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\markboth{\qquad Shafarevich--Tate groups of modular motives
\hfill}{\hfill Neil Dummigan, William Stein and Mark Watkins \qquad}
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% [Shafarevich--Tate groups of modular motives]
\title{Constructing elements in Shafarevich--Tate groups of modular
motives}
\author{Neil Dummigan \and William Stein \and Mark Watkins}
\date{}
% \subjclass{11F33, 11F67, 11G40.}
% \keywords{modular form, $L$-function, visibility, Bloch--Kato
% conjecture, Shafarevich--Tate group.}
\begin{document}
\maketitle
\addcontentsline{toc}{chapter}{Neil Dummigan, William Stein and Mark
Watkins, Constructing elements in Shafarevich--Tate groups of modular
motives}
% \tableofcontents
\begin{abstract}
We study Shafarevich--Tate groups of motives attached to modular
forms on $\Gamma_0(N)$ of weight $>2$. 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 would imply 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 $\le12$. Our methods build upon the
idea of visibility due to Cremona and Mazur, but in the context of
motives rather than abelian varieties.
\end{abstract}
\maketitle
\section{Introduction}
Let $E$ be an elliptic curve defined over $\QQ$ and $L(E,s)$
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 $N\le 5500$, 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
$L(f,k/2)\ne0$. It turns out that this forces $L(g,s)$ to vanish
to order $\ge2$ 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 multi\-plicity
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).
\clearpage
\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 $k\ge2$ 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
$$\rho_{\lambda}\colon\Gal(\Qbar/\QQ)\to \Aut(V_{\lambda}),$$
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}, we can construct $V_{\lambda}$ 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 $p\ne\ell$ (including $p=\infty$), we
let
$$
H^1_f(\QQ_p,V_{\lambda}(j))=\ker \Bigl(H^1(D_p,V_{\lambda}(j))\to
H^1(I_p,V_{\lambda}(j))\Bigr).
$$
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
$$
H^1_f(\QQ_{\ell},V_{\lambda}(j))=\ker
\Bigl(H^1(D_{\ell},V_{\lambda}(j))\to
H^1(D_{\ell},V_{\lambda}(j)\otimes_{\QQ_{\ell}} B\cris)\Bigr)
$$
(see \cite{BK}, Section~1 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
$$
0 \to T_{\lambda}(j) \to
V_{\lambda}(j) \xrightarrow{\ \pi\ } A_{\lambda}(j) \to0.
$$
Let
$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
We then define the $\lambda$-Selmer group
$H^1_f(\QQ,A_{\lambda}(j))$ as 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
$$
\Sha(j)=\bigoplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/
\pi_*H^1_f(\QQ,V_{\lambda}(j)).
$$
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~$E$. We can choose an abelian variety $B$
in the isogeny class whose endomorphism ring
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~$B$ (here we assume finiteness of the latter, or just take the
quotient by its maximal divisible subgroup). To see this one uses
\cite{BK}, 3.11 for $\ell=p$. For $\ell\ne p$,
$H^1_f(\QQ_p,V_{\ell})=0$.
Considering the formal group, we can represent every class in
$B(\QQ_p)/\ell B(\QQ_p)$ by an $\ell$-power torsion point
in $B(\QQ_p)$, so that it maps to zero in $H^1(\QQ_p,A_{\ell})$.
Define the group of global torsion points
$$
\Gamma_{\QQ}=\bigoplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).
$$
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}
{From} now on, we assume for convenience that $N\ge3$. 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 such a way that $T_B$ and $T\dR\otimes_{O_E}O_E[1/Nk!]$
agree respectively with 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 nonconstant coefficients, of
modular curves. (See especially \cite{DFG}, 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
$\mathfrak{E}^{k-2}$). We realise $M(N)(\CC)$ as the disjoint union
of $\varphi(N)$ copies of the quotient
$\Gamma(N)\backslash\mathfrak{H}^*$ (where $\mathfrak{H}^*$ is the
completed upper half plane), and let $\tau$ be a variable on
$\mathfrak{H}$, so that 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 $2\pi i f(\tau)\,d\tau\wedge dz_1\wedge\cdots\wedge dz_{k-2}$
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 $L(f,k/2)\ne0$. 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 $l\ne p$, $\ord_{\lambda}(c_p(j))$ is
defined to be
\begin{multline*}
\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{multline*}
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 Faltings \cite{Fa1}, Theorem~5.6.)
