CoCalc Shared Fileswww / papers / motive_visibility / dsw_5.tex
Author: William A. Stein
1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2%
3% motive_visibility.tex
4%
5% 25 August 2002
6%
7% Project of William Stein, Neil Dummigan, Mark Watkins
8%
9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
10
11\documentclass{amsart}
12\usepackage{amssymb}
13\usepackage{amsmath}
14\usepackage{amscd}
15
16\newcommand{\edit}[1]{\footnote{#1}\marginpar{\hfill {\sf\thefootnote}}}
17
18\newtheorem{prop}{Proposition}[section]
19\newtheorem{defi}[prop]{Definition}
20\newtheorem{conj}[prop]{Conjecture}
21\newtheorem{lem}[prop]{Lemma}
22\newtheorem{thm}[prop]{Theorem}
23\newtheorem{cor}[prop]{Corollary}
24\newtheorem{examp}[prop]{Example}
25\newtheorem{remar}[prop]{Remark}
26\def\id{\mathop{\mathrm{ id}}\nolimits}
27\DeclareMathOperator{\Ker}{\mathrm {Ker}}
28\DeclareMathOperator{\Aut}{{\mathrm {Aut}}}
29\renewcommand{\Im}{{\mathrm {Im}}}
30\DeclareMathOperator{\ord}{ord}
31\DeclareMathOperator{\End}{End}
32\DeclareMathOperator{\Hom}{Hom}
33\DeclareMathOperator{\Mor}{Mor}
34\DeclareMathOperator{\Norm}{Norm}
35\DeclareMathOperator{\Nm}{Nm}
36\DeclareMathOperator{\tr}{tr}
37\DeclareMathOperator{\Tor}{Tor}
38\DeclareMathOperator{\Sym}{Sym}
39\DeclareMathOperator{\Hol}{Hol}
40\DeclareMathOperator{\vol}{vol}
41\DeclareMathOperator{\tors}{tors}
42\DeclareMathOperator{\cris}{cris}
43\DeclareMathOperator{\length}{length}
44\DeclareMathOperator{\dR}{dR}
45\DeclareMathOperator{\lcm}{lcm}
46\DeclareMathOperator{\Frob}{Frob}
47\def\rank{\mathop{\mathrm{ rank}}\nolimits}
48\newcommand{\Gal}{\mathrm {Gal}}
49\newcommand{\Spec}{{\mathrm {Spec}}}
50\newcommand{\Ext}{{\mathrm {Ext}}}
51\newcommand{\res}{{\mathrm {res}}}
52\newcommand{\Cor}{{\mathrm {Cor}}}
53\newcommand{\AAA}{{\mathbb A}}
54\newcommand{\CC}{{\mathbb C}}
55\newcommand{\RR}{{\mathbb R}}
56\newcommand{\QQ}{{\mathbb Q}}
57\newcommand{\ZZ}{{\mathbb Z}}
58\newcommand{\NN}{{\mathbb N}}
59\newcommand{\EE}{{\mathbb E}}
60\newcommand{\TT}{{\mathbb T}}
61\newcommand{\HHH}{{\mathbb H}}
62\newcommand{\pp}{{\mathfrak p}}
63\newcommand{\qq}{{\mathfrak q}}
64\newcommand{\FF}{{\mathbb F}}
65\newcommand{\KK}{{\mathbb K}}
66\newcommand{\GL}{\mathrm {GL}}
67\newcommand{\SL}{\mathrm {SL}}
68\newcommand{\Sp}{\mathrm {Sp}}
69\newcommand{\Br}{\mathrm {Br}}
70\newcommand{\Qbar}{\overline{\mathbb Q}}
71\newcommand{\Xbar}{\overline{X}}
72\newcommand{\Ebar}{\overline{E}}
73\newcommand{\sbar}{\overline{s}}
74\newcommand{\nf}[1]{\mbox{\bf #1}}
75\newcommand{\fbar}{\overline{f}}
76
77% ---- SHA ----
78\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
79  \newcommand{\textcyr}[1]{%
80    {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
81     \selectfont #1}}
82\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
83
84\newcommand{\HH}{{\mathfrak H}}
85\newcommand{\aaa}{{\mathfrak a}}
86\newcommand{\bb}{{\mathfrak b}}
87\newcommand{\dd}{{\mathfrak d}}
88\newcommand{\ee}{{\mathbf e}}
89\newcommand{\Fbar}{\overline{F}}
90\newcommand{\CH}{\mathrm {CH}}
91
92\begin{document}
93\title[Shafarevich-Tate Groups of Modular Motives]{Constructing Elements in Shafarevich-Tate Groups of Modular Motives}
94\author{Neil Dummigan}
95\author{William Stein}
96\author{Mark Watkins}
97\date{August 24th, 2002}
98\subjclass{11F33, 11F67, 11G40.}
99
100\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
101Shafarevich-Tate group.}
102
103\address{University of Sheffield\\ Department of Pure
104Mathematics\\
105Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
106U.K.}
108One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
110University Park\\State College, PA 16802\\ U.S.A.}
111
112\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
113\email{watkins@math.psu.edu}
114
115
116\begin{abstract}
117
118We study Shafarevich-Tate groups of motives attached to modular forms
119on $\Gamma_0(N)$ of weight bigger than~$2$.  We deduce a criterion for
120the existence of nontrivial elements of these Shafarevich-Tate groups,
121and give $16$ examples in which the Beilinson-Bloch conjecture implies
122the existence of such elements.  We also use modular symbols and
123observations about Tamagawa numbers to compute nontrivial conjectural
124lower bounds on the orders of the Shafarevich-Tate groups of modular
125motives of low level and weight at most $12$.  Our methods build upon
126Mazur's idea of visibility, but in the context of motives instead of
127abelian varieties.
128\end{abstract}
129
130\maketitle
131
132\section{Introduction}
133Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
134be the associated $L$-function. The conjecture of Birch and
135Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$
136at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
137also gives an interpretation of the leading term in the Taylor
138expansion in terms of various quantities, including the order of
139the Shafarevich-Tate group of $E$.
140
141Cremona and Mazur \cite{CM} look, among all strong Weil elliptic
142curves over $\QQ$ of conductor $N\leq 5500$, at those with
143non-trivial Shafarevich-Tate group (according to the Birch and
144Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
145group has predicted elements of prime order $m$. In most cases
146they find another elliptic curve, often of the same conductor,
147whose $m$-torsion is Galois-isomorphic to that of the first one,
148and which has positive rank. The rational points on the second elliptic
149curve produce classes in the common $H^1(\QQ,E[m])$. They show
150\cite{CM2} that these lie in the Shafarevich-Tate group of the
151first curve, so rational points on one curve explain elements of
152the Shafarevich-Tate group of the other curve.
153
154The Bloch-Kato conjecture \cite{BK} is the generalisation to
155arbitrary motives of the leading term part of the Birch and
156Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
157\cite{B} generalises the part about the order of vanishing at the
158central point, identifying it with the rank of a certain Chow
159group.
160
161This paper is a partial generalisation of \cite{CM} and \cite{AS}
162from abelian varieties over $\QQ$ associated to modular forms of
163weight~$2$ to the motives attached to modular forms of higher weight.
164It also does for congruences between modular forms of equal weight
165what \cite{Du2} did for congruences between modular forms of different
166weights.
