_{\dR}$ and $T_B$, locally at~$\lambda$. (We shall mainly be
concerned with the $q$-part of the Bloch-Kato conjecture, where~$q$
is a prime of good reduction. For such primes, the de Rham
conjecture follows from Theorem 5.6 of \cite{Fa1}.)
Strictly speaking, the conjecture in \cite{BK} is only given for
$E=\QQ$. We have taken here the obvious generalisation of a slight
rearrangement of (5.15.1) of \cite{BK}. The Bloch-Kato conjecture
has been reformulated and generalised by Fontaine and Perrin-Riou,
who work with general $E$, though that is not really the point of
their work. In Section 11 of \cite{Fo2} it is sketched how to
deduce the original conjecture from theirs, in the
case $E=\QQ$.
\begin{lem}\label{vol}
$\vol_{\infty}/\aaa^{\pm}=c(2\pi i)^{k/2}\aaa^{\pm}\Omega_{\pm}$, with $c\in E$ and
$\ord_{\lambda}(c)=0$ for $\lambda\nmid Nk!$.
\end{lem}
\begin{proof}
We note that $\vol_{\infty}$ is equal to $(2\pi i)^{k/2}$ times the determinant of the
period map from $F^{k/2}V_{\dR}\otimes\CC$ to
$V_B^{\pm}\otimes\CC$, with respect to lattices dual to those we
used above in the definition of $\vol_{\infty}$ (c.f. the last
paragraph of 1.7 of \cite{De2}). We are using here natural
pairings. Meanwhile, $\Omega_{\pm}$ is the determinant of the same map with
respect to the lattices $F^{k/2}T_{\dR}$ and $O_E\delta_f^{\pm}$.
Recall that the index of $O_E\delta_f^{\pm}$ in
$T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then the proof is completed
by noting that, locally away from primes dividing $Nk!$, the index
of $T_{\dR}$ in its dual is equal to the index of $T_B$ in its
dual, both being equal to the ideal denoted~$\eta$ in \cite{DFG2}.
\end{proof}
\begin{remar} Note that the ``quantities'' $\aaa^{\pm}\Omega_{\pm}$ and
$\vol_{\infty}/\aaa^{\pm}$ are independent of the choice of $\delta_f^{\pm}$.
\end{remar}
\begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer.
Then the fractional ideal $c_p(j)$ is supported at most on
divisors of~$p$.
\end{lem}
\begin{proof}
As on p.~30 of \cite{Fl1}, for odd $l\neq p$,
$\ord_{\lambda}(c_p(j))$ is the length of the finite
$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
torsion free.
\end{proof}
\begin{lem}\label{local1}
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not
congruent modulo $\qq$ (for Fourier coefficients of index coprime
to $Nq$) to any newform of weight~$k$, trivial character, and
level dividing $N/p$. Then $\ord_{\qq}(c_p(j))=0$ for all
integers~$j$.
\end{lem}
\begin{proof}
There is a natural injective map from
$V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$ to $H^0(I_p,A_{\qq}(j))$
(i.e., $A_{\qq}(j)^{I_p}$). Consideration of $\qq$-torsion shows
that
$$
\dim_{O_E/\qq} H^0(I_p,A[\qq](j))\geq \dim_{E_{\qq}}
H^0(I_p,V_{\qq}(j)).
$$ To prove the lemma it suffices to show that
$$
\dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)),
$$
since this ensures that $H^0(I_p,A_{\qq}(j))=
V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$, hence that
$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p})$.
If the dimensions differ then, given that $f$ is not congruent
modulo $\qq$ to a newform of level dividing $N/p$, Condition (b) of
Proposition~2.3 of \cite{L} is satisfied. If Condition (a) was not
satisfied then Proposition~2.2 of \cite{L} would imply that $f$
was congruent modulo $\qq$ to a twist of level dividing $N/p$.
Since Condition (c) is clearly also satisfied, we are in a situation
covered by one of the three cases in Proposition~2.3 of \cite{L}.
Since $p\not\equiv -1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded,
so $A[\qq](j)$ is unramified at $p$ and $\ord_p(N)=1$. (Here we
are using Carayol's result that $N$ is the prime-to-$q$ part of
the conductor of $V_{\qq}$ \cite{Ca1}.) But then Theorem~1 of
\cite{JL} (which uses the condition $q>k$) implies the existence
of a newform of weight~$k$, trivial character and level dividing
$N/p$, congruent to~$g$ modulo $\qq$, for Fourier coefficients of
index coprime to $Nq$. This contradicts our hypotheses.
\end{proof}
\begin{remar}
For an example of what can be done when~$f$ is congruent to
a form of lower level, see the first example in Section~\ref{sec:other_ex}
below.
\end{remar}
\begin{lem}\label{at q}
If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then
$\ord_{\qq}(c_q)=0$.
\end{lem}
\begin{proof}
It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an
application, at the end of Section~2.2, of the results of
\cite{Fa1}) that $T_{\qq}$ is the
$O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered
module $T_{\dR}\otimes O_{\qq}$ by the functor they call
$\mathbb{V}$. (This property is part of the definition of an
$S$-integral premotivic structure given in Section~1.2 of
\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
in Theorem~4.1 of \cite{BK} follows from the first paragraph of
2(h) of \cite{Fa1}.)