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 \cite{BK}, (5.15.1). 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. \cite{Fo2}, Section~11 sketches 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 $\vol_{\infty}$ (cf.\ \cite{De2}, last
paragraph of 1.7). Here we are using 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}\rm 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 \cite{Fl1}, p.~30, for odd $l\ne p$,
$\ord_{\lambda}(c_p(j))$ 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 $\,p\not\equiv \pm 1\pmod{q}$. 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
$$
\dim_{O_E/\qq} H^0(I_p,A[\qq](j))\ge\dim_{E_{\qq}}
H^0(I_p,V_{\qq}(j)).
$$ 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 $H^0(I_p,A_{\qq}(j))=
V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$, and therefore that
$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p})$.
Suppose that Condition (b) of \cite{L}, Proposition~2.3 is not
satisfied. Then there exists a character
$\chi\colon\Gal(\Qbar/\QQ)\to O_{\qq}^{\times}$ of $q$-power
order such that the $p$-part of the conductor of
$V_{\qq}\otimes\chi$ is strictly smaller than that of $V_{\qq}$.
Let $f_{\chi}$ denote the {\em newform}, of level dividing $N/p$,
associated with $V_{\qq}\otimes\chi$. The character of $f_{\chi}$
has conductor at worst $p$. Since $\chi$ has conductor $p$ and
$q$-power order, $p\equiv 1\pmod{q}$, so by hypothesis $p^2\nmid
N$. Hence $f_{\chi}$ has level coprime to $p$ and must have
trivial character. Then the existence of $f_{\chi}$ contradicts
our hypotheses.
Suppose now that
$$
\dim_{O_E/\qq} H^0(I_p,A[\qq](j))>\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)),
$$ (if not, there is nothing to prove). If Condition (a) of \cite{L},
Proposition~2.3
were not satisfied then \cite{L}, Proposition~2.2 would imply
the existence of an impossible twist, as in the previous
paragraph. (Here we are also using \cite{L}, Proposition~1.1.)
Since Condition (c) is clearly also satisfied, we are in a
situation covered by one of the three cases in \cite{L}, Proposition~2.3.
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 \cite{JL},
Theorem~1 (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}\rm
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 \cite{DFG}, Lemma~5.7 (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 \cite{DFG}, Section~1.2.)
Given this, the lemma follows from \cite{BK}, Theorem~4.1(iii). (That
$\mathbb{V}$ is the same as the functor used in \cite{BK}, Theorem~4.1
follows from \cite{Fa1}, first paragraph of 2(h).)
\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
$L(f,k/2)\aaa^{\pm}/\vol_{\infty}\ne0$ 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 $k\ge2$ 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 $q\nmid N\varphi(N)k!$. 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:
$$L(f,k/2)\ne0 \qquad\text{and}\qquad L(g,k/2)=0.$$
\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 \cite{V},
Section~1. Note the apparent typographical error in \cite{V},
Theorem~1.13 which should presumably refer to ``Condition 2''. Since
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
$\ord_{\qq}\bigl(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm})\bigr)>0$, 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 $0\le j\le k-2$ 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 $\sD_0$ be the group of divisors of degree zero
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
integer $r\ge0$, 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
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\sD_0,P_{k-2}(R))$
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
Via the isomorphism (8) of \cite{V}, Section~1.5 combined with
the argument of \cite{V}, 1.7, 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 $q\nmid \varphi(N)$. It follows from the
last line of \cite{St}, Section~4.2 that, up to some small
factorials which do not matter locally at $\qq$,
\[
\Phi_f^{\pm}([\infty]-[0])
=\sum^{k-2}_{\substack{j=0,\\ j\equiv (k/2)-1\,\mathrm{(mod\,2)}}}
r_f(j)X^jY^{k-2-j}.
\]
Since $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
\[
\Delta_f^{\pm}([\infty]-[0])
=\sum^{k-2}_{\substack{j=0,\\ j\equiv (k/2)-1\,\mathrm{(mod\,2)}}}
(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.
\]
The
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
show is divisible by $\qq$.