167
168We consider the situation where two newforms~$f$ and~$g$, both of
169even weight $k>2$ and level~$N$, are congruent modulo a maximal ideal
170$\qq$ of odd residue characteristic, and $L(g,k/2)=0$ but
171$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish
172to order at least $2$ at $s=k/2$.  In Section~\ref{sec:examples},
173we give sixteen
174examples (all with $k=4$ and $k=6$), and in each $\qq$ divides the
175numerator of the algebraic number $L(f,k/2)/\vol_{\infty}$, where
176$\vol_{\infty}$ is a certain canonical period.
177
178In fact, we show how this divisibility may be deduced from the
179vanishing of $L(g,k/2)$ using recent work of Vatsal \cite{V}. The
180point is, the congruence between$f$ and~$g$ leads to a congruence
181between suitable algebraic parts'' of the special values
182$L(f,k/2)$ and $L(g,k/2)$. In slightly more detail, a multiplicity
183one result of Faltings and Jordan shows that the congruence of
184Fourier expansions leads to a congruence of certain associated
185cohomology classes. These are then identified with the modular
186symbols which give rise to the algebraic parts of special values.
187If $L(g,k/2)$ vanishes then the congruence implies that
188$L(f,k/2)/\vol_{\infty}$ must be divisible by $\qq$.
189
190The Bloch-Kato conjecture sometimes implies that the Shafarevich-Tate
191group attached to~$f$ has nonzero $\qq$-torsion. Under certain
192hypotheses and assumptions, the most substantial of which is the
193Beilinson-Bloch conjecture relating the vanishing of $L(g,k/2)$ to the
194existence of algebraic cycles, we are able to construct some of the
195predicted elements of~$\Sha$ using the Galois-theoretic
196interpretation of the congruences to transfer elements from a Selmer
197group for~$g$ to a Selmer group for~$f$.
198One might say that algebraic cycles for one motive explain elements
199of~$\Sha$ for the other.
200%In proving the local
201%conditions at primes dividing the level, and also in examining the
202%local Tamagawa factors at these primes, we make use of a higher weight
203%level-lowering result due to Jordan and Livn\'e \cite{JL}.
204
205We also compute data which, assuming the Bloch-Kato
206conjecture, provides lower bounds for the orders of numerous
207Shafarevich-Tate group (see Section~\ref{sec:invis}).
208%Our data is consistent
209%with the fact \cite{Fl2} that the part of $\#\Sha$ coprime to the
210%congruence modulus is necessarily a perfect square (assuming that~$\Sha$
211%is finite).
212
213\section{Motives and Galois representations}
214Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
215$\Gamma_0(N)$, with coefficients in an algebraic number field~$E$,
216which is necessarily totally real.  Let~$\lambda$ be any finite prime
217of~$E$, and let~$\ell$ denote its residue characteristic.  A theorem
218of Deligne \cite{De1} implies the existence of a two-dimensional
219vector space $V_{\lambda}$ over $E_{\lambda}$, and a continuous
220representation
221$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
222such that
223\begin{enumerate}
224\item $\rho_{\lambda}$ is unramified at~$p$ for all primes~$p$ not dividing
225$lN$, and
226\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
227characteristic polynomial of $\Frob_p^{-1}$ acting on
228$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
229\end{enumerate}
230
231Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
232the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
233There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
234both $2$-dimensional $E$-vector spaces. For details of the
235construction see \cite{Sc}. The de Rham realisation has a Hodge
236filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset 237F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
238cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
239cohomology.
240For each prime $\lambda$, there is a natural isomorphism
241$V_B\otimes E_{\lambda}\simeq V_{\lambda}$. We may choose a
242$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
243each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
244Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
245There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
246which amounts to multiplying the action of $\Frob_p$ by $p^j$.
247
248Following \cite{BK} (Section 3), for $p\neq l$ (including
249$p=\infty$) let
250$$251H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow 252H^1(I_p,V_{\lambda}(j))). 253$$
254The subscript~$f$ stands for finite
255part'', $D_p$ is a decomposition subgroup at a prime above~$p$,
256$I_p$ is the inertia subgroup, and the cohomology is for
257continuous cocycles and coboundaries. For $p=l$ let
258$$259H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow 260H^1(D_l,V_{\lambda}(j)\otimes B_{\cris})) 261$$
262(see Section 1 of
263\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
264$B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
265elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
266in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes~$p$.
267
268There is a natural exact sequence
269$$270 \begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}. 271$$
272Let
273$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
274Define the $\lambda$-Selmer group
275$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
276$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
277$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes~$p$. Note that the
278condition at $p=\infty$ is superfluous unless $l=2$. Define the
279Shafarevich-Tate group
280$$281 \Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/ 282 \pi_*H^1_f(\QQ,V_{\lambda}(j)). 283$$
284Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any
285prime ideal~$\lambda$ is the length of the $\lambda$-component of
286$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,
287and write~$\Sha$ for~$\Sha(j)$.
288
289Define the group of global torsion points
290$$291 \Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)). 292$$
293This is analogous to the group of rational torsion points on an
294elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in
295which the exponent of any prime ideal~$\lambda$ is the length of
296the $\lambda$-component of $\Gamma_{\QQ}$.
297
298\section{Canonical periods}
299We assume from now on for convenience that $N\geq 3$. We need to
300choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
301and de Rham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
302in a way such that $T_B$ and $T_{\dR}\otimes_{O_E}O_E[1/Nk!]$
303agree with (respectively) the $O_E$-lattice $\mathfrak{M}_{f,B}$
304and the $O_E[1/Nk!]$-lattice $\mathfrak{M}_{f,\dR}$ defined in
305\cite{DFG} using cohomology, with non-constant coefficients, of
306modular curves. (In \cite{DFG}, see especially Sections 2.2 and
3075.4, and the paragraph preceding Lemma 2.3.)
308
309For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
310module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
311$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes 312E_{\lambda}\simeq V_{\lambda}$. Then the $O_{\lambda}$ module
313$T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable.
314
315Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
316generalised elliptic curves with full level-$N$ structure. Let
317$\mathfrak{E}$ be the universal generalised elliptic curve over
318$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product
319of $\mathfrak{E}$ over $M(N)$. (The motive $M_f$ is constructed
320using a projector on the cohomology of a desingularisation of
321$\mathfrak{E}^{k-2}$. Realising $M(N)(\CC)$ as the disjoint union
322of $\phi(N)$ copies of the quotient
323$\Gamma(N)\backslash\mathfrak{H}^*$ (where $\mathfrak{H}^*$ is the
324completed upper half plane), and letting $\tau$ be a variable on
325$\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is isomorphic to
326the elliptic curve with period lattice generated by $1$ and
327$\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a variable on
328the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the fibre product.
329Then $2\pi i f(\tau)\,d\tau\wedge dz_1\wedge\ldots\wedge dz_{k-2}$
330is a well-defined differential form on (a desingularisation of)
331$\mathfrak{E}^{k-2}$ and naturally represents a generating element
332of $F^{k-1}T_{\dR}$. (At least we can make our choices locally at
333primes dividing $Nk!$ so that this is the case.) We shall call
334this element $e(f)$.
335
336Under the de Rham isomorphism between $V_{\dR}\otimes\CC$ and
337$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
338a natural action of complex conjugation on $V_B$, breaking it up
339into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
340Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
341to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
342$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
343are rank one $O_E$-modules, but not necessarily free, since the
344class number of $O_E$ may be greater than one. Choose nonzero
345elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
346the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
347$\Omega_f^{\pm}$ by $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$.