\end{proof}
\begin{lem}
If $A[\lambda]$ is an
irreducible representation of $\Gal(\Qbar/\QQ)$,
then
$$\ord_{\lambda}(\#\Gamma_{\QQ})=0.$$
\end{lem}
\begin{proof}
This follows trivially from the definition.
\end{proof}
Putting together the above lemmas we arrive at the following:
\begin{prop}\label{sha}
Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that
$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$,
where $\qq\mid q$. Assume the same hypotheses as in Lemma
\ref{local1} for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which
locally at $\qq$ are as in the previous section. If
$L(f,k/2)\aaa^{\pm}/\vol_{\infty}\neq 0$ then the Bloch-Kato
conjecture predicts that
$$
\ord_{\qq}(\#\Sha)=\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty}).
$$
\end{prop}
\section{Congruences of special values}
Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
large enough to contain all the coefficients $a_n$ and $b_n$.
Suppose that $\qq\mid q$ is a prime of~$E$ such that $f\equiv
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Assume
that $A[\qq]$ is an irreducible representation of
$\Gal(\Qbar/\QQ)$, and that $q\nmid N\phi(N)k!$. Choose
$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates
$T_B^{\pm}$ locally at $\qq$. Make two further assumptions:
$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$
\begin{prop} \label{div}
With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
\end{prop}
\begin{proof} This is based on some of the ideas used in Section 1 of
\cite{V}. Note the apparent typo in Theorem~1.13 of \cite{V},
which presumably should refer to ``Condition 2''. Since
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm
1=(-1)^{(k/2)-1}$. It is well known, and easy to prove, that
$$
\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).
$$
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
where the integral is taken along the positive imaginary axis,
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
Thus we are reduced
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
Let $\mathcal{D}_0$ be the group of divisors of degree zero
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
integer $r\geq 0$, let $P_r(R)$ be the additive group of
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
groups have a natural action of $\Gamma_1(N)$. Let
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
Via the isomorphism (8) in Section~1.5 of \cite{V}, combined with
the argument in 1.7 of \cite{V}, the cohomology class
$\omega_f^{\pm}$ corresponds to a modular symbol $\Phi_f^{\pm}\in
S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_{E,\qq})$. We are
now dealing with cohomology over $X_1(N)$ rather than $M(N)$,
which is why we insist that $q\nmid \phi(N)$. It follows from the
last line of Section~4.2 of \cite{St} that, up to some small
factorials which do not matter locally at $\qq$,
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since
$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
show is divisible by $\qq$.
Similarly
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of
$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
Therefore it would suffice to show that, for some $\mu\in O_E$,
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
$\qq$ in $S_{\Gamma_1(N)}(k,O_{E,\qq})$. It suffices to show that,
for some $\mu\in O_E$, the element
$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
non-constant coefficients. This would be the case if
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
one-dimensional subspace upon reduction modulo~$\qq$. But this is
a consequence of Theorem 2.1(1) of \cite{FJ} (for which we need
the irreducibility of $A[\qq]$).
\end{proof}
\begin{remar}\label{sign}
The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are
equal. They are determined by the eigenvalue of the
Atkin-Lehner involution~$W_N$,
which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and
$b_N$ are each $N^{k/2-1}$ times this sign and~$\qq$ has residue
characteristic coprime to $2N$. The common sign in the functional
equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of
$W_N$ acting on~$f$ and~$g$.
\end{remar}
This is analogous to the remark at the end of Section~3 of \cite{CM},
which shows that if~$\qq$ has odd residue characteristic and
$L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then $L(g,s)$ must vanish to order
at least two at $s=k/2$. Note that Maeda's conjecture
implies that there are no examples of~$g$ of
level one with positive sign in their functional equation such that
$L(g,k/2)=0$ (see \cite{CF}).
\section{Constructing elements of the Shafarevich-Tate group}
Let~$f$,~$g$ and $\qq$ be as in the first paragraph of the
previous section. In the previous section we showed how the
congruence between $f$ and $g$ relates the vanishing of $L(g,k/2)$
to the divisibility by $\qq$ of an ``algebraic part'' of
$L(f,k/2)$. Conjecturally the former is associated with the
existence of certain algebraic cycles (for $M_g$) while the latter
is associated with the existence of certain elements of the
Shafarevich-Tate group (for $M_f$, as we saw in \S 4). In this
section we show how the congruence, interpreted in terms of Galois
representations, provides a direct link between algebraic cycles
and the Shafarevich-Tate group.
For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
Recall that $L(g,k/2)=0$ and $L(f,k/2)\neq 0$. Since the sign in
the functional equation for $L(g,s)$ is positive (this follows
from $L(f,k/2)\neq 0$, see Remark \ref{sign}), the order of
vanishing of $L(g,s)$ at $s=k/2$ is at least $2$. According to the
Beilinson-Bloch conjecture \cite{B,Be}, the order of vanishing of
$L(g,s)$ at $s=k/2$ is the rank of the group
$\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational rational equivalence
classes of null-homologous, algebraic cycles of codimension $k/2$
on the motive $M_g$. (This generalises the part of the
Birch--Swinnerton-Dyer conjecture which says that for an elliptic
curve $E/\QQ$, the order of vanishing of $L(E,s)$ at $s=1$ is
equal to the rank of the Mordell-Weil group $E(\QQ)$.)
