Author: William A. Stein
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2%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
3%
4% dsw.tex
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6% 29 August 2002
7%
8% Project of William Stein, Neil Dummigan, Mark Watkins
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11
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25\theoremstyle{definition}
26\newtheorem{defi}[prop]{Definition}
27\theoremstyle{remark}
28\newtheorem{examp}[prop]{Example}
29\newtheorem{remar}[prop]{Remark}
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81
82% ---- SHA ----
83\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
84  \newcommand{\textcyr}{%
85    {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
86     \selectfont #1}}
87\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
88
89\newcommand{\HH}{{\mathfrak H}}
90\newcommand{\aaa}{{\mathfrak a}}
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94\newcommand{\Fbar}{\overline{F}}
95\newcommand{\CH}{\mathrm {CH}}
96
97\begin{document}
98\title[Shafarevich-Tate Groups of Modular Motives]{Constructing Elements in Shafarevich-Tate Groups of Modular Motives}
99\author{Neil Dummigan}
100\author{William Stein}
101\author{Mark Watkins}
102\date{August 29th, 2002}
103\subjclass{11F33, 11F67, 11G40.}
104
105\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
106Shafarevich-Tate group.}
107
108\address{University of Sheffield\\ Department of Pure
109Mathematics\\
110Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
111U.K.}
113One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
115University Park\\State College, PA 16802\\ U.S.A.}
116
117\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
118\email{watkins@math.psu.edu}
119
120
121\begin{abstract}
122
123We study Shafarevich-Tate groups of motives attached to modular
124forms on $\Gamma_0(N)$ of weight bigger than~$2$.  We deduce a
125criterion for the existence of nontrivial elements of these
126Shafarevich-Tate groups, and give $16$ examples in which a strong
127form of the Beilinson-Bloch conjecture implies the existence of
128such elements. We also use modular symbols and observations about
129Tamagawa numbers to compute nontrivial conjectural lower bounds on
130the orders of the Shafarevich-Tate groups of modular motives of
131low level and weight at most $12$.  Our methods build upon the
132idea of visibility due to Cremona and Mazur, but in the context of
134\end{abstract}
135
136\maketitle
137
138\section{Introduction}
139Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
140be the associated $L$-function. The conjecture of Birch and
141Swinnerton-Dyer \cite{BSD} predicts that the order of vanishing of $L(E,s)$
142at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
143also gives an interpretation of the leading term in the Taylor
144expansion in terms of various quantities, including the order of
145the Shafarevich-Tate group of~$E$.
146
147Cremona and Mazur \cite{CM} look, among all strong Weil elliptic
148curves over $\QQ$ of conductor $N\leq 5500$, at those with
149nontrivial Shafarevich-Tate group (according to the Birch and
150Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
151group has predicted elements of prime order~$p$. In most cases
152they find another elliptic curve, often of the same conductor,
153whose $p$-torsion is Galois-isomorphic to that of the first one,
154and which has positive rank. The rational points on the second elliptic
155curve produce classes in the common $H^1(\QQ,E[p])$. They show
156\cite{CM2} that these lie in the Shafarevich-Tate group of the
157first curve, so rational points on one curve explain elements of
158the Shafarevich-Tate group of the other curve.
159
160The Bloch-Kato conjecture \cite{BK} is the generalisation to
161arbitrary motives of the leading term part of the Birch and
162Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
163\cite{B} generalises the part about the order of vanishing at the
164central point, identifying it with the rank of a certain Chow
165group.
166
167This paper is a partial generalisation of \cite{CM} and \cite{AS}
168from abelian varieties over $\QQ$ associated to modular forms of
169weight~$2$ to the motives attached to modular forms of higher weight.
170It also does for congruences between modular forms of equal weight
171what \cite{Du2} did for congruences between modular forms of different
172weights.
173
174We consider the situation where two newforms~$f$ and~$g$, both of
175even weight $k>2$ and level~$N$, are congruent modulo a maximal
176ideal $\qq$ of odd residue characteristic, and $L(g,k/2)=0$ but
177$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish
178to order at least $2$ at $s=k/2$.  In Section~\ref{sec:examples},
179we give sixteen such examples (all with $k=4$ and $k=6$), and in
180each example, we find that $\qq$ divides the numerator of the
181algebraic number $L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$
182is a certain canonical period.