348
349\section{The Bloch-Kato conjecture}\label{sec:bkconj}
350Let $L(f,s)$ be the $L$-function attached to~$f$. For
351$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with
352Euler product
353$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but
354there is an analytic continuation given by an integral, as
355described in the next section. Suppose that $L(f,k/2)\neq 0$. The
356Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the
357following equality of fractional ideals of~$E$:
358$$359 \frac{L(f,k/2)}{\vol_{\infty}}= 360 \left(\prod_pc_p(k/2)\right) 361 \frac{\#\Sha}{\aaa^{\pm}(\#\Gamma_{\QQ})^2}. 362$$
363(Strictly speaking, the conjecture in \cite{BK}
364is only given for $E=\QQ$.) Here, $\pm$
365represents the parity of $(k/2)-1$, and $\vol_{\infty}$ is equal
366to $(2\pi i)^{k/2}$ multiplied by the determinant of the
367isomorphism $V_B^{\pm}\otimes\CC\simeq 368(V_{\dR}/F^{k/2})\otimes\CC$, calculated with respect to the
369lattices $O_E\delta_f^{\pm}$ and the image of $T_{\dR}$. For
370$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be
371\begin{align*}
372\length&\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-
373  \ord_{\lambda}(P_p(p^{-j}))\\
374=&\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).
375\end{align*}
376
377We omit the definition of $\ord_{\lambda}(c_p(j))$ for
378$\lambda\mid p$, which requires one to assume Fontaine's de Rham
379conjecture (\cite{Fo}, Appendix A6), and depends on the choices of
380$T_{\dR}$ and $T_B$, locally at~$\lambda$. (We shall mainly be
381concerned with the $q$-part of the Bloch-Kato conjecture, where~$q$
382is a prime of good reduction. For such primes, the de Rham
383conjecture follows from Theorem 5.6 of \cite{Fa1}.)
384
385\begin{lem}\label{vol}
386$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and
387$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.
388\end{lem}
389\begin{proof}
390$\vol_{\infty}$ is also equal to the determinant of the period map
391from $F^{k/2}V_{\dR}\otimes\CC$ to $V_B^{\pm}\otimes\CC$, with
392respect to lattices dual to those we used above in the definition
393of $\vol_{\infty}$ (c.f. the last paragraph of 1.7 of \cite{De2}).
394We are using here natural pairings. Recall that the index of
395$O_E\delta_f^{\pm}$ in $T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then
396the proof is completed by noting that, locally away from primes
397dividing $Nk!$, the index of $T_{\dR}$ in its dual is equal to the
398index of $T_B$ in its dual, both being equal to the ideal
399denoted~$\eta$ in \cite{DFG2}.
400\end{proof}
401\begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer.
402Then the fractional ideal $c_p(j)$ is supported at most on
403divisors of~$p$.
404\end{lem}
405\begin{proof}
406As on p.~30 of \cite{Fl1}, for odd $l\neq p$,
407$\ord_{\lambda}(c_p(j))$ is the length of the finite
408$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
409where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
410trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
411torsion free.
412\end{proof}
413
414\begin{lem}\label{local1}
415Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
416is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
417$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
418that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not congruent
419modulo $\qq$ to any newform of weight~$k$, trivial character and
420level dividing $N/p$. Then any integer~$j$,
421$\ord_{\qq}(c_p(j))=0$.
422\end{lem}
423\begin{proof}
424It suffices to show that
425$$426 \dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)), 427$$
428since this ensures that
429$H^0(I_p,A_{\qq}(j))=V_{\qq}^{I_p}/T_{\qq}^{I_p}$, hence that
430$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}^{I_p}/T_{\qq}^{I_p})$. If
431the dimensions differ then, given that $f$ is not congruent modulo
432$\qq$ to a newform of level dividing $N/p$, Proposition 2.2 of
433\cite{L} shows that we are in the situation covered by one of the
434three cases in Proposition 2.3 of \cite{L}. Since $p\not\equiv 435-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is
436unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's
437result that $N$ is the prime-to-$q$ part of the conductor of
438$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses
439the condition $q>k$) implies the existence of a newform of weight
440$k$, trivial character and level dividing $N/p$, congruent to~$g$
441modulo $\qq$. This contradicts our hypotheses.
442\end{proof}
443
444\begin{remar}
445For an example of what can be done when~$f$ is congruent to
446a form of lower level, see the first example in Section~\ref{sec:other_ex}
447below.
448\end{remar}
449
450\begin{lem}\label{at q}
451If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then
452$\ord_{\qq}(c_q)=0$.
453\end{lem}
454\begin{proof}
455It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an
456application, at the end of Section~2.2, of the results of
457\cite{Fa1}) that $T_{\qq}$ is the
458$O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered
459module $T_{\dR}\otimes O_{\qq}$ by the functor they call
460$\mathbb{V}$. (This property is part of the definition of an
461$S$-integral premotivic structure given in Section~1.2 of
462\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
463of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
464in Theorem~4.1 of \cite{BK} follows from the first paragraph of
4652(h) of \cite{Fa1}.)
466\end{proof}
467
468\begin{lem}
469If $A[\lambda]$ is an
470irreducible representation of $\Gal(\Qbar/\QQ)$,
471then
472$$\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0.$$
473\end{lem}
474This follows trivially from the definition.
475
476Putting together the above lemmas we arrive at the following:
477\begin{prop}\label{sha}
478Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that $A[\qq]$
479is an irreducible representation of $\Gal(\Qbar/\QQ)$, where $\qq\mid q$.
480Assume the same hypotheses as in Lemma \ref{local1},
481for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$
482are as in the previous section. If
483$$484 \ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty})>0 485$$
486(with numerator nonzero) then the Bloch-Kato conjecture
487predicts that
488$$489 \ord_{\qq}(\#\Sha)>0. 490$$
491\end{prop}
492
493\section{Congruences of special values}
494Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
495weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
496large enough to contain all the coefficients $a_n$ and $b_n$.
497Suppose that $\qq\mid q$ is a prime of~$E$ such that $f\equiv 498g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$, and assume
499that $q\nmid N\phi(N)k!$.
500Choose $\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
501$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates
502$T_B^{\pm}$ locally at $\qq$.
503Make two further assumptions:
504$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$
505
506\begin{prop} \label{div}
507With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
508\end{prop}
509\begin{proof} This is based on some of the ideas used in Section 1 of
510\cite{V}.  Note the apparent typo in Theorem~1.13 of \cite{V},
511which presumably should refer to Condition 2''. Since
512$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
513$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm 5141=(-1)^{(k/2)-1}$. It is well known, and easy to prove, that
515$$516 \int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s). 517$$
518Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
519$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
520where the integral is taken along the positive imaginary axis,
521then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
522Thus we are reduced
523to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
524
525Let $\mathcal{D}_0$ be the group of divisors of degree zero
526supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
527integer $r\geq 0$, let $P_r(R)$ be the additive group of
528homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
529groups have a natural action of $\Gamma_1(N)$. Let
530$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
531be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
532
533Via the isomorphism (8) in Section~1.5 of \cite{V}, combined with
534the argument in 1.7 of \cite{V}, the cohomology class
535$\omega_f^{\pm}$ corresponds to a modular symbol $\Phi_f^{\pm}\in 536S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
537element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$.