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
get (assuming also the Beilinson-Bloch conjecture) a subspace of
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
Sections~1 and~6.5 of \cite{Fo2}. We shall call it the ``strong''
Beilinson-Bloch conjecture.
Similarly, if $L(f,k/2)\neq 0$ then we expect that
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
coincides with the $\qq$-part of $\Sha$.
\begin{thm}\label{local}
Let $q\nmid N$ be a prime satisfying $q>k$. Let~$r$ be the
dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$. Suppose that $A[\qq]$ is
an irreducible representation of $\Gal(\Qbar/\QQ)$ and that for no
prime $p\mid N$ is $f$ congruent modulo $\qq$ (for Fourier
coefficients of index coprime to $Nq$) to a newform of weight~$k$,
trivial character and level dividing $N/p$. Suppose that, for all
primes $p\mid N$, $\,p\not\equiv -w_p\pmod{q}$, with $p\not\equiv
-1\pmod{q}$ if $p^2\mid N$. (Here $w_p$ is the common eigenvalue
of the Atkin-Lehner involution $W_p$ acting on $f$ and $g$.) Then
the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
$\FF_{\qq}$-rank at least $r$.
\end{thm}
\begin{proof}
The theorem is trivially true if $r=0$, so we assume that $r>0$.
It follows easily from our hypothesis that the rank of the free
part of $H^1_f(\QQ,T'_{\qq}(k/2))$ is~$r$. The natural map from
$H^1_f(\QQ,T'_{\qq}(k/2))/\qq H^1_f(\QQ,T'_{\qq}(k/2))$ to
$H^1(\QQ,A'[\qq](k/2))$ is injective. Take a nonzero class $c$ in
the image, which has $\FF_{\qq}$-rank $r$. Choose $d\in
H^1_f(\QQ,T'_{\qq}(k/2))$ mapping to $c$. Consider the
$\Gal(\Qbar/\QQ)$-cohomology of the short exact sequence
$$
\begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD},
$$
where~$\pi$ is multiplication by a uniformising element of
$O_{\qq}$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial.
Hence $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
we get a nonzero, $\qq$-torsion class $\gamma\in
H^1(\QQ,A_{\qq}(k/2))$.
Our aim is to show that $\res_p(\gamma)\in
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
\vspace{1em}
\noindent{\bf Case (1)} $p\nmid qN$:
Consider the $I_p$-cohomology of the short exact sequence above.
Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p,
A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
follows from the fact that $d\in H^1_f(\QQ,T'_{\qq}(k/2))$ that
the image in $H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is
zero, hence that the restriction of~$c$ to
$H^1(I_p,A'[\qq](k/2))\simeq H^1(I_p,A[\qq](k/2))$ is zero. Hence
the restriction of $\gamma$ to $H^1(I_p,A_{\qq}(k/2))$ is also
zero. By line~3 of p.~125 of \cite{Fl2},
$H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just contained in)
the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$ to
$H^1(I_p,A_{\qq}(k/2))$, so we have shown that $\res_p(\gamma)\in
H^1_f(\QQ_p,A_{\qq}(k/2))$.
\vspace{1em}
\noindent{\bf Case (2)} $p\mid N$:
First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
It suffices to show that
$$\hspace{3.5em}
\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),
$$
since then the natural map from $H^0(I_p,V'_{\qq}(k/2))$ to
$H^0(I_p, A'_{\qq}(k/2))$ is surjective; this may be done as in
the proof of Lemma \ref{local1}. It follows as above that the
image of $c\in H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is
zero. Then $\res_p(c)$ comes from
$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
order of this group is the same as the order of the group
$H^0(\QQ_p,A[\qq](k/2))$ (this is Lemma 1 of \cite{W}), which we
claim is trivial. By the work of Carayol \cite{Ca1}, the level $N$
is the conductor of $V_{\qq}(k/2)$, so $p\mid N$ implies that
$V_{\qq}(k/2)$ is ramified at $p$, hence $\dim
H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim
H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
consider the case where this common dimension is $1$. The
(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha
p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication
by~$\alpha$ on the one-dimensional space $H^0(I_p,V_{\qq})$. It
follows from Theor\'eme A of \cite{Ca1} that this is the same as
the Euler factor at $p$ of $L(f,s)$. By Theorems 3(ii) and 5 of
\cite{AL}, it then follows that $p^2\nmid N$ and
$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$) as
$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
$\res_p(\gamma)=0$ and certainly lies in
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
\vspace{1em}
\noindent{\bf Case (3)} $p=q$:
Since $q\nmid N$ is a prime of good reduction for the motive
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
$V'_{\qq}$ have the same dimension, where
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
filtered module $T_{\dR}\otimes O_{\qq}$. Since also $q>k$, we may
now prove, in the same manner as Proposition 9.2 of \cite{Du3},
that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$. For the
convenience of the reader, we give some details.
In Lemma 4.4 of \cite{BK}, a cohomological functor $\{h^i\}_{i\geq
0}$ is constructed on the Fontaine-Lafaille category of filtered
Dieudonn\'e modules over $\ZZ_q$. $h^i(D)=0$ for all $i\geq2$ and
all $D$, and $h^i(D)=\Ext^i(1_{FD},D)$ for all $i$ and $D$, where
$1_{FD}$ is the ``unit'' filtered Dieudonn\'e module.