183
184In fact, we show how this divisibility may be deduced from the
185vanishing of $L(g,k/2)$ using recent work of Vatsal \cite{V}. The
186point is, the congruence between$f$ and~$g$ leads to a congruence
187between suitable algebraic parts'' of the special values
188$L(f,k/2)$ and $L(g,k/2)$. In slightly more detail, a multiplicity
189one result of Faltings and Jordan shows that the congruence of
190Fourier expansions leads to a congruence of certain associated
191cohomology classes. These are then identified with the modular
192symbols which give rise to the algebraic parts of special values.
193If $L(g,k/2)$ vanishes then the congruence implies that
194$L(f,k/2)/\vol_{\infty}$ must be divisible by $\qq$.
195
196The Bloch-Kato conjecture sometimes then implies that the
197Shafarevich-Tate group $\Sha$ attached to~$f$ has nonzero
198$\qq$-torsion. Under certain hypotheses and assumptions, the most
199substantial of which is the Beilinson-Bloch conjecture relating
200the vanishing of $L(g,k/2)$ to the existence of algebraic cycles,
201we are able to construct some of the predicted elements of~$\Sha$
202using the Galois-theoretic interpretation of the congruences to
203transfer elements from a Selmer group for~$g$ to a Selmer group
204for~$f$. One might say that algebraic cycles for one motive
205explain elements of~$\Sha$ for the other, or that we use
206congruences to link the Beilinson-Bloch conjecture for one motive
207with the Bloch-Kato conjecture for the other.
208%In proving the local
209%conditions at primes dividing the level, and also in examining the
210%local Tamagawa factors at these primes, we make use of a higher weight
211%level-lowering result due to Jordan and Livn\'e \cite{JL}.
212
213We also compute data which, assuming the Bloch-Kato conjecture,
214provides lower bounds for the orders of numerous Shafarevich-Tate
215groups (see Section~\ref{sec:invis}).
216%Our data is consistent
217%with the fact \cite{Fl2} that the part of $\#\Sha$ coprime to the
218%congruence modulus is necessarily a perfect square (assuming that~$\Sha$
219%is finite).
220
221\section{Motives and Galois representations}
222This section and the next provide definitions of some of the
223quantities appearing later in the Bloch-Kato conjecture. Let
224$f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
225$\Gamma_0(N)$, with coefficients in an algebraic number field~$E$,
226which is necessarily totally real. Let~$\lambda$ be any finite
227prime of~$E$, and let~$\ell$ denote its residue characteristic.  A
228theorem of Deligne \cite{De1} implies the existence of a
229two-dimensional vector space $V_{\lambda}$ over $E_{\lambda}$, and
230a continuous representation
231$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
232such that
233\begin{enumerate}
234\item $\rho_{\lambda}$ is unramified at~$p$ for all primes~$p$
235not dividing~$lN$, and
236\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
237characteristic polynomial of $\Frob_p^{-1}$ acting on
238$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
239\end{enumerate}
240
241Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
242the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
243There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
244both $2$-dimensional $E$-vector spaces. For details of the
245construction see \cite{Sc}. The de Rham realisation has a Hodge
246filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset 247F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
248cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
249cohomology.
250For each prime $\lambda$, there is a natural isomorphism
251$V_B\otimes E_{\lambda}\simeq V_{\lambda}$. We may choose a
252$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
253each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
254Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
255There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
256which amounts to multiplying the action of $\Frob_p$ by $p^j$.
257
258Following \cite{BK} (Section 3), for $p\neq l$ (including
259$p=\infty$) let
260$$261H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow 262H^1(I_p,V_{\lambda}(j))). 263$$
264The subscript~$f$ stands for finite
265part'', $D_p$ is a decomposition subgroup at a prime above~$p$,
266$I_p$ is the inertia subgroup, and the cohomology is for
267continuous cocycles and coboundaries. For $p=l$ let
268$$269H^1_f(\QQ_l,V_{\lambda}(j))=\ker 270(H^1(D_l,V_{\lambda}(j))\rightarrow 271H^1(D_l,V_{\lambda}(j)\otimes_{\QQ_l} B_{\cris})) 272$$
273(see Section 1 of
274\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
275$B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
276elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
277in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes~$p$.