538Inverting $N\phi(N)$ takes into account the fact that we are now
539dealing with cohomology over $X_1(N)$ rather than $M(N)$. It
540follows from the last line of Section~4.2 of \cite{St} that, up to
541some small factorials which do not matter locally at $\qq$,
542$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv 543(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since
544$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
545$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv 546(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
547coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
548show is divisible by $\qq$.
549Similarly
550$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv 551(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of
552$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
553Therefore it would suffice to show that, for some $\mu\in O_E$,
554the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
555$\qq$ in $S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$. It suffices to show
556that, for some $\mu\in O_E$, the element
557$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
558considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
559non-constant coefficients. This would be the case if
560$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
561one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
562a consequence of Theorem 2.1(1) of \cite{FJ}.
563\end{proof}
564\begin{remar}\label{sign}
565The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are
566equal. They are determined by the eigenvalue of the involution $W_N$,
567which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and
568$b_N$ are each $N^{k/2-1}$ times this sign and~$\qq$ has residue
569characteristic coprime to $2N$.  The common sign in the functional
570equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of
571$W_N$ acting on~$f$ and~$g$.
572\end{remar}
573
574This is analogous to the remark at the end of Section~3 of \cite{CM},
575which shows that if~$\qq$ has odd residue characteristic and
576$L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then $L(g,s)$ must vanish to order
577at least two at $s=k/2$.  Note that Maeda's conjecture
578implies that there are no examples of~$g$ of
580$L(g,k/2)=0$ (see \cite{CF}).
581
582\section{Constructing elements of the Shafarevich-Tate group}
583Let~$f$ and~$g$ be as in the first paragraph of the previous
584section. For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
585$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
586$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
587is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
588the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
589irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
590
591Suppose that $L(g,k/2)=0$. If the sign in the functional equation
592is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
593\ref{sign}), this implies that the order of vanishing of $L(g,s)$
594at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
595conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
596is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational
597rational equivalence classes of null-homologous,
598algebraic cycles  of codimension $k/2$
599on the motive $M_g$. (This generalises the part
600of the Birch--Swinnerton-Dyer conjecture which says that for an
601elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at
602$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)
603
604Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
605to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
606subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
607If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
608get (assuming also the Beilinson-Bloch conjecture) a subspace of
609$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
610vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
611conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
612equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
613follow from the conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
614Sections~1 and~6.5 of \cite{Fo2}.
615
616Similarly, if $L(f,k/2)\neq 0$ then we expect that
617$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
618coincides with the $\qq$-part of $\Sha$.
619\begin{thm}\label{local}
620Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
621$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
622$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$ and
623that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a
624newform of weight~$k$, trivial character and level dividing $N/p$.
625Suppose that, for all primes $p\mid N$, $\,p\not\equiv 626-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here
627$w_p$ is the common eigenvalue of the Atkin-Lehner involution
628$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of
629$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
630\end{thm}
631
632\begin{proof}
633Take a nonzero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
634continuity and rescaling we may assume that it lies in
635$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq 636H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction (note that
637$T'_{\qq}(k/2)/\qq T'_{\qq}(k/2)\simeq A'[\qq](k/2)$) we get a
638nonzero class $c\in H^1(\QQ,A'[\qq](k/2))\simeq 639H^1(\QQ,A[\qq](k/2))$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$
640is trivial. It follows that $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
641$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
642we get a nonzero, $\qq$-torsion class $\gamma\in 643H^1(\QQ,A_{\qq}(k/2))$.
644
645Our aim is to show that $\res_p(\gamma)\in 646H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
647consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
648
649\begin{enumerate}
650\item {\bf $p\nmid qN$. }
651
652Consider the $I_p$-cohomology of the short exact sequence
653$$654 \begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD}, 655$$
656where~$\pi$ is multiplication by a uniformising element of
657$O_{\qq}$. Since in this case $A'_{\qq}(k/2)$ is unramified at
658$p$, $H^0(I_p, A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is
659$\qq$-divisible. Therefore $H^1(I_p,A'[\qq](k/2))$ (which,
660remember, is the same as $H^1(I_p,A[\qq](k/2))$) injects into
661$H^1(I_p,A'_{\qq}(k/2))$. It follows from the fact that $d\in 662H^1_f(\QQ,V'_{\qq}(k/2))$ that the image in
663$H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is zero, hence
664that the restriction of~$c$ (to $H^1(I_p,A'[\qq](k/2))\simeq 665H^1(I_p,A[\qq](k/2))$) is zero. Hence the restriction of $\gamma$
666to $H^1(I_p,A_{\qq}(k/2))$ is also zero. By line~3 of p.~125 of
667\cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
668contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
669to $H^1(I_p,A_{\qq}(k/2))$,  so we have shown that
670$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
671
672\item {\bf $p\mid N$. }
673
674First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
675It suffices to show that
676$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
677since then the natural map from
678$H^0(I_p,V'_{\qq}(k/2))$ to $H^0(I_p, A'_{\qq}(k/2))$ is
679surjective; this may be done as in the proof of Lemma
680\ref{local1}. It follows as above that the image of $c\in 681H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is zero. Then
682$\res_p(c)$ comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by
683inflation-restriction. The order of this group is the same as the
684order of the group $H^0(\QQ_p,A[\qq](k/2))$, which we claim is
685trivial. By the work of Carayol \cite{Ca1}, $p\mid N$ implies that
686$V_{\qq}(k/2)$ is ramified at $p$, so $\dim 687H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim 688H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
689consider the case where this common dimension is $1$. The
690(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha 691p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication by~$\alpha$
692on the one-dimensional space $H^0(I_p,V_{\qq}(k/2))$. It
693follows from Theor\'eme A of \cite{Ca1} that this is the same as
694the Euler factor at $p$ of $L(f,s)$. By the work of Atkin and
695Lehner \cite{AL}, it then follows that $p^2\nmid N$ and
696$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
697$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
698$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$ as
699$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
700$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
701$\res_p(\gamma)=0$ and certainly lies in
702$H^1_f(\QQ_p,A_{\qq}(k/2))$.
703
704\item {\bf $p=q$. }
705
706Since $q\nmid N$ is a prime of good reduction for the motive
707$M_g$, $\,V'_{\qq}$ is a crystalline representation of
708$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
709$V'_{\qq}$ have the same dimension, where
710$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q} 711B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
712As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
713the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
714filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we
715may now prove, in the same manner as Proposition 9.2 of
716\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.
717\end{enumerate}
718\end{proof}
719
720Theorem~2.7 of \cite{AS} is concerned with verifying local
721conditions in the case $k=2$, where~$f$ and~$g$ are associated
722with abelian varieties~$A$ and~$B$. (Their theorem also applies to
723abelian varieties over number fields.) Our restriction outlawing
724congruences modulo $\qq$ with cusp forms of lower level is
725analogous to theirs forbidding~$q$ from dividing Tamagawa factors
726$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic
727curve with $\ord_l(j(A))<0$, consideration of a Tate
728parametrisation shows that if $q\mid c_{A,l}$, i.e., if
729$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
730at~$l$.)
731
732In this paper we have encountered two technical problems which we
733dealt with in quite similar ways:
734\begin{enumerate}
735\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
736\item proving local conditions at primes $p\mid N$, for an element
737of $\qq$-torsion.