Similarly
\[
\Phi_g^{\pm}([\infty]-[0])
=\sum^{k-2}_{\substack{j=0,\\ j\equiv (k/2)-1\,\mathrm{(mod\,2)}}}
r_g(j)X^jY^{k-2-j}.
\]
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
nonconstant 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 \cite{FJ}, Theorem 2.1(1) (for which we need
the irreducibility of $A[\qq]$).
\end{proof}
\begin{remar}\label{sign}\rm
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 \cite{CM}, remark at the end of Section~3, which
shows that if~$\qq$ has odd residue characteristic and
$L(f,k/2)\ne0$ 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 Chebotarev 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 $L(f,k/2)\ne0$. Since the sign in
the functional equation for $L(g,s)$ is positive (this follows
from $L(f,k/2)\ne0$, 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 \cite{Ne2}, 3.1 and 3.2.
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 $C^i_{\lambda}(M)$ of
\cite{Fo2}, Sections~1 and~6.5. We shall call it the ``strong''
Beilinson--Bloch conjecture.
Similarly, if $L(f,k/2)\ne0$ 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
$p\not\equiv\pm1\pmod{q}$ 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
$$
0 \to A[\qq](k/2) \to A_{\qq}(k/2) \xrightarrow{\ \pi\ } A_{\qq}(k/2)
\to 0,
$$
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 that
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
we get a nonzero $\qq$-torsion class $\gamma\in
H^1(\QQ,A_{\qq}(k/2))$.
%%%%%%%%%%%%%%%%%%%%%%%%%
% On p.99, l.-10 is an exact sequence which shouldn't really have the
% primes on the As. The trouble with just deleting them is that when it
% is referred to on l.-2 (p.99) they should be there. Since things with
% primes are associated with g while things without them are associated
% with f, perhaps this could be corrected by making l.-11 "...cohomology
% of the analogue for $f$ of the short exact sequence". This is a bit
% weird but it will serve the purpose of preventing the reader from
% thinking that we have made a mistake, and losing confidence while
% reading the next few lines. It also avoids repeating the exact sequence
% (plus primes) on l.-1, which would have typesetting implications for
% the following page.
%%%%%%%%%%%%%%%%%%%%%%%%%
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 $p=q$.
\subsubsection*{Case 1, $p\nmid qN$:}
Consider the $I_p$-cohomology of the short exact sequence
$$
0 \to A'[\qq](k/2) \to A'_{\qq}(k/2) \xrightarrow{\ \pi\ } A'_{\qq}(k/2)
\to 0
$$
(the analogue for $g$ of the 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 \cite{Fl2}, line~3 of p.~125,
$H^1_f(\QQ_p,A_{\qq}(k/2))$ 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 $\res_p(\gamma)\in
H^1_f(\QQ_p,A_{\qq}(k/2))$.
\subsubsection*{Case 2, $p\mid N$:}
We first show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
It suffices to show that
$$
\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),
$$
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 \cite{W}, Lemma~1), 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 \cite{Ca1}, Theor\'eme~A that this is the same as
the Euler factor at $p$ of $L(f,s)$. By \cite{AL}, Theorems~3(ii) and~5,
it then follows that $p^2\nmid N$ and
$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
$W_pf=w_pf$. We twist by $k/2$, so that $\Frob_p^{-1}$ 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
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
\subsubsection*{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
$\Gal(\Qbar_q/\QQ_q)$, which means that $D\cris(V'_{\qq})$ 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 \cite{Fa1}, Theorem 5.6.)
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 \cite{Du3}, Proposition~9.2,
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 \cite{BK}, Lemma~4.4, a cohomological functor $\{h^i\}_{i\ge0}$ 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 \cite{BK}, Lemma~4.5(c),
$$
h^1(D)\simeq H^1_e(\QQ_q,T_{\qq}),
$$
where
$$
H^1_e(\QQ_q,T_{\qq})=\ker(H^1(\QQ_q,T_{\qq})\to
H^1(\QQ_q,V_{\qq})/H^1_e(\QQ_q,V_{\qq}))
$$
and
$$
H^1_e(\QQ_q,V_{\qq})=\ker(H^1(\QQ_q,V_{\qq})\to
H^1(\QQ_q,B\cris^{f=1}\otimes_{\QQ_q} V_{\qq})).