278
279There is a natural exact sequence
280$$281\begin{CD}0@>>>T_{\lambda}(j)@>>> 282V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}. 283$$
284Let
285$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
286Define the $\lambda$-Selmer group
287$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
288$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
289$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes~$p$. Note that the
290condition at $p=\infty$ is superfluous unless $l=2$. Define the
291Shafarevich-Tate group
292$$293 \Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/ 294 \pi_*H^1_f(\QQ,V_{\lambda}(j)). 295$$
296Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any
297prime ideal~$\lambda$ is the length of the $\lambda$-component of
298$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,
299and write~$\Sha$ for~$\Sha(k/2)$. It depends on the choice of
300$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
301each $V_{\lambda}$. But if $A[\lambda]$ is irreducible then
302$T_{\lambda}$ is unique up to scaling and the $\lambda$-part of
303$\Sha$ is independent of choices.
304
305In the case $k=2$ the motive comes from an (self-dual)
306isogeny class of abelian varieties over $\QQ$, with endomorphism
307algebra containing $E$. If one chooses an abelian variety $B$ in
308the isogeny class and takes all the $T_{\lambda}(1)$ to be
309$\lambda$-adic Tate modules, then what we have defined above
310coincides with the usual Shafarevich-Tate group of $B$. To see
311this one uses 3.11 of \cite{BK}, for $l=p$. For $l\neq p$,
312$H^1(\QQ_p,V_l)=0$. Considering the formal group, every class in
313$B(\QQ_p)/lB(\QQ_p)$ is represented by an $l$-power torsion point
314in $B(\QQ_p)$, so maps to zero in $H^1(\QQ_p,A_l)$.
315
316Define the group of global torsion points
317$$318 \Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)). 319$$
320This is analogous to the group of rational torsion points on an
321elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in
322which the exponent of any prime ideal~$\lambda$ is the length of
323the $\lambda$-component of $\Gamma_{\QQ}$.
324
325\section{Canonical periods}
326We assume from now on for convenience that $N\geq 3$. We need to
327choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
328and de Rham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
329in a way such that $T_B$ and $T_{\dR}\otimes_{O_E}O_E[1/Nk!]$
330agree with (respectively) the $O_E$-lattice $\mathfrak{M}_{f,B}$
331and the $O_E[1/Nk!]$-lattice $\mathfrak{M}_{f,\dR}$ defined in
332\cite{DFG} using cohomology, with non-constant coefficients, of
333modular curves. (In \cite{DFG}, see especially Sections 2.2 and
3345.4, and the paragraph preceding Lemma 2.3.)
335
336For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
337module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
338$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes 339E_{\lambda}\simeq V_{\lambda}$. Then the $O_{\lambda}$-module
340$T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable.
341
342Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
343generalised elliptic curves with full level-$N$ structure. Let
344$\mathfrak{E}$ be the universal generalised elliptic curve over
345$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product
346of $\mathfrak{E}$ over $M(N)$. (The motive $M_f$ is constructed
347using a projector on the cohomology of a desingularisation of
348$\mathfrak{E}^{k-2}$). Realising $M(N)(\CC)$ as the disjoint union
349of $\phi(N)$ copies of the quotient
350$\Gamma(N)\backslash\mathfrak{H}^*$ (where $\mathfrak{H}^*$ is the
351completed upper half plane), and letting $\tau$ be a variable on
352$\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is isomorphic to
353the elliptic curve with period lattice generated by $1$ and
354$\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a variable on
355the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the fibre product.
356Then $2\pi i f(\tau)\,d\tau\wedge dz_1\wedge\ldots\wedge dz_{k-2}$
357is a well-defined differential form on (a desingularisation of)
358$\mathfrak{E}^{k-2}$ and naturally represents a generating element
359of $F^{k-1}T_{\dR}$. (At least we can make our choices locally at
360primes dividing $Nk!$ so that this is the case.) We shall call
361this element $e(f)$.