738\end{enumerate}
739If our only interest was in testing the Bloch-Kato conjecture at
740$\qq$, we could have made these problems cancel out, as in Lemma
7418.11 of \cite{DFG}, by weakening the local conditions. However, we
742have chosen not to do so, since we are also interested in the
743Shafarevich-Tate group, and since the hypotheses we had to assume
744are not particularly strong. Note that, since $A[\qq]$ is
745irreducible, the $\qq$-part of $\Sha$ does not depend on the
746choice of $T_{\qq}$.
747
748\section{Examples and Experiments}
749\label{sec:examples}
750This section contains tables and numerical examples that illustrate
751the main themes of this paper.  In Section~\ref{sec:vistable}, we
752explain Table~\ref{tab:newforms}, which contains~$16$ examples of
753pairs $f,g$ such that the Beilinson-Bloch conjecture and
754Theorem~\ref{local} together imply the existence of nontrivial
755elements of the
756Shafarevich-Tate group of the motive attached to~$f$.
757Section~\ref{sec:howdone} outlines the higher-weight modules symbols
758computations that were used in making Table~\ref{tab:newforms}.
759Section~\ref{sec:invis} discusses Table~\ref{tab:invisforms},
760which summarizes the results
761of an extensive computation of conjectural orders of Shafarevich-Tate
762groups for modular motives of low level and weight.
763Section~\ref{sec:other_ex} gives specific examples in which various
764hypotheses fail.
765
766\subsection{Visible $\Sha$ Table~\ref{tab:newforms}}\label{sec:vistable}
767\begin{table}
768\caption{\label{tab:newforms}Visible $\Sha$}\vspace{-3ex}
769
770$$771\begin{array}{|c|c|c|c|c|}\hline 772 g & \deg(g) & f & \deg(f) & q's \\\hline 773\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\ 774\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\ 775\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\ 776\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\ 777\vspace{-2ex} & & & & \\ 778\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\ 779\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\ 780\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\ 781\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\ 782\vspace{-2ex} & & & & \\ 783\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\ 784\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\ 785\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\ 786\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\ 787\vspace{-2ex} & & & & \\ 788\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\ 789\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\ 790\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\ 791\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\ 792\hline 793\end{array} 794$$
795\end{table}
796
797
798Table~\ref{tab:newforms} on page~\pageref{tab:newforms} lists
799sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
800along with at least one prime~$q$ such that there is a prime
801$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
802$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
803The notation is as follows.
804The first column contains a label whose structure is
805\begin{center}
806{\bf [Level]k[Weight][GaloisOrbit]}
807\end{center}
808This label determines a newform $g=\sum a_n q^n$, up to Galois
809conjugacy. For example, \nf{127k4C} denotes a newform in the third
810Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
811orbits are ordered first by the degree of $\QQ(\ldots, a_n, 812\ldots)$, then by the sequence of absolute values $|\mbox{\rm 813Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
814being first in the event that the two absolute values are equal,
815and the first Galois orbit is denoted {\bf A}, the second {\bf B},
816and so on. The second column contains the degree of the field
817$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
818contain~$f$ and its degree, respectively. The fifth column
819contains at least one prime~$q$ such that there is a prime
820$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
821hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
822satisfied for~$f$,~$g$, and~$\qq$.
823
824For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
825prime $q$ appears in the $q$-column, meaning $q^2$ divides the
826order of the group $S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp})$, defined
827at the end of 7.3 below.
828
829
830We describe the first line of Table~\ref{tab:newforms}
831in more detail.  See the next section for further details
832on how the computations were performed.
833
834Using modular symbols, we find that there is a newform
835$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots 836\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
837the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
838also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
839coefficients generate a number field~$K$ of degree~$17$, and by
840computing the image of the modular symbol $XY\{0,\infty\}$ under
841the period mapping, we find that $L(f,2)\neq 0$.  The newforms~$f$
842and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
843characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
844both equal to
845$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 846+ \cdots\in \FF_{43}[[q]].$$
847
848There is no form in the Eisenstein subspaces of
849$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
850$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
851$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
852prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
853level~$1$ form of weight~$4$. Thus we have checked the hypotheses
854of Theorem~\ref{local}, so if $r$ is the dimension of
855$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
856$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
857
858Recall that since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that
859$r\geq 2$. Then, since $L(f,k/2)\neq 0$, we expect that the
860$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to
861the $\qq$-torsion subgroup of $\Sha$. Admitting these assumptions,
862we have constructed the $\qq$-torsion in $\Sha$ predicted by the
863Bloch-Kato conjecture.
864
865For particular examples of elliptic curves one can often find and
866write down rational points predicted by the Birch and
867Swinnerton-Dyer conjecture. It would be nice if likewise one could
868explicitly produce algebraic cycles predicted by the
869Beilinson-Bloch conjecture in the above examples. Since
870$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
8710.3.2 of \cite{Z}), so ought to be trivial in
872$\CH_0^{k/2}(M_g)\otimes\QQ$.
873
874\subsection{How the computation was performed}\label{sec:howdone}
875We give a brief summary of how the computation was performed.  The
876algorithms that we used were implemented by the second author, and
877most are a standard part of MAGMA (see \cite{magma}).
878
879Let~$g$,~$f$, and~$q$ be some data from a line of
880Table~\ref{tab:newforms} and let~$N$ denote the level of~$g$. We
881verified the existence of a congruence modulo~$q$, that
882$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq 8830$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
884not arise from any $S_k(\Gamma_0(N/p))$, as follows:
885
886To prove there is a congruence, we showed that the corresponding
887{\em integral} spaces of modular symbols satisfy an appropriate
888congruence, which forces the existence of a congruence on the
889level of Fourier expansions.  We showed that $\rho_{g,\qq}$ is
890irreducible by computing a set that contains all possible residue
891characteristics of congruences between~$g$ and any Eisenstein
892series of level dividing~$N$, where by congruence, we mean a
893congruence for all Fourier coefficients of index~$n$ with
894$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
895form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
896listing a basis of such~$h$ and finding the possible congruences,
897where again we disregard the Fourier coefficients of index not
898coprime to~$N$.
899
900To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
901modular symbol ${\mathbf 902e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$ under a map
903with the same kernel as the period mapping, and found that the
904image was~$0$.  The period mapping sends the modular
905symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
906so that ${\mathbf e}$ maps to~$0$ implies that
907$L(g,\frac{k}{2})=0$. In a similar way, we verified that
908$L(f,\frac{k}{2})\neq 0$.  Next, we checked that $W_N(g) 909=(-1)^{k/2} g$ which, because of the functional equation, implies
910that $L'(g,\frac{k}{2})=0$. Table~\ref{tab:newforms} is of
911independent interest because it includes examples of modular forms
912of even weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
913the functional equation.  We found no such examples of weights
914$\geq 8$.
915
916\subsection{Conjecturally nontrivial $\Sha$}\label{sec:invis}
917In this section we apply some of the results of
918Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders
919of Shafarevich-Tate groups of many modular motives. The results of
920this section suggest that~$\Sha$ of a modular motive is usually not
921visible at level~$N$'', i.e., explained by congruences at level~$N$,
922which agrees with the observations of \cite{CM} and \cite{AS}. For
923example, when $k>6$ we find many examples of conjecturally
924nontrivial~$\Sha$ but no examples of nontrivial visible~$\Sha$.