Now let $D=T_{\dR}\otimes O_{\qq}$ and $D'=T'_{\dR}\otimes
O_{\qq}$. By Lemma 4.5 (c) of \cite{BK},
$$
\hspace{3.5em} h^1(D)\simeq H^1_e(\QQ_q,T_{\qq}),
$$
where
$$
\hspace{3.5em}H^1_e(\QQ_q,T_{\qq})=\ker(H^1(\QQ_q,T_{\qq})\rightarrow
H^1(\QQ_q,V_{\qq})/H^1_e(\QQ_q,V_{\qq}))
$$
and
$$
\hspace{3.5em}H^1_e(\QQ_q,V_{\qq})=\ker(H^1(\QQ_q,V_{\qq})\rightarrow
H^1(\QQ_q,B_{\cris}^{f=1}\otimes_{\QQ_q} V_{\qq})).
$$ Likewise
$h^1(D')\simeq H^1_e(\QQ_q,T'_{\qq}).$ When applying results of
\cite{BK} we view $D$, $T_{\qq}$ etc. simply as $\ZZ_q$-modules,
forgetting the $O_{\qq}$-structure.
For an integer $j$ let $D(j)$ be $D$ with the Hodge filtration
shifted by $j$. Then
$$\hspace{3.5em}
h^1(D(j))\simeq H^1_e(\QQ_q,T_{\qq}(j))
$$
(as long as $k-p+1\pi >>h^1(D'(k/2))@>>>h^1(D'(k/2)/\qq D'(k/2))\\
@VVV@VVV@VVV\\
H^1(\QQ_q,T'_{\qq}(k/2))@>\pi
>>H^1(\QQ_q,T'_{\qq}(k/2))@>>>H^1(\QQ_q,A'[\qq](k/2)).
\end{CD}$$
The vertical arrows are all inclusions and we know that the image
of $h^1(D'(k/2))$ in $H^1(\QQ_q,T'_{\qq}(k/2))$ is exactly
$H^1_f(\QQ_q,T'_{\qq}(k/2))$. The top right horizontal map is
surjective since $h^2(D'(k/2))=0$.
The class $\res_q(c)\in H^1(\QQ_q,A'[\qq](k/2))$ is in the image
of $H^1_f(\QQ_q,T'_{\qq}(k/2))$, by construction, and therefore is
in the image of $h^1(D'(k/2)/\qq D'(k/2))$. By the fullness and
exactness of the Fontaine-Lafaille functor \cite{FL} (see Theorem
4.3 of \cite{BK}), $D'(k/2)/\qq D'(k/2)$ is isomorphic to
$D(k/2)/\qq D(k/2)$.
It follows that the class $\res_q(c)\in H^1(\QQ_q,A[\qq](k/2))$ is
in the image of $h^1(D(k/2)/\qq D(k/2))$ by the vertical map in
the exact sequence analogous to the above. Since the map from
$h^1(D(k/2))$ to $h^1(D(k/2)/\qq D(k/2))$ is surjective,
$\res_q(c)$ lies in the image of $H^1_f(\QQ_q,T_{\qq}(k/2))$. From
this it follows that $\res_q(\gamma)\in
H^1_f(\QQ_q,A_{\qq}(k/2))$, as desired.
\end{proof}
Theorem~2.7 of \cite{AS} is concerned with verifying local
conditions in the case $k=2$, where~$f$ and~$g$ are associated
with abelian varieties~$A$ and~$B$. (Their theorem also applies to
abelian varieties over number fields.) Our restriction outlawing
congruences modulo $\qq$ with cusp forms of lower level is
analogous to theirs forbidding~$q$ from dividing Tamagawa factors
$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic
curve with $\ord_l(j(A))<0$, consideration of a Tate
parametrisation shows that if $q\mid c_{A,l}$, i.e., if
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
at~$l$.)
In this paper we have encountered two technical problems which we
dealt with in quite similar ways:
\begin{enumerate}
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
\item proving local conditions at primes $p\mid N$, for an element
of $\qq$-torsion.
\end{enumerate}
If our only interest was in testing the Bloch-Kato conjecture at
$\qq$, we could have made these problems cancel out, as in Lemma
8.11 of \cite{DFG}, by weakening the local conditions. However, we
have chosen not to do so, since we are also interested in the
Shafarevich-Tate group, and since the hypotheses we had to assume
are not particularly strong. Note that, since $A[\qq]$ is
irreducible, the $\qq$-part of $\Sha$ does not depend on the
choice of $T_{\qq}$.
\section{Examples and Experiments}
\label{sec:examples} This section contains tables and numerical
examples that illustrate the main themes of this paper. In
Section~\ref{sec:vistable}, we explain Table~\ref{tab:newforms},
which contains~$16$ examples of pairs $f,g$ such that the strong
Beilinson-Bloch conjecture and Theorem~\ref{local} together imply
the existence of nontrivial elements of the Shafarevich-Tate group
of the motive attached to~$f$. Section~\ref{sec:howdone} outlines
the higher-weight modular symbol computations that were used in
making Table~\ref{tab:newforms}. Section~\ref{sec:invis} discusses
Table~\ref{tab:invisforms}, which summarizes the results of an
extensive computation of conjectural orders of Shafarevich-Tate
groups for modular motives of low level and weight.