362
363Under the de Rham isomorphism between $V_{\dR}\otimes\CC$ and
364$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
365a natural action of complex conjugation on $V_B$, breaking it up
366into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
367Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
368to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
369$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
370are rank one $O_E$-modules, but not necessarily free, since the
371class number of $O_E$ may be greater than one. Choose nonzero
372elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
373the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
374$\Omega_f^{\pm}$ by $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$.
375
376\section{The Bloch-Kato conjecture}\label{sec:bkconj}
377In this section we extract from the Bloch-Kato conjecture for
378$L(f,k/2)$ a prediction about the order of the Shafarevich-Tate
379group, by analysing the other terms in the formula.
380
381Let $L(f,s)$ be the $L$-function attached to~$f$. For
382$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with
383Euler product
384$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but
385there is an analytic continuation given by an integral, as
386described in the next section. Suppose that $L(f,k/2)\neq 0$. The
387Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the
388following equality of fractional ideals of~$E$:
389$$390 \frac{L(f,k/2)}{\vol_{\infty}}= 391 \left(\prod_pc_p(k/2)\right) 392 \frac{\#\Sha}{\aaa^{\pm}(\#\Gamma_{\QQ})^2}. 393$$
394Here, {\bf and from this point onwards, }$\pm$ represents the
395parity of $(k/2)-1$. The quantity
396$\vol_{\infty}$ is equal to $(2\pi i)^{k/2}$
397multiplied by the determinant of the isomorphism
398$V_B^{\pm}\otimes\CC\simeq (V_{\dR}/F^{k/2})\otimes\CC$,
399calculated with respect to the lattices $O_E\delta_f^{\pm}$ and
400the image of $T_{\dR}$. For $l\neq p$, $\ord_{\lambda}(c_p(j))$ is
401defined to be
402\begin{align*}
403\length&\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-
404  \ord_{\lambda}(P_p(p^{-j}))\\
405=&\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).
406\end{align*}
407
408We omit the definition of $\ord_{\lambda}(c_p(j))$ for
409$\lambda\mid p$, which requires one to assume Fontaine's de Rham
410conjecture (\cite{Fo}, Appendix A6), and depends on the choices of
411$T_{\dR}$ and $T_B$, locally at~$\lambda$. (We shall mainly be
412concerned with the $q$-part of the Bloch-Kato conjecture, where~$q$
413is a prime of good reduction. For such primes, the de Rham
414conjecture follows from Theorem 5.6 of \cite{Fa1}.)
415
416Strictly speaking, the conjecture in \cite{BK} is only given for
417$E=\QQ$. We have taken here the obvious generalisation of a slight
418rearrangement of (5.15.1) of \cite{BK}. The Bloch-Kato conjecture
419has been reformulated and generalised by Fontaine and Perrin-Riou,
420who work with general $E$, though that is not really the point of
421their work. In Section 11 of \cite{Fo2} it is sketched how to
422deduce the original conjecture from theirs, though only in the
423case $E=\QQ$.
424\begin{lem}\label{vol}
425$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and
426$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.
427\end{lem}
428\begin{proof}
429We note that $\Omega_{\pm}$ is equal to the determinant of the
430period map from $F^{k/2}V_{\dR}\otimes\CC$ to
431$V_B^{\pm}\otimes\CC$, with respect to lattices dual to those we
432used above in the definition of $\vol_{\infty}$ (c.f. the last
433paragraph of 1.7 of \cite{De2}). We are using here natural
434pairings. Recall that the index of $O_E\delta_f^{\pm}$ in
435$T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then the proof is completed
436by noting that, locally away from primes dividing $Nk!$, the index
437of $T_{\dR}$ in its dual is equal to the index of $T_B$ in its
438dual, both being equal to the ideal denoted~$\eta$ in \cite{DFG2}.
439\end{proof}
440\begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer.
441Then the fractional ideal $c_p(j)$ is supported at most on
442divisors of~$p$.