925
926For any newform~$f$, let $L(M_f,s) = \prod_{i=1}^{d} L(f^{(i)},s)$
927where $f^{(i)}$ runs over the $\Gal(\Qbar/\QQ)$-conjugates of~$f$.
928Let~$T$ be the complex torus
929$\CC^d/\mathcal{L}$, where the lattice $\mathcal{L}$ is defined by
930integrating integral cuspidal modular symbols against the conjugates
931of~$f$.  Let $\Omega_{M_f}$ denote the volume of the $-1$
932eigenspace $T^{-}=\{z \in T : \overline{z}=-z\}$ for complex
933conjugation on~$T$.
934
935
936{\begin{table}
937\vspace{-2ex}
938\caption{\label{tab:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}
939\vspace{-4ex}
940
941$$942\begin{array}{|c|c|c|c|}\hline 943f & \deg(f) & B\,\, (\text{\Sha bound})& \text{all odd congruence primes}\\\hline 944\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\ 945\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\ 946\nf{263k4B} & 39 & 41^{2} & 263 \\ 947\nf{269k4C} & 39 & 23^{2} & 269 \\ 948\nf{271k4B} & 39 & 29^{2} & 271 \\ 949\nf{281k4B} & 40 & 29^{2} & 281 \\ 950\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\ 951\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\ 952\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\ 953\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\ 954\hline 955\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\ 956\nf{101k6B} & 24 & 17^{2} & 101 \\ 957\nf{103k6B} & 24 & 23^{2} & 103 \\ 958\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\ 959\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\ 960\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\ 961\nf{157k6B} & 34 & 251^{2} & 157 \\ 962\nf{167k6B} & 40 & 41^{2} & 167 \\ 963\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\ 964\nf{173k6B} & 39 & 71^{2} & 173 \\ 965\nf{181k6B} & 40 & 107^{2} & 181 \\ 966\nf{191k6B} & 46 & 85091^{2} & 191 \\ 967\nf{193k6B} & 41 & 31^{2} & 193 \\ 968\nf{199k6B} & 46 & 200329^2 & 199 \\ 969\hline 970\nf{47k8B} & 16 & 19^{2} & 47 \\ 971\nf{59k8B} & 20 & 29^{2} & 59 \\ 972\nf{67k8B} & 20 & 29^{2} & 67 \\ 973\nf{71k8B} & 24 & 379^{2} & 71 \\ 974\nf{73k8B} & 22 & 197^{2} & 73 \\ 975\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\ 976\nf{79k8B} & 25 & 307^{2} & 79 \\ 977\nf{83k8B} & 27 & 1019^{2} & 83 \\ 978\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\ 979\nf{89k8B} & 29 & 44491^{2} & 89 \\ 980\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\ 981\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\ 982\nf{103k8B} & 32 & 75367^{2} & 103 \\ 983\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\ 984\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\ 985\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\ 986\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\ 987\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\ 988\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\ 989\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\ 990\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\ 991\hline 992\end{array} 993$$
994\end{table}
995\begin{table}
996$$997\begin{array}{|c|c|c|c|}\hline 998f & \deg(f) & B\,\, (\text{\Sha bound})& \text{all odd congruence primes}\\\hline 999\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\ 1000\nf{121k8G} & 12 & 13^{2} & 3, 11 \\ 1001\nf{121k8H} & 12 & 19^{2} & 5, 11 \\ 1002\nf{125k8D} & 16 & 179^{2} & 5 \\ 1003\nf{127k8B} & 39 & 59^{2} & 127 \\ 1004\nf{128k8F} & 4 & 11^{2} & 1 \\ 1005\nf{131k8B} & 43 & 241^{2} \!\cdot\! 817838201^{2}&131\\ 1006\nf{134k8C} & 11 & 61^{2} & 11, 17, 41, 67, 71, 421, 2356138931854759 \\ 1007\nf{137k8B} & 42 & 71^{2} \!\cdot\! 749093^{2} & 137 \\ 1008\nf{139k8B} & 43 & 47^{2} \!\cdot\! 89^{2} \!\cdot\! 1021^{2} & 139 \\ 1009\nf{141k8C} & 14 & 13^{2} & 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\ 1010\nf{142k8B} & 10 & 11^{2} & 3, 53, 71, 56377, 1965431024315921873 \\ 1011\nf{143k8C} & 19 & 307^{2} & 3, 11, 13, 89, 199, 409, 178397, 639259, 17440535 101297287 \\ 1013\nf{143k8D} & 21 & 109^{2} & 3, 7, 11, 13, 61, 79, 103, 173, 241, 769, 36583 1014\\ 1015\nf{145k8C} & 17 & 29587^{2} & 5, 11, 29, 107, 251623, 393577, 518737, 9837145 1016699 \\ 1017\nf{146k8C} & 12 & 3691^{2} & 11, 73, 269, 503, 1673540153, 11374452082219 \\ 1018\nf{148k8B} & 11 & 19^{2} & 3, 37 \\ 1019\nf{149k8B} & 47 & 11^{4} \!\cdot\! 40996789^{2} & 149\\ 1020\hline 1021 1022\nf{43k10B} & 17 & 449^{2} & 43 \\ 1023\nf{47k10B} & 20 & 2213^{2} & 47 \\ 1024\nf{53k10B} & 21 & 673^{2} & 53 \\ 1025\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\ 1026\nf{59k10B} & 25 & 37^{2} & 59 \\ 1027\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\ 1028\nf{64k10K} & 2 & 19^{2} & 3 \\ 1029\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\ 1030\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\ 1031\nf{71k10B} & 30 & 1103^{2} & 71 \\ 1032 1033\hline 1034\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\ 1035\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\ 1036\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\ 1037\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\ 1038\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\ 1039\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\ 1040\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\ 1041\nf{49k12H} & 12 & 271^{2} & 7 \\ 1042\hline 1043\end{array} 1044$$
1045\end{table}
1046
1047The following conjecture is probably not difficult to prove, but we
1048haven't given a proof, so we formally state it as a conjecture, then
1049assume it.
1050\begin{conj}\label{conj:lrat}
1051If $p\nmid Nk!$ is a prime then the $p$-parts of
1052$$1053 \frac{L(M_f,k/2)}{\Omega_{M_f}}\qquad\text{and}\qquad 1054 \Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\right) 1055$$
1056are equal, where $\vol_\infty$ is as in Section~\ref{sec:bkconj}.
1057\end{conj}
1058For the rest of this section, {\em we officially assume the Bloch-Kato
1059conjecture and Conjecture~\ref{conj:lrat}.}
1060
1061Let~$\mathcal{S}$ be the set of newforms with~level $N$ and weight~$k$
1062satisfying either $k=4$ and $N\leq 321$, or $k=6$ and $N\leq 199$, or
1063$k=8$ and $N\leq 133$, or $k=10$ and $N\leq 72$, or $k=12$ and $N\leq 106449$.  Given $f\in \mathcal{S}$, let~$B$ be the lower bound on $\#\Sha$ defined
1065as follows:
1066\begin{enumerate}
1067\item Let $L_1$ be the rational number $L(M_f,k/2)/\Omega_{M_f}$.
1068      If $L_1=0$ let $B=1$ and terminate.
1069\item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$.