$$
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
$$
h^1(D(j))\simeq H^1_e(\QQ_q,T_{\qq}(j))
$$
(provided that $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 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 \cite{BK},
Theorem~4.3), $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}
\cite{AS}, Theorem~2.7 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 \cite{DFG},
Lemma~8.11, 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 illustrating 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 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{Table \ref{tab!newforms}: visible $\protect\Sha$}
\label{sec!vistable}
Table~\ref{tab!newforms}
% on page~\pageref{tab!newforms}
lists sixteen pairs of newforms~$f$ and~$g$
\begin{table}[ht]
\parindent=0cm \tabcolsep=.6em \renewcommand{\arraystretch}{1.4}
\newcommand{\entry}[6]{
$#1$ & \hfil #2 \hfil & $#3$ & \hfil #4 \hfil & #5 \\[#6pt]
}
\centering
\begin{tabular}{|p{1.6cm}|p{1.1cm}|p{1.6cm}|p{1.1cm}|p{1.8cm}|}
\hline
\hfil$g$\hfil & $\deg g$ & \hfil$f$\hfil & $\deg f$ & possible $q$ \\
\hline
\entry{\nf{127k4A}} {1} {\nf{127k4C}} {17} {43} {0}
\entry{\nf{159k4B}} {1} {\nf{159k4E}} {16} {5, 23} {0}
\entry{\nf{365k4A}} {1} {\nf{365k4E}} {18} {29} {0}
\entry{\nf{369k4B}} {1} {\nf{369k4I}} {9} {13} {8}
\entry{\nf{453k4A}} {1} {\nf{453k4E}} {23} {17} {0}
\entry{\nf{465k4B}} {1} {\nf{465k4I}} {7} {11} {0}
\entry{\nf{477k4B}} {1} {\nf{477k4L}} {12} {73} {0}
\entry{\nf{567k4B}} {1} {\nf{567k4H}} {8} {23} {8}
\entry{\nf{581k4A}} {1} {\nf{581k4E}} {34} {$19^2$} {0}
\entry{\nf{657k4A}} {1} {\nf{657k4C}} {7} {5} {0}
\entry{\nf{657k4A}} {1} {\nf{657k4G}} {12} {5} {0}
\entry{\nf{681k4A}} {1} {\nf{681k4D}} {30} {59} {8}
\entry{\nf{684k4C}} {1} {\nf{684k4K}} {4} {$7^2$} {0}
\entry{\nf{95k6A}} {1} {\nf{95k6D}} {9} {31, 59} {0}
\entry{\nf{122k6A}} {1} {\nf{122k6D}} {6} {73} {0}
\entry{\nf{260k6A}} {1} {\nf{260k6E}} {4} {17} {0}
\hline
\end{tabular}
\caption{Visible $\protect\Sha$}
\label{tab!newforms}
\end{table}
(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)\ge2$ while $L(f,k/2)\ne0$. It uses the following
notation: 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))$. 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} 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, which means that $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. The next section gives 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))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
coefficients generate a number field~$K$ of degree~$17$, and by
computing the image of the modular symbol $XY\{0,\infty\}$ under
the period mapping, we find that $L(f,2)\ne0$. The newforms~$f$
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
both equal to
$$\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)\ge2$, we expect that
$r\ge2$. Then, since $L(f,k/2)\ne0$, 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 \cite{Z},
Corollary~0.3.2), 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
al\-gorithms 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,k/2)=L'(g,k/2)=0$ and $L(f,k/2)\ne0$, 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,k/2)=0$, we computed the image of the
modular symbol $\be=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~$\be$ to a nonzero multiple of $L(g,\frac{k}{2})$,
so that $\be$ maps to~$0$ implies that
$L(g,k/2)=0$. In a similar way, we verified that
$L(f,k/2)\ne0$. Next, we checked that $W_N(g)=(-1)^{k/2} g$
which, because of the functional equation, implies
that $L'(g,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 $k/2$ that is not forced by
the functional equation. We found no such examples of weights
$\ge8$.
\subsection{Conjecturally nontrivial $\protect\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$'', that is, not explained by congruences at
level~$N$ (compare 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}\sL$, where $\sL$ is the lattice 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$.