443\end{lem}
444\begin{proof}
445As on p.~30 of \cite{Fl1}, for odd $l\neq p$,
446$\ord_{\lambda}(c_p(j))$ is the length of the finite
447$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
448where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
449trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
450torsion free.
451\end{proof}
452
453\begin{lem}\label{local1}
454Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
455is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
456$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
457that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not
458congruent modulo $\qq$ to any newform of weight~$k$, trivial
459character, and level dividing $N/p$. Then $\ord_{\qq}(c_p(j))=0$
460for all integers~$j$.
461\end{lem}
462\begin{proof}
463There is a natural injective map from
464$V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$ to $H^0(I_p,A[\qq](j))$.
465Consideration of $\qq$-torsion shows that
466$$467 \dim_{O_E/\qq} H^0(I_p,A[\qq](j))\geq \dim_{E_{\qq}} 468 H^0(I_p,V_{\qq}(j)). 469$$ To prove the lemma it suffices to show that
470$$471 \dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)), 472$$
473since this ensures that $H^0(I_p,A_{\qq}(j))= 474V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$, hence that
475$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p})$.
476If the dimensions differ then, given that $f$ is not congruent
477modulo $\qq$ to a newform of level dividing $N/p$, Proposition 2.2
478of \cite{L} shows that we are in the situation covered by one of
479the three cases in Proposition 2.3 of \cite{L}. Since $p\not\equiv 480-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is
481unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's
482result that $N$ is the prime-to-$q$ part of the conductor of
483$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses
484the condition $q>k$) implies the existence of a newform of weight
485$k$, trivial character and level dividing $N/p$, congruent to~$g$
486modulo $\qq$. This contradicts our hypotheses.
487\end{proof}
488
489\begin{remar}
490For an example of what can be done when~$f$ is congruent to
491a form of lower level, see the first example in Section~\ref{sec:other_ex}
492below.
493\end{remar}
494
495\begin{lem}\label{at q}
496If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then
497$\ord_{\qq}(c_q)=0$.
498\end{lem}
499\begin{proof}
500It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an
501application, at the end of Section~2.2, of the results of
502\cite{Fa1}) that $T_{\qq}$ is the
503$O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered
504module $T_{\dR}\otimes O_{\qq}$ by the functor they call
505$\mathbb{V}$. (This property is part of the definition of an
506$S$-integral premotivic structure given in Section~1.2 of
507\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
508of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
509in Theorem~4.1 of \cite{BK} follows from the first paragraph of
5102(h) of \cite{Fa1}.)
511\end{proof}
512
513\begin{lem}
514If $A[\lambda]$ is an
515irreducible representation of $\Gal(\Qbar/\QQ)$,
516then
517$$\ord_{\lambda}(\#\Gamma_{\QQ})=0.$$
518\end{lem}
519\begin{proof}
520This follows trivially from the definition.
521\end{proof}
522
523Putting together the above lemmas we arrive at the following:
524\begin{prop}\label{sha}
525Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that
526$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$,
527where $\qq\mid q$. Assume the same hypotheses as in Lemma
528\ref{local1} for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which
529locally at $\qq$ are as in the previous section. If
530$L(f,k/2)\aaa^{\pm}/\vol_{\infty}\neq 0$ then the Bloch-Kato
531conjecture predicts that
532$$533 \ord_{\qq}(\#\Sha)=\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty}). 534$$
535\end{prop}
536
537\section{Congruences of special values}
538Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
539weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
540large enough to contain all the coefficients $a_n$ and $b_n$.