1070\item Let $L_3$ be the part of $L_2$ that is coprime to
1071      $p+1$ for every prime~$p$ such that $p^2\mid N$.
1072\item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic
1073      of any prime of
1074      congruence between~$f$ and a form of weight~$k$ and
1075      lower level. (By congruence here, we mean a congruence for coefficients
1076      $a_n$ with $n$ coprime to the level of~$f$.)
1077\item Let $B$ be the part of $L_4$ coprime to the residue characteristic
1078      of any prime of congruence
1079      between~$f$ and an Eisenstein series.  (This eliminates
1080      residue characteristics of reducible representations.)
1081\end{enumerate}
1082Proposition~\ref{sha} implies that if
1083$\ord_p(B) > 0$, then $\ord_p(\#\Sha) > 0$.
1084
1085We computed~$B$ for every newform in~$\mathcal{S}$.  There are many
1086examples in which $L_3$ is large, but~$B$ is not, and this is because
1087of Tamagawa factors. For example, {\bf 39k4C} has $L_3=19$, but $B=1$
1088because of a $19$-congruence with a form of level~$13$; in this case
1089we must have $19\mid c_{13}(2)$, where $c_{13}(2)$ is as in
1090Section~\ref{sec:bkconj}.  See Section~\ref{sec:other_ex} for more
1091details.  Also note that in every example~$B$ is a perfect square,
1092which is consistent with the fact \cite{Fl2} that the part of the
1093order of $\Sha$ coprime to the modular degree is necessarily a perfect
1094square.  That our computed value of~$B$ should be a square is not a
1095priori obvious.
1096
1098of rings of integers, so our definition of~$B$ is overly conservative.
1099For example,~$5$ occurs in row~$2$ of Table~\ref{tab:newforms} but not
1100in Table~\ref{tab:invisforms}, because \nf{159k4E} is Eisenstein at
1101some prime above~$5$, but the prime of congruences of
1102characteristic~$5$ between \nf{159k4B} and \nf{159k4E} is not
1103Eisenstein.
1104
1105
1106The newforms for which $B>1$ are given in
1107Table~\ref{tab:invisforms}. The second column of the table records
1108the degree of the field generated by the Fourier coefficients
1109of~$f$.  The third contains~$B$.  Let~$W$ be the intersection of
1110the span of all conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and
1111$W^{\perp}$ the Petersson orthogonal complement of~$W$ in
1112$S_k(\Gamma_0(N),\ZZ)$. Then the fourth column contains the odd
1113prime divisors of $\#(S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp}))$, which
1114are exactly the possible primes of congruence for~$f$. We place a
1115$*$ next to the four entries of Table~\ref{tab:invisforms} that
1116also occur in Table~\ref{tab:newforms}.
1117
1118\subsection{Examples in which hypotheses fail}\label{sec:other_ex}
1119We have some other examples where forms of
1120different levels are congruent.
1121However, Remark~\ref{sign} does not
1122apply, so that one of the forms could have an odd functional
1123equation, and the other could have an even functional equation.
1124For instance, we have a $19$-congruence between the
1125newforms $g=\nf{13k4A}$ and $f=\nf{39k4C}$ of Fourier
1126coefficients coprime to $39$.
1127Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$
1128has {\it odd} functional equation.
1129Here~$f$ fails the condition about not being congruent
1130to a form of lower level, so in Lemma~\ref{local1} it is possible that
1131$\ord_{\qq}(c_{19}(2))>0$. In fact this does happen. Because
1132$V'_{\qq}$ (attached to~$g$ of level $81$) is unramified at $p=19$,
1133$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
1134two-dimensional. As in (2) of the proof of Theorem~\ref{local},
1135one of the eigenvalues of $\Frob_p^{-1}$ acting on this
1136two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
1137$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
1138$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
1139$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
1140$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
1141Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
1142our example here with $p=7$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
1143non-trivial when $w_p=-1$, so (2) of the proof of Theorem~\ref{local}
1144does not work. This is just as well, since had it
1145worked we would have expected
1146$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
1147shows not to be the case.
1148
1149In the following example, the divisibility between the levels is the
1150other way round.  There is a $7$-congruence between $g=\nf{122k6A}$
1151and $f=\nf{61k6B}$, both $L$-functions have even functional equation,
1152and $L(g,3)=0$. In the proof of Theorem~\ref{local},
1153there is a problem with the local condition at $p=2$. The map from
1154$H^1(I_2,A'[\qq](3))$ to $H^1(I_2,A'_{\qq}(3))$ is not necessarily
1155injective, but its kernel is at most one dimensional, so we still get
1156the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
1157$\FF_{\qq}$-rank at least~$1$ (assuming $r\geq 2$), and thus get
1158(probable) elements of $\Sha$ for \nf{61k6B}. In particular, these
1159elements of $\Sha$ are {\it invisible} at level 61. When the levels
1160are different we are no longer able to apply Theorem 2.1 of
1161\cite{FJ}. However, we still have the congruences of integral modular
1162symbols required to make the proof of Proposition \ref{div} go
1163through. Indeed, as noted above, the congruences of modular forms were
1164found by producing congruences of modular symbols. Despite these
1165congruences of modular symbols, Remark 5.3 does not apply, since there
1166is no reason to suppose that $w_N=w_{N'}$, where $N$ and $N'$ are the
1167distinct levels.
1168
1169Finally, there are two examples where we have a form $g$ with even
1170functional equation such that $L(g,k/2)=0$, and a congruent form
1171$f$ which has odd functional equation; these are a 23-congruence
1172between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
1173between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
1174$\ord_{s=2}L(f,s)=1$, it ought to be the case that
1175$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
1176similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
1177the appropriate modification of Theorem \ref{local} does not
1178necessarily provide us with non-trivial $\qq$-torsion in $\Sha$.
1179It only tells us that the $\qq$-torsion subgroup of
1180$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It
1181could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$
1182appears in the conjectural formula for the first derivative of the
1183complex $L$ function, evaluated at $s=k/2$, but in combination
1184with a regulator that we have no way of calculating.
1185
1186Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
1187associated with $f$ and $g$ by the construction of Mazur, Tate and
1188Teitelbaum \cite{MTT}, each divided by a suitable canonical
1189period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
1190quite clear what to make of this. This divisibility may be proved
1191as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
1192times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
1193normalised) are congruent $\bmod{\,\qq}$, as a result of the
1194congruence between the modular symbols out of which they are
1195constructed. Integrating an appropriate function against these
1196measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
1197to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
1198since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
1199where the signs in the functional equations of $L(g,s)$ and
1200$L_q(g,s)$ are the same, positive in this instance. (According to
1201the proposition in Section 18 of \cite{MTT}, the signs differ
1202precisely when $L_q(g,s)$ has a trivial zero'' at $s=k/2$.)
1203
1204We also found some examples for which the conditions of Theorem~\ref{local}
1205were not met.  For example, we have a $7$-congruence between
1206\nf{639k4B} and \nf{639k4H}, but $w_{71}=-1$, so that $71\equiv 1207-w_{71}\pmod{7}$. There is a similar problem with a $7$-congruence
1208between \nf{260k6A} and \nf{260k6E} --- here $w_{13}=1$ so that
1209$13\equiv -w_{13}\pmod{7}$. Finally, there is a $5$-congruence between
1210\nf{116k6A} and \nf{116k6D}, but here the prime~$5$ is less than the
1211weight~$6$.