\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 $\sL$ defined before the lemma is obtained (up to divisors
of $Nk!$) by pairing the cohomology modular symbols
$\Phi_{f^{(i)}}^{\pm}$ (as in \S5) with the homology modular
symbols in~$H$; equivalently, since the pairing factors
through the map $H\to I$, the lattice $\sL$ is obtained
by pairing with the elements of~$I$.
For $1\le i\le 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 $\bigoplus_{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\le i\le 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 then have 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}\rm
The newform $f=\nf{319k4C}$ is congruent to one of its Galois conjugates
modulo~$17$ and $17$ divides $L(M_f/\QQ,k/2)/\Omega_{M_f/\QQ}$, so the
lemma and our computations say nothing about whether $17$ divides
$\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right)$ or otherwise.
\end{remar}
Let~$\sS$ be the set of newforms with~level $N$ and
weight~$k$ satisfying either $k=4$ and $N\le 321$, or $k=6$ and
$N\le 199$, or $k=8$ and $N\le 149$, or $k=10$ and $N\le 72$,
or $k=12$ and $N\le 49$. Given $f\in \sS$, 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\pm 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$, trivial character 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~$\sS$. 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} on
pp.~\pageref{begin_table}--\pageref{end_table}. 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 nonconjugate 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)\ne0$, 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})\ge3$, 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\ge2$), 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 \cite{FJ}, Theorem~2.1.
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))\ge2$, 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 \cite{MTT}, Section 18, 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 nontrivial 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.
\clearpage
\begin{table}[ht]
\parindent=0cm \tabcolsep=.6em \renewcommand{\arraystretch}{1.3}
\newcommand{\entry}[4]{
$#1$ & \hfil $#2$ \hfil & $#3$ & #4 \\ }
\centering
\begin{tabular}{|p{1.8cm}|p{.9cm}|p{3.2cm}|p{5.6cm}|}
\hline
\entry{f} {\kern-1mm\deg f\kern-1mm} {\hbox{$B$ (bound for
$\Sha$)}} {all odd congruence primes}
\hline
\entry{\nf{127k4C}*} {17} {43^2} {43, 127} \label{begin_table}
\entry{\nf{159k4E}*} {8} {23^2} {3, 5, 11, 23, 53, 13605689}
\entry{\nf{263k4B}} {39} {41^2} {263}
\entry{\nf{269k4C}} {39} {23^2} {269}
\entry{\nf{271k4B}} {39} {29^2} {271}
\entry{\nf{281k4B}} {40} {29^2} {281}
\entry{\nf{295k4C}} {16} {7^2} {3, 5, 11, 59, 101, 659, 70791023}
\entry{\nf{299k4C}} {20} {29^2} {13, 23, 103, 20063, 21961}
% \entry{\nf{319k4C}} {19} {17^2} {3, 11, 23, 29, 37, 3181, 434348087}
% 319k4C removed since Lemma not satisfied.
\entry{\nf{321k4C}} {16} {13^2} {3, 5, 107, 157, 12782373452377}
\hline
\entry{\nf{95k6D}*} {9} {31^2 \!\cdot\! 59^2} {3, 5, 17, 19, 31, 59, 113,
26701}
\entry{\nf{101k6B}} {24} {17^2} {101}