541Suppose that $\qq\mid q$ is a prime of~$E$ such that $f\equiv 542g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Assume
543that $A[\qq]$ is an irreducible representation of
544$\Gal(\Qbar/\QQ)$, and that $q\nmid N\phi(N)k!$. Choose
545$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
546$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates
547$T_B^{\pm}$ locally at $\qq$. Make two further assumptions:
548$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$
549
550\begin{prop} \label{div}
551With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
552\end{prop}
553\begin{proof} This is based on some of the ideas used in Section 1 of
554\cite{V}.  Note the apparent typo in Theorem~1.13 of \cite{V},
555which presumably should refer to Condition 2''. Since
556$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
557$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm 5581=(-1)^{(k/2)-1}$. It is well known, and easy to prove, that
559$$560 \int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s). 561$$
562Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
563$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
564where the integral is taken along the positive imaginary axis,
565then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
566Thus we are reduced
567to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
568
569Let $\mathcal{D}_0$ be the group of divisors of degree zero
570supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
571integer $r\geq 0$, let $P_r(R)$ be the additive group of
572homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
573groups have a natural action of $\Gamma_1(N)$. Let
574$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
575be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
576
577Via the isomorphism (8) in Section~1.5 of \cite{V}, combined with
578the argument in 1.7 of \cite{V}, the cohomology class
579$\omega_f^{\pm}$ corresponds to a modular symbol $\Phi_f^{\pm}\in 580S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
581element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_{E,\qq})$. We are
582now dealing with cohomology over $X_1(N)$ rather than $M(N)$,
583which is why we insist that $q\nmid \phi(N)$. It follows from the
584last line of Section~4.2 of \cite{St} that, up to some small
585factorials which do not matter locally at $\qq$,
586$$\Phi_f^{\pm}([\infty]-)=\sum_{j=0,j\equiv 587(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since
588$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
589$$\Delta_f^{\pm}([\infty]-)=\sum_{j=0,j\equiv 590(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
591coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
592show is divisible by $\qq$.
593Similarly
594$$\Phi_g^{\pm}([\infty]-)=\sum_{j=0,j\equiv 595(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of
596$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
597Therefore it would suffice to show that, for some $\mu\in O_E$,
598the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
599$\qq$ in $S_{\Gamma_1(N)}(k,O_{E,\qq})$. It suffices to show that,
600for some $\mu\in O_E$, the element
601$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
602considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
603non-constant coefficients. This would be the case if
604$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
605one-dimensional subspace upon reduction modulo~$\qq$. But this is
606a consequence of Theorem 2.1(1) of \cite{FJ} (for which we need
607the irreducibility of $A[\qq]$).
608\end{proof}
609\begin{remar}\label{sign}
610The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are
611equal. They are determined by the eigenvalue of the
612Atkin-Lehner involution~$W_N$,
613which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and
614$b_N$ are each $N^{k/2-1}$ times this sign and~$\qq$ has residue
615characteristic coprime to $2N$.  The common sign in the functional
616equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of
617$W_N$ acting on~$f$ and~$g$.
618\end{remar}
619
620This is analogous to the remark at the end of Section~3 of \cite{CM},
621which shows that if~$\qq$ has odd residue characteristic and
622$L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then $L(g,s)$ must vanish to order
623at least two at $s=k/2$.  Note that Maeda's conjecture
624implies that there are no examples of~$g$ of
626$L(g,k/2)=0$ (see \cite{CF}).
627
628\section{Constructing elements of the Shafarevich-Tate group}
629Let~$f$,~$g$ and $\qq$ be as in the first paragraph of the
630previous section. In the previous section we showed how the
631congruence between $f$ and $g$ relates the vanishing of $L(g,k/2)$
632to the divisibility by $\qq$ of an algebraic part'' of
633$L(f,k/2)$. Conjecturally the former is associated with the
634existence of certain algebraic cycles (for $M_g$) while the latter
635is associated with the existence of certain elements of the
636Shafarevich-Tate group (for $M_f$, as we saw in \S 4). In this
637section we show how the congruence, interpreted in terms of Galois
638representations, provides a direct link between algebraic cycles
639and the Shafarevich-Tate group.
640
641For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
642$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
643$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
644is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
645the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
646irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
647
648Recall that $L(g,k/2)=0$ and $L(f,k/2)\neq 0$. Since the sign in
649the functional equation for $L(g,s)$ is positive (this follows
650from $L(f,k/2)\neq 0$, see Remark \ref{sign}), the order of
651vanishing of $L(g,s)$ at $s=k/2$ is at least $2$. According to the
652Beilinson-Bloch conjecture \cite{B}, the order of vanishing of
653$L(g,s)$ at $s=k/2$ is the rank of the group
654$\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational rational equivalence
655classes of null-homologous, algebraic cycles of codimension $k/2$
656on the motive $M_g$. (This generalises the part of the
657Birch--Swinnerton-Dyer conjecture which says that for an elliptic
658curve $E/\QQ$, the order of vanishing of $L(E,s)$ at $s=1$ is
659equal to the rank of the Mordell-Weil group $E(\QQ)$.)