1212
1213\begin{thebibliography}{AL}
1214\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
1215$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
1216\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
1217Shafarevich-Tate groups of abelian varieties, preprint.
1218\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
1219{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
1220\bibitem[BCP]{magma}
1221W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
1222  {T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
1223  235--265, Computational algebra and number theory (London, 1993).
1224\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
1225of motives, The Grothendieck Festschrift Volume I, 333--400,
1226Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
1227\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques
1228associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
1229\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
1230\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
1231Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em
1232Duke Math. J. }{\bf 59 }(1989), 785--801.
1233\bibitem[CM1]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
1234Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
123513--28.
1236\bibitem[CM2]{CM2} J. E. Cremona, B. Mazur, Appendix to A. Agashe,
1237W. Stein, Visible evidence for the Birch and Swinnerton-Dyer
1238conjecture for modular abelian varieties of rank zero, preprint.
1239\bibitem[CF]{CF} B. Conrey, D. Farmer, On the non-vanishing of
1240$L_f(s)$ at the center of the critical strip, preprint.
1241\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
1242$\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math.
1243{\bf 179, } 139--172, Springer, 1969.
1244\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
1245d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
1246part 2, 313--346.
1247\bibitem[DFG1]{DFG} F. Diamond, M. Flach, L. Guo, Adjoint motives
1248of modular forms and the Tamagawa number conjecture, preprint.
1249{{\sf
1250http://www.andromeda.rutgers.edu/\~{\mbox{}}liguo/lgpapers.html}}
1251\bibitem[DFG2]{DFG2} F. Diamond, M. Flach, L. Guo, The Bloch-Kato
1252conjecture for adjoint motives of modular forms, {\em Math. Res.
1253Lett. }{\bf 8 }(2001), 437--442.
1254\bibitem[Du1]{Du3} N. Dummigan, Symmetric square $L$-functions and
1255Shafarevich-Tate groups, {\em Experiment. Math. }{\bf 10 }(2001),
1256383--400.
1257\bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and
1258Selmer groups, {\em Math. Res. Lett. }{\bf 8 }(2001), 479--494.
1259\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
1260Galois representations, {\em in }Algebraic analysis, geometry and
1261number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
1262Press, Baltimore, 1989.
1263\bibitem[FJ]{FJ} G. Faltings, B. Jordan, Crystalline cohomology
1264and $\GL(2,\QQ)$, {\em Israel J. Math. }{\bf 90 }(1995), 1--66.
1265\bibitem[Fl1]{Fl2} M. Flach, A generalisation of the Cassels-Tate
1266pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.
1267\bibitem[Fl2]{Fl1} M. Flach, On the degree of modular parametrisations,
1268S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
1269ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
1270Boston Berlin, 1993.
1271\bibitem[Fo1]{Fo} J.-M. Fontaine, Sur certains types de
1272repr\'esentations $p$-adiques du groupe de Galois d'un corps
1273local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
1274}{\bf 115 }(1982), 529--577.
1275\bibitem[Fo2]{Fo2} J.-M. Fontaine, Valeurs sp\'eciales des
1276fonctions $L$ des motifs, S\'eminaire Bourbaki, Vol. 1991/92. {\em
1277Ast\'erisque }{\bf 206 }(1992), Exp. No. 751, 4, 205--249.
1278\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture epsilon''
1279for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
128051--56.
1281\bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois
1282representations coming from modular forms, {\em J. Number Theory
1283}{\bf 31 }(1989), 133--141.
1284\bibitem[MTT]{MTT} B. Mazur, J. Tate, J. Teitelbaum, On $p$-adic
1285analogues of the conjectures of Birch and Swinnerton-Dyer, {\em
1286Invent. Math. }{\bf 84 }(1986), 1--48.
1287\bibitem[Ne]{Ne2} J. Nekov\'ar, $p$-adic Abel-Jacobi maps and $p$-adic
1288heights. The arithmetic and geometry of algebraic cycles (Banff,
1289AB, 1998), 367--379, CRM Proc. Lecture Notes, 24, Amer. Math.
1290Soc., Providence, RI, 2000.
1291\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
1292{\em Invent. Math. }{\bf 100 }(1990), 419--430.
1293\bibitem[St]{St} G. Stevens, $\Lambda$-adic modular forms of
1294half-integral weight and a $\Lambda$-adic Shintani lifting.
1295Arithmetic geometry (Tempe, AZ, 1993), 129--151, Contemp. Math.,
1296174, Amer. Math. Soc., Providence, RI, 1994.
1297\bibitem[V]{V} V. Vatsal, Canonical periods and congruence
1298formulae, {\em Duke Math. J. }{\bf 98 }(1999), 397--419.
1299\bibitem[Z]{Z} S. Zhang, Heights of Heegner cycles and derivatives of $L$-series,
1300{\em Invent. Math. }{\bf 130 }(1997), 99--152.
1301\end{thebibliography}
1302
1303
1304\end{document}
1305127k4A    43  127k4C   17   [43]
1306159k4A  5,23  159k4E    8   [5]x[23]
1307365k4B    29  365k4E   18   [29]x[5] (extra factor of 5 divides the level)
1308369k4A  5,13  369k4J    9   [5]x[13]x[2]
1309453k4A  5,17  453k4E   23   [5]x[17]
1310453k4A    23  151k4A   30   Odd func eq for g
1311465k4A    11  465k4H    7   [11]x[5]x[2]
1312477k4A    73  477k4M   12   [73]x[2]
1313567k4A    23  567k4I    8   [23]x[3]
131481k4A     13  567k4L   12   Odd func eq for f, Theorem 4.1 gives nothing.
1315581k4A 19,19  581k4E   34   [19^2]x[4] (possible BIG Sha, as 19^2 divides MD)
1316639k4A     7  639k4H   12   [7]
1317657k4A     5  657k4C    7   [5]x[3]x[2] (see next note)
1318657k4A     5  657k4G   12   [5]x[4] (does 657k4A make both these visible?)
1319681k4A    43  227k4A   23   Odd func eq for g
1320681k4A    59  681k4D   30   [59]x[3]x[2]
1321684k4C   7,7  684k4K    4   [7^2]x[2] (see note to 581k4A)
132295k6A  31,59   95k6D    9   [31]x[59]
1323116k6A     5  116k6D    6   [5]x[29]x[2]
1324122k6A     7  61k6B    14   7^2 appears in L(61k6B,3)
1325122k6A    73  122k6C    6   [73]x[3] (guess that 3 is a bad prime now)
1326260k6A  7,17  260k6E    4   [7]x[17]x[4] <-- Did not compute MD or LROP
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339\nf{263k4B} &  & 41^2 & \\
1340\nf{269k4C} &  & 23^2 & \\
1341\nf{271k4B} &  & 29^2 &\\
1342\nf{281k4B} &  & 29^2\\
1343\hline
1344\nf{101k6B} &  & 17^2 & 101\\
1345\nf{103k6B} &  & 23^2\\
1346\nf{111k6C} &  & 11^2\\
1347\nf{153k6G} &  & 7^2\\
1348\nf{157k6B} &  & 252^2\\
1349\nf{167k6B} &  & 41^2\\
1350\nf{172k6B} &  & 7^2\\
1351\nf{173k6B} &  & 71^2\\
1352\nf{181k6B} &  & 107^2\\
1353\hline
1354\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571\\
1355