Section~\ref{sec:other_ex} gives specific examples in which
various hypotheses fail. Note that in \S 7 ``modular symbol'' has
a different meaning from in \S 5, being related to homology rather
than cohomology. For precise definitions see \cite{SV}.
\subsection{Visible $\Sha$ Table~\ref{tab:newforms}}\label{sec:vistable}
\begin{table}
\caption{\label{tab:newforms}Visible $\Sha$}\vspace{-3ex}
$$
\begin{array}{|c|c|c|c|c|}\hline
g & \deg(g) & f & \deg(f) & q\text{'}s \\\hline
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
\vspace{-2ex} & & & & \\
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
\vspace{-2ex} & & & & \\
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\
\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\
\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\
\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\
\vspace{-2ex} & & & & \\
\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\
\hline
\end{array}
$$
\end{table}
Table~\ref{tab:newforms} on page~\pageref{tab:newforms} lists
sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
along with at least one prime~$q$ such that there is a prime
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
The notation is as follows.
The first column contains a label whose structure is
\begin{center}
{\bf [Level]k[Weight][GaloisOrbit]}
\end{center}
This label determines a newform $g=\sum a_n q^n$, up to Galois
conjugacy. For example, \nf{127k4C} denotes a newform in the third
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
\ldots)$, then by the sequence of absolute values $|\mbox{\rm
Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
being first in the event that the two absolute values are equal,
and the first Galois orbit is denoted {\bf A}, the second {\bf B},
and so on. The second column contains the degree of the field
$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
contain~$f$ and its degree, respectively. The fifth column
contains at least one prime~$q$ such that there is a prime
$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
satisfied for~$f$,~$g$, and~$\qq$.
For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
prime $q$ appears in the $q$-column, meaning $q^2$ divides the
order of the group $S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp})$, defined
at the end of 7.3 below.
We describe the first line of Table~\ref{tab:newforms}
in more detail. See the next section for further details
on how the computations were performed.
Using modular symbols, we find that there is a newform
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
coefficients generate a number field~$K$ of degree~$17$, and by
computing the image of the modular symbol $XY\{0,\infty\}$ under
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
both equal to
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7
+ \cdots\in \FF_{43}[[q]].$$
There is no form in the Eisenstein subspaces of
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
of Theorem~\ref{local}, so if $r$ is the dimension of
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
Recall that since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that
$r\geq 2$. Then, since $L(f,k/2)\neq 0$, we expect that the
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to
the $\qq$-torsion subgroup of $\Sha$. Admitting these assumptions,
we have constructed the $\qq$-torsion in $\Sha$ predicted by the
Bloch-Kato conjecture.
For particular examples of elliptic curves one can often find and
write down rational points predicted by the Birch and
Swinnerton-Dyer conjecture. It would be nice if likewise one could
explicitly produce algebraic cycles predicted by the
Beilinson-Bloch conjecture in the above examples. Since
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
0.3.2 of \cite{Z}), so ought to be trivial in
$\CH_0^{k/2}(M_g)\otimes\QQ$.
\subsection{How the computation was performed}\label{sec:howdone}
We give a brief summary of how the computation was performed. The
algorithms that we used were implemented by the second author, and
most are a standard part of MAGMA (see \cite{magma}).
Let~$g$,~$f$, and~$q$ be some data from a line of
Table~\ref{tab:newforms} and let~$N$ denote the level of~$g$. We
verified the existence of a congruence modulo~$q$, that
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
To prove there is a congruence, we showed that the corresponding
{\em integral} spaces of modular symbols satisfy an appropriate
congruence, which forces the existence of a congruence on the
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
irreducible by computing a set that contains all possible residue
characteristics of congruences between~$g$ and any Eisenstein
series of level dividing~$N$, where by congruence, we mean a
congruence for all Fourier coefficients of index~$n$ with
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
listing a basis of such~$h$ and finding the possible congruences,
where again we disregard the Fourier coefficients of index not
coprime to~$N$.
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
modular symbol ${\mathbf e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$
under a map with the same kernel as the period mapping, and found that the
image was~$0$. The period mapping sends the modular
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
so that ${\mathbf e}$ maps to~$0$ implies that
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g)
=(-1)^{k/2} g$ which, because of the functional equation, implies
that $L'(g,\frac{k}{2})=0$. Table~\ref{tab:newforms} is of
independent interest because it includes examples of modular forms
of even weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
the functional equation. We found no such examples of weights
$\geq 8$.
\subsection{Conjecturally nontrivial $\Sha$}\label{sec:invis}
In this section we apply some of the results of
Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders
of Shafarevich-Tate groups of many modular motives. The results of
this section suggest that~$\Sha$ of a modular motive is usually not
``visible at level~$N$'', i.e., explained by congruences at level~$N$,
which agrees with the observations of \cite{CM} and \cite{AS}. For
example, when $k>6$ we find many examples of conjecturally
nontrivial~$\Sha$ but no examples of nontrivial visible~$\Sha$.
For any newform~$f$, let $L(M_f/\QQ,s) = \prod_{i=1}^{d}
L(f^{(i)},s)$ where $f^{(i)}$ runs over the
$\Gal(\Qbar/\QQ)$-conjugates of~$f$. Let~$T$ be the complex torus
$\CC^d/(2\pi i)^{k/2}\mathcal{L}$, where the lattice $\mathcal{L}$
is defined by integrating integral cuspidal modular symbols (for
$\Gamma_0(N)$) against the conjugates of~$f$. Let
$\Omega_{M_f/\QQ}$ denote the volume of the $(-1)^{(k/2)-1}$
eigenspace $T^{\pm}=\{z \in T : \overline{z}=(-1)^{(k/2)-1}z\}$
for complex conjugation on~$T$.