660
661Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
662to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
663subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
664If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
665get (assuming also the Beilinson-Bloch conjecture) a subspace of
666$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
667vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
668conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
669equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
670follow from the conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
671Sections~1 and~6.5 of \cite{Fo2}. We shall call it the strong''
672Beilinson-Bloch conjecture.
673
674Similarly, if $L(f,k/2)\neq 0$ then we expect that
675$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
676coincides with the $\qq$-part of $\Sha$.
677\begin{thm}\label{local}
678Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
679$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
680$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$ and
681that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a
682newform of weight~$k$, trivial character and level dividing $N/p$.
683Suppose that, for all primes $p\mid N$, $\,p\not\equiv 684-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here
685$w_p$ is the common eigenvalue of the Atkin-Lehner involution
686$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of
687$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
688\end{thm}
689
690\begin{proof}
691It follows easily from our hypothesis that the rank of the free
692part of $H^1_f(\QQ,T'_{\qq}(k/2))$ is $r$. The natural map from
693$H^1_f(\QQ,T'_{\qq}(k/2))/\qq H^1_f(\QQ,T'_{\qq}(k/2))$ to
694$H^1(\QQ,A'[\qq](k/2))$ is injective. Take a nonzero class $c$ in
695the image, which has $\FF_{\qq}$-rank $r$. Choose $d\in 696H^1_f(\QQ,T'_{\qq}(k/2))$ mapping to $c$. Consider the
697$\Gal(\Qbar/\QQ)$-cohomology of the short exact sequence
698$$699 \begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD}, 700$$
701where~$\pi$ is multiplication by a uniformising element of
702$O_{\qq}$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial.
703Hence $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
704$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
705we get a nonzero, $\qq$-torsion class $\gamma\in 706H^1(\QQ,A_{\qq}(k/2))$.
707
708Our aim is to show that $\res_p(\gamma)\in 709H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
710consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
711
712\begin{enumerate}
713\item {\bf $p\nmid qN$. }
714
715Consider the $I_p$-cohomology of the short exact sequence above.
716
717Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p, 718A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
719$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
720$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
721follows from the fact that $d\in H^1_f(\QQ,T'_{\qq}(k/2))$ that
722the image in $H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is
723zero, hence that the restriction of~$c$ to
724$H^1(I_p,A'[\qq](k/2))\simeq H^1(I_p,A[\qq](k/2))$ is zero. Hence
725the restriction of $\gamma$ to $H^1(I_p,A_{\qq}(k/2))$ is also
726zero. By line~3 of p.~125 of \cite{Fl2},
727$H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just contained in)
728the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$ to
729$H^1(I_p,A_{\qq}(k/2))$,  so we have shown that $\res_p(\gamma)\in 730H^1_f(\QQ_p,A_{\qq}(k/2))$.
731
732\item {\bf $p\mid N$. }
733
734First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
735It suffices to show that
736$$\hspace{3.5em} 737\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)), 738$$
739since then the natural map from $H^0(I_p,V'_{\qq}(k/2))$ to
740$H^0(I_p, A'_{\qq}(k/2))$ is surjective; this may be done as in
741the proof of Lemma \ref{local1}. It follows as above that the
742image of $c\in H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is
743zero. Then $\res_p(c)$ comes from
744$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
745order of this group is the same as the order of the group
746$H^0(\QQ_p,A[\qq](k/2))$ (this is Lemma 1 of \cite{W}), which we
747claim is trivial. By the work of Carayol \cite{Ca1}, the level $N$
748is the conductor of $V_{\qq}(k/2)$, so $p\mid N$ implies that
749$V_{\qq}(k/2)$ is ramified at $p$, hence $\dim 750H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim 751H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
752consider the case where this common dimension is $1$. The
753(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha 754p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication
755by~$\alpha$ on the one-dimensional space $H^0(I_p,V_{\qq})$. It
756follows from Theor\'eme A of \cite{Ca1} that this is the same as
757the Euler factor at $p$ of $L(f,s)$. By Theorems 3(ii) and 5 of
758\cite{AL}, it then follows that $p^2\nmid N$ and
759$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
760$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
761$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$) as
762$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
763$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
764$\res_p(\gamma)=0$ and certainly lies in
765$H^1_f(\QQ_p,A_{\qq}(k/2))$.