\entry{\nf{103k6B}} {24} {23^2} {103}
\entry{\nf{111k6C}} {9} {11^2} {3, 37, 2796169609}
\entry{\nf{122k6D}*} {6} {73^2} {3, 5, 61, 73, 1303196179}
\entry{\nf{153k6G}} {5} {7^2} {3, 17, 61, 227}
\entry{\nf{157k6B}} {34} {251^2} {157}
\entry{\nf{167k6B}} {40} {41^2} {167}
\entry{\nf{172k6B}} {9} {7^2} {3, 11, 43, 787}
\entry{\nf{173k6B}} {39} {71^2} {173}
\entry{\nf{181k6B}} {40} {107^2} {181}
\entry{\nf{191k6B}} {46} {85091^2} {191}
\entry{\nf{193k6B}} {41} {31^2} {193}
\entry{\nf{199k6B}} {46} {200329^2} {199}
\hline
\end{tabular}
\end{table}
\clearpage
\begin{table}[ht]
\parindent=0cm \tabcolsep=.6em \renewcommand{\arraystretch}{1.3}
\newcommand{\entry}[4]{
$#1$ & \hfil $#2$ \hfil & $#3$ & #4 \\ }
\centering \begin{tabular}{|p{1.8cm}|p{.9cm}|p{3.2cm}|p{5.6cm}|}
\hline
\entry{f} {\kern-1mm\deg f\kern-1mm} {\hbox{$B$ (bound for
$\Sha$)}} {all odd congruence primes}
\hline
\entry{\nf{47k8B}} {16} {19^2} {47}
\entry{\nf{59k8B}} {20} {29^2} {59}
\entry{\nf{67k8B}} {20} {29^2} {67}
\entry{\nf{71k8B}} {24} {379^2} {71}
\entry{\nf{73k8B}} {22} {197^2} {73}
\entry{\nf{74k8C}} {6} {23^2} {37, 127, 821, 8327168869}
\entry{\nf{79k8B}} {25} {307^2} {79}
\entry{\nf{83k8B}} {27} {1019^2} {83}
\entry{\nf{87k8C}} {9} {11^2} {3, 5, 7, 29, 31, 59, 947, 22877, \hfill
\newline 3549902897}
\entry{\nf{89k8B}} {29} {44491^2} {89}
\entry{\nf{97k8B}} {29} {11^2 \!\cdot\! 277^2} {97}
\entry{\nf{101k8B}} {33} {19^2 \!\cdot\! 11503^2} {101}
\entry{\nf{103k8B}} {32} {75367^2} {103}
\entry{\nf{107k8B}} {34} {17^2 \!\cdot\! 491^2} {107}
\entry{\nf{109k8B}} {33} {23^2 \!\cdot\! 229^2} {109}
\entry{\nf{111k8C}} {12} {127^2} {3, 7, 11, 13, 17, 23, 37, 6451, \hfill
\newline 18583, 51162187}
\entry{\nf{113k8B}} {35} {67^2 \!\cdot\! 641^2} {113}
\entry{\nf{115k8B}} {12} {37^2} {3, 5, 19, 23, 572437, \hfill
\newline 5168196102449}
\entry{\nf{117k8I}} {8} {19^2} {3, 13, 181}
\entry{\nf{118k8C}} {8} {37^2} {5, 13, 17, 59, 163, \hfill
\newline 3923085859759909}
\entry{\nf{119k8C}} {16} {1283^2} {3, 7, 13, 17, 109, 883, 5324191,
\hfill \newline 91528147213}
\entry{\nf{121k8F}} {6} {71^2} {3, 11, 17, 41}
\entry{\nf{121k8G}} {12} {13^2} {3, 11}
\entry{\nf{121k8H}} {12} {19^2} {5, 11}
\entry{\nf{125k8D}} {16} {179^2} {5}
\entry{\nf{127k8B}} {39} {59^2} {127}
\hline
\end{tabular}
\end{table}
\clearpage
\begin{table}[ht]
\parindent=0cm \tabcolsep=.6em \renewcommand{\arraystretch}{1.3}
\newcommand{\entry}[4]{
$#1$ & \hfil $#2$ \hfil & $#3$ & #4 \\ }
\centering \begin{tabular}{|p{1.8cm}|p{.9cm}|p{3.2cm}|p{5.6cm}|}
\hline
\entry{f} {\kern-1mm\deg f\kern-1mm} {\hbox{$B$ (bound for
$\Sha$)}} {all odd congruence primes}
\hline
\entry{\nf{128k8F}} {4} {11^2} {1}
\entry{\nf{131k8B}} {43} {241^2\cdot 817838201^2} {131}
\entry{\nf{134k8C}} {11} {61^2} {11, 17, 41, 67, 71, 421, \hfill
\newline 2356138931854759}
\entry{\nf{137k8B}} {42} {71^2\cdot 749093^2} {137}
\entry{\nf{139k8B}} {43} {47^2\cdot 89^2\cdot 1021^2} {139}
\entry{\nf{141k8C}} {14} {13^2} {3, 5, 7, 47, 4639, 43831013, \hfill
\newline 4047347102598757}
\entry{\nf{142k8B}} {10} {11^2} {3, 53, 71, 56377, \hfill
\newline 1965431024315921873}