\newpage
{\begin{table}
\vspace{-2ex}
\caption{\label{tab:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}
\vspace{-4ex}
$$
\begin{array}{|c|c|c|c|}\hline
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\
\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\
\nf{263k4B} & 39 & 41^{2} & 263 \\
\nf{269k4C} & 39 & 23^{2} & 269 \\
\nf{271k4B} & 39 & 29^{2} & 271 \\
\nf{281k4B} & 40 & 29^{2} & 281 \\
\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\
\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\
%\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\
% 319k4C removed since Lemma not satisfied.
\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\
\hline
\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\
\nf{101k6B} & 24 & 17^{2} & 101 \\
\nf{103k6B} & 24 & 23^{2} & 103 \\
\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\
\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\
\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\
\nf{157k6B} & 34 & 251^{2} & 157 \\
\nf{167k6B} & 40 & 41^{2} & 167 \\
\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\
\nf{173k6B} & 39 & 71^{2} & 173 \\
\nf{181k6B} & 40 & 107^{2} & 181 \\
\nf{191k6B} & 46 & 85091^{2} & 191 \\
\nf{193k6B} & 41 & 31^{2} & 193 \\
\nf{199k6B} & 46 & 200329^2 & 199 \\
\hline
\nf{47k8B} & 16 & 19^{2} & 47 \\
\nf{59k8B} & 20 & 29^{2} & 59 \\
\nf{67k8B} & 20 & 29^{2} & 67 \\
\nf{71k8B} & 24 & 379^{2} & 71 \\
\nf{73k8B} & 22 & 197^{2} & 73 \\
\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\
\nf{79k8B} & 25 & 307^{2} & 79 \\
\nf{83k8B} & 27 & 1019^{2} & 83 \\
\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\
\nf{89k8B} & 29 & 44491^{2} & 89 \\
\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\
\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\
\nf{103k8B} & 32 & 75367^{2} & 103 \\
\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\
\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\
\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\
\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\
\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\
\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\
\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\
\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\
\hline
\end{array}
$$
\end{table}
\begin{table}
$$
\begin{array}{|c|c|c|c|}\hline
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\
\nf{121k8G} & 12 & 13^{2} & 3, 11 \\
\nf{121k8H} & 12 & 19^{2} & 5, 11 \\
\nf{125k8D} & 16 & 179^{2} & 5 \\
\nf{127k8B} & 39 & 59^{2} & 127 \\
\nf{128k8F} & 4 & 11^{2} & 1 \\
\nf{131k8B} & 43 & 241^{2} \!\cdot\! 817838201^{2}&131\\
\nf{134k8C} & 11 & 61^{2} & 11, 17, 41, 67, 71, 421, 2356138931854759 \\
\nf{137k8B} & 42 & 71^{2} \!\cdot\! 749093^{2} & 137 \\
\nf{139k8B} & 43 & 47^{2} \!\cdot\! 89^{2} \!\cdot\! 1021^{2} & 139 \\
\nf{141k8C} & 14 & 13^{2} & 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\
\nf{142k8B} & 10 & 11^{2} & 3, 53, 71, 56377, 1965431024315921873 \\
\nf{143k8C} & 19 & 307^{2} & 3, 11, 13, 89, 199, 409, 178397,
639259, 17440535
97287 \\
\nf{143k8D} & 21 & 109^{2} & 3, 7, 11, 13, 61, 79, 103, 173, 241,
769, 36583
\\
\nf{145k8C} & 17 & 29587^{2} & 5, 11, 29, 107, 251623, 393577,
518737, 9837145
699 \\
\nf{146k8C} & 12 & 3691^{2} & 11, 73, 269, 503, 1673540153, 11374452082219 \\
\nf{148k8B} & 11 & 19^{2} & 3, 37 \\
\nf{149k8B} & 47 & 11^{4} \!\cdot\! 40996789^{2} & 149\\
\hline
\nf{43k10B} & 17 & 449^{2} & 43 \\
\nf{47k10B} & 20 & 2213^{2} & 47 \\
\nf{53k10B} & 21 & 673^{2} & 53 \\
\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\
\nf{59k10B} & 25 & 37^{2} & 59 \\
\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\
\nf{64k10K} & 2 & 19^{2} & 3 \\
\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\
\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\
\nf{71k10B} & 30 & 1103^{2} & 71 \\
\hline
\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\
\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\
\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\
\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\
\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\
\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\
\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\
\nf{49k12H} & 12 & 271^{2} & 7 \\
\hline
\end{array}
$$
\end{table}
\begin{lem}\label{lem:lrat}
Suppose that $p\nmid Nk!$ is such that~$f$ is not congruent to any of its
Galois conjugates modulo a prime dividing~$p$. Then the $p$-parts
of
$$
\frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}\qquad\text{and}\qquad
\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right)
$$
are equal, where $\vol_\infty$ is as in Section~\ref{sec:bkconj}.