\entry{\nf{143k8C}} {19} {307^2} {3, 11, 13, 89, 199, 409, 178397, \hfill
\newline 639259, 17440535 97287}
\entry{\nf{143k8D}} {21} {109^2} {3, 7, 11, 13, 61, 79, 103, 173, 241,
\hfill \newline 769, 36583}
\entry{\nf{145k8C}} {17} {29587^2} {5, 11, 29, 107, 251623, 393577,
\hfill \newline 518737, 9837145 699}
\entry{\nf{146k8C}} {12} {3691^2} {11, 73, 269, 503, 1673540153, \hfill
\newline 11374452082219}
\entry{\nf{148k8B}} {11} {19^2} {3, 37}
\entry{\nf{149k8B}} {47} {11^{4}\cdot 40996789^2} {149}
\hline
\entry{\nf{43k10B}} {17} {449^2} {43}
\entry{\nf{47k10B}} {20} {2213^2} {47}
\entry{\nf{53k10B}} {21} {673^2} {53}
\entry{\nf{55k10D}} {9} {71^2} {3, 5, 11, 251, 317, 61339, \hfill
\newline 19869191}
\entry{\nf{59k10B}} {25} {37^2} {59}
\entry{\nf{62k10E}} {7} {23^2} {3, 31, 101, 523, 617, 41192083}
\entry{\nf{64k10K}} {2} {19^2} {3}
\entry{\nf{67k10B}} {26} {191^2\cdot 617^2} {67}
\entry{\nf{68k10B}} {7} {83^2} {3, 7, 17, 8311}
\entry{\nf{71k10B}} {30} {1103^2} {71}
\hline
\end{tabular}
\end{table}
\clearpage
\begin{table}[ht]
\parindent=0cm \tabcolsep=.6em \renewcommand{\arraystretch}{1.3}
\newcommand{\entry}[4]{
$#1$ & \hfil $#2$ \hfil & $#3$ & #4 \\ }
\centering \begin{tabular}{|p{1.8cm}|p{.9cm}|p{3.2cm}|p{5.6cm}|}
\hline
\entry{f} {\kern-1mm\deg f\kern-1mm} {\hbox{$B$ (bound for
$\Sha$)}} {all odd congruence primes}
\hline
\entry{\nf{19k12B}} {9} {67^2} {5, 17, 19, 31, 571}
\entry{\nf{31k12B}} {15} {67^2\cdot 71^2} {31, 13488901}
\entry{\nf{35k12C}} {6} {17^2} {5, 7, 23, 29, 107, 8609, 1307051}
\entry{\nf{39k12C}} {6} {73^2} {3, 13, 1491079, 3719832979693}
\entry{\nf{41k12B}} {20} {54347^2} {7, 41, 3271, 6277}
\entry{\nf{43k12B}} {20} {212969^2} {43, 1669, 483167}
\entry{\nf{47k12B}} {23} {24469^2} {17, 47, 59, 2789}
\entry{\nf{49k12H}} {12} {271^2} {7}
\hline
\end{tabular}
\caption{\label{tab!invisforms}Conjecturally nontrivial $\protect\Sha$
(mostly invisible)}
\label{end_table}
\end{table}
% \clearpage
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\bigskip
\noindent
Neil Dummigan, \\
University of Sheffield, \\
Department of Pure Mathematics, \\
Hicks Building, \\
Hounsfield Road, \\
Sheffield, S3 7RH, U.K. \\
e-mail: n.p.dummigan@shef.ac.uk
\medskip
\noindent
William Stein, \\
Harvard University, \\
Department of Mathematics, \\
One Oxford Street, \\
Cambridge, MA 02138, U.S.A. \\
e-mail: was@math.harvard.edu
\medskip
\noindent
Mark Watkins, \\
Penn State Mathematics Department, \\
University Park, \\
State College, PA 16802, U.S.A. \\
e-mail: watkins@math.psu.edu
\end{document}
{\begin{table}
\vspace{-2ex}
\caption{\label{tab!invisforms}Conjecturally nontrivial $\protect\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}
$$
\end{table}
\end{document}
editing scrap
Original Table 1
\begin{table}[ht]
$$
\renewcommand{\arraystretch}{1.3}
\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 \\[8pt]
\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 \\[8pt]
\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 \\[8pt]
\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}
$$
\caption{Visible $\protect\Sha$} % \vspace{-3ex}
\label{tab!newforms}
\end{table}