\end{lem}
\begin{proof} Let~$H$ be the $\ZZ$-module of all
integral cuspidal modular symbols for $\Gamma_0(N)$. Let~$I$ be the
image of~$H$ under projection into
the submodule of $H\otimes\QQ$ corresponding
to~$f$ and its Galois conjugates. Note that~$I$ is not necessarily
contained in~$H$, but it is contained in $H\otimes \ZZ[\frac{1}{m}]$
where~$m$ is divisible by the residue
characteristics of any primes of congruence between~$f$ and cuspforms
of weight~$k$ for $\Gamma_0(N)$ which are not Galois conjugate to~$f$.
The lattice $\mathcal{L}$ defined in the paragraph before
the lemma is (up to divisors of $Nk!$)
obtained by pairing the cohomology modular symbols
$\Phi_{f^{(i)}}^{\pm}$ (as in \S 5) with the homology modular
symbols in~$H$; equivalently, since the pairing factors
through the map $H\to I$, the lattice $\mathcal{L}$ is obtained
by pairing with the elements of~$I$.
For $1\leq i\leq d$ let
$I_i$ be the $O_E$-module generated by the image of the projection
of~$I$ into $I\otimes E$ corresponding to $f^{(i)}$.
The finite
index of $I\otimes O_E$ in $\oplus_{i=1}^d I_i$ is divisible only
by primes of congruence between $f$ and its Galois conjugates. Up
to these primes, $\Omega_{M_f/\QQ}/(2\pi i)^{((k/2)-1)d}$ is then
a product of the $d$ quantities obtained by pairing
$\Phi_{f^{(i)}}^{\pm}$ with $I_i$, for $1\leq i\leq d$. (These quantities
inhabit a kind of tensor product of $\CC^*$ over $E^*$ with the
group of fractional
ideals of $E$.) Bearing in
mind the last line of \S 3, we see that these quantities are the
$\aaa^{\pm}\Omega^{\pm}_{f^{(i)}}$, up to divisors of $Nk!$.
Now we may apply Lemma \ref{vol}. We have then a
factorisation of the left hand side which shows it to be equal to the
right hand side, to the extent claimed by the lemma. Note that
$\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}$ has an interpretation in terms
of integral modular symbols, as in \S 5, and just gets Galois-conjugated when
one replaces $f$ by some $f^{(i)}$.
\end{proof}
\begin{remar}
The newform $f=\nf{319k4C}$ is congruent to one of its Galois conjugates
modulo~$17$ and $17\mid \frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}$ so the lemma
and our computations
say nothing about whether or not $17$ divides
$\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right)$.
\end{remar}
Let~$\mathcal{S}$ be the set of newforms with~level $N$ and
weight~$k$ satisfying either $k=4$ and $N\leq 321$, or $k=6$ and
$N\leq 199$, or $k=8$ and $N\leq 149$, or $k=10$ and $N\leq 72$,
or $k=12$ and $N\leq 49$. Given $f\in \mathcal{S}$, let~$B$ be
defined as follows:
\begin{enumerate}
\item Let $L_1$ be the numerator of the
rational number $L(M_f/\QQ,k/2)/\Omega_{M_f/\QQ}$.
If $L_1=0$ let $B=1$ and terminate.
\item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$.
\item Let $L_3$ be the part of $L_2$ that is coprime to
$p+1$ for every prime~$p$ such that $p^2\mid N$.
\item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic
of any prime of
congruence between~$f$ and a form of weight~$k$ and
lower level. (By congruence here, we mean a congruence for coefficients
$a_n$ with $n$ coprime to the level of~$f$.)
\item Let $L_5$ be the part of $L_4$ coprime to the residue characteristic
of any prime of congruence
between~$f$ and an Eisenstein series. (This eliminates
residue characteristics of reducible representations.)
\item Let $B$ be the part of $L_5$ coprime to the residue characteristic
of any prime of congruence between $f$ and any one of its Galois
conjugates.
\end{enumerate}
Proposition~\ref{sha} and Lemma~\ref{lem:lrat} imply that if
$\ord_p(B) > 0$ then, according
to the Bloch-Kato conjecture, $\ord_p(\#\Sha)=\ord_p(B) > 0$.
We computed~$B$ for every newform in~$\mathcal{S}$. There are
many examples in which $L_3$ is large, but~$B$ is not, and this is
because of Tamagawa factors. For example, {\bf 39k4C} has
$L_3=19$, but $B=1$ because of a $19$-congruence with a form of
level~$13$; in this case we must have $19\mid c_{3}(2)$, where
$c_{3}(2)$ is as in Section~\ref{sec:bkconj}. See
Section~\ref{sec:other_ex} for more details. Also note that in
every example~$B$ is a perfect square, which, away from congruence
primes, is as predicted by the existence of Flach's generalised
Cassels-Tate pairing \cite{Fl2}. (Note that if $A[\lambda]$ is
irreducible then the lattice $T_{\lambda}$ is at worst a scalar
multiple of its dual, so the pairing shows that the order of the
$\lambda$-part of $\Sha$, if finite, is a square.) That our
computed value of~$B$ should be a square is not {\it a priori}
obvious.
For simplicity, we discard residue characteristics instead of primes
of rings of integers, so our definition of~$B$ is overly conservative.
For example,~$5$ occurs in row~$2$ of Table~\ref{tab:newforms} but not
in Table~\ref{tab:invisforms}, because \nf{159k4E} is Eisenstein at
some prime above~$5$, but the prime of congruences of
characteristic~$5$ between \nf{159k4B} and \nf{159k4E} is not
Eisenstein.
The newforms for which $B>1$ are given in
Table~\ref{tab:invisforms}. The second column of the table records the
degree of the field generated by the Fourier coefficients of~$f$. The
third contains~$B$. Let~$W$ be the intersection of the span of all
conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and $W^{\perp}$ the
Petersson orthogonal complement of~$W$ in $S_k(\Gamma_0(N),\ZZ)$. The
fourth column contains the odd prime divisors of
$\#(S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp}))$, which are exactly the
possible primes of congruence between~$f$ and non-conjugate cusp forms
of the same weight and level. We place a~$*$ next to the four entries
of Table~\ref{tab:invisforms} that also occur in
Table~\ref{tab:newforms}.
\subsection{Examples in which hypotheses fail}\label{sec:other_ex}
We have some other examples where forms of different levels are
congruent (for Fourier coefficients of index coprime to the
levels). However, Remark~\ref{sign} does not apply, so that one of
the forms could have an odd functional equation, and the other
could have an even functional equation. For instance, we have a
$19$-congruence between the newforms $g=\nf{13k4A}$ and
$f=\nf{39k4C}$ of Fourier coefficients of index coprime to $39$.
Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$ has {\it odd}
functional equation. Here~$f$ fails the condition about not being
congruent to a form of lower level, so in Lemma~\ref{local1} it is
possible that $\ord_{\qq}(c_{3}(2))>0$. In fact this does happen.
Because $V'_{\qq}$ (attached to~$g$ of level $13$) is unramified
at $p=3$, $H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
two-dimensional. As in (2) of the proof of Theorem~\ref{local},
one of the eigenvalues of $\Frob_p^{-1}$ acting on this
two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
our example here with $p=3$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
nontrivial when $w_p=-1$, so (2) of the proof of
Theorem~\ref{local} does not work. This is just as well, since had
it worked we would have expected
$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
shows not to be the case.
In the following example, the divisibility between the levels is
the other way round. There is a $7$-congruence between
$g=\nf{122k6A}$ and $f=\nf{61k6B}$, both $L$-functions have even
functional equation, and $L(g,3)=0$. In the proof of
Theorem~\ref{local}, there is a problem with the local condition
at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to
$H^1(I_2,A'_{\qq}(3))$ is not necessarily injective, but its
kernel is at most one dimensional, so we still get the
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
$\FF_{\qq}$-rank at least~$1$ (assuming $r\geq 2$), and thus get
elements of $\Sha$ for \nf{61k6B} (assuming all along the strong
Beilinson-Bloch conjecture). In particular, these elements of
$\Sha$ are {\it invisible} at level 61. When the levels are
different we are no longer able to apply Theorem 2.1 of \cite{FJ}.
However, we still have the congruences of integral modular symbols
required to make the proof of Proposition \ref{div} go through.
Indeed, as noted above, the congruences of modular forms were
found by producing congruences of modular symbols. Despite these
congruences of modular symbols, Remark~\ref{sign} does not apply,
since there is no reason to suppose that $w_N=w_{N'}$, where $N$
and $N'$ are the distinct levels.
Finally, there are two examples where we have a form $g$ with even
functional equation such that $L(g,k/2)=0$, and a congruent form
$f$ which has odd functional equation; these are a 23-congruence
between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
$\ord_{s=2}L(f,s)=1$, it ought to be the case that
$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
the appropriate modification of Theorem \ref{local} (with strong
Beilinson-Bloch conjecture) does not necessarily provide us with
nontrivial $\qq$-torsion in $\Sha$. It only tells us that the
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ has
$\FF_{\qq}$-rank at least $1$. It could all be in the image of
$H^1_f(\QQ,V_{\qq}(2))$. $\Sha$ appears in the conjectural formula
for the first derivative of the complex $L$ function, evaluated at
$s=k/2$, but in combination with a regulator that we have no way
of calculating.
Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
associated with $f$ and $g$ by the construction of Mazur, Tate and
Teitelbaum \cite{MTT}, each divided by a suitable canonical
period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
quite clear what to make of this. This divisibility may be proved
as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
normalised) are congruent $\bmod{\,\qq}$, as a result of the
congruence between the modular symbols out of which they are
constructed. Integrating an appropriate function against these
measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
where the signs in the functional equations of $L(g,s)$ and
$L_q(g,s)$ are the same, positive in this instance. (According to
the proposition in Section 18 of \cite{MTT}, the signs differ
precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)
We also found some examples for which the conditions of
Theorem~\ref{local} were not met. For example, we have a
$7$-congruence between \nf{639k4B} and \nf{639k4H}, but
$w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a
similar problem with a $7$-congruence between \nf{260k6A} and
\nf{260k6E} --- here $w_{13}=1$ so that $13\equiv
-w_{13}\pmod{7}$. According to Propositions \ref{div} and
\ref{sha}, Bloch-Kato still predicts that the $\qq$-part of $\Sha$
is non-trivial in these examples. Finally, there is a
$5$-congruence between \nf{116k6A} and \nf{116k6D}, but here the
prime~$5$ is less than the weight~$6$ so Propositions \ref{div}
and \ref{sha} (and even Lemma~\ref{lem:lrat}) do not apply.
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\end{document}