766
767\item {\bf $p=q$. }
768
769Since $q\nmid N$ is a prime of good reduction for the motive
770$M_g$, $\,V'_{\qq}$ is a crystalline representation of
771$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
772$V'_{\qq}$ have the same dimension, where
773$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q} 774B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
775As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
776the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
777filtered module $T_{\dR}\otimes O_{\qq}$. Since also $q>k$, we may
778now prove, in the same manner as Proposition 9.2 of \cite{Du3},
779that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$. For the
780convenience of the reader, we give some details.
781
782In Lemma 4.4 of \cite{BK}, a cohomological functor $\{h^i\}_{i\geq 7830}$ is constructed on the Fontaine-Lafaille category of filtered
784Dieudonn\'e modules over $\ZZ_q$. $h^i(D)=0$ for all $i\geq2$ and
785all $D$, and $h^i(D)=\Ext^i(1_{FD},D)$ for all $i$ and $D$, where
786$1_{FD}$ is the unit'' filtered Dieudonn\'e module.
787
788Now let $D=T_{\dR}\otimes O_{\qq}$ and $D'=T_{\dR}\otimes 789O_{\qq}$. By Lemma 4.5 (c) of \cite{BK},
790$$791 \hspace{3.5em} h^1(D)\simeq H^1_e(\QQ_q,T_{\qq}), 792$$
793where
794$$795 \hspace{3.5em}H^1_e(\QQ_q,T_{\qq})=\ker(H^1(\QQ_q,T_{\qq})\rightarrow 796 H^1(\QQ_q,V_{\qq})/H^1_e(\QQ_q,V_{\qq})) 797$$
798and
799$$800\hspace{3.5em}H^1_e(\QQ_q,V_{\qq})=\ker(H^1(\QQ_q,V_{\qq})\rightarrow 801H^1(\QQ_q,B_{\cris}^{f=1}\otimes_{\QQ_q} V_{\qq})). 802$$ Likewise
803$h^1(D')\simeq H^1_e(\QQ_q,T'_{\qq}).$ When applying results of
804\cite{BK} we view $D$, $T_{\qq}$ etc. simply as $\ZZ_q$-modules,
805forgetting the $O_{\qq}$-structure.
806
807For an integer $j$ let $D(j)$ be $D$ with the Hodge filtration
808shifted by $j$. Then
809$$\hspace{3.5em} 810 h^1(D(j))\simeq H^1_e(\QQ_q,T_{\qq}(j)) 811$$
812(as long as $k-p+1<j<p-1$, so that $D(j)$ satisfies the hypotheses
813of Lemma 4.5 of \cite{BK}). By Corollary 3.8.4 of \cite{BK},
814$$815 \hspace{3.5em} 816 H^1_f(\QQ_q,V_{\qq}(j))/H^1_e(\QQ_q,V_{\qq}(j))\simeq 817 (D(j)\otimes_{\ZZ_q}\QQ_q)/(1-f)(D(j)\otimes_{\ZZ_q}\QQ_q), 818$$
819where $f$ is the Frobenius operator on crystalline cohomology. By
8201.2.4(ii) of \cite{Sc}, and the Weil conjectures,
821$H^1_e(\QQ_q,V_{\qq}(j))=H^1_f(\QQ_q,V_{\qq}(j))$, since $j\neq 822(k-1)/2$. Similarly
823$H^1_e(\QQ_q,V'_{\qq}(j))=H^1_f(\QQ_q,V'_{\qq}(j))$.
824
825We have
826$$\hspace{3.5em}h^1(D(k/2))\simeq H^1_f(\QQ_q,T_{\qq}(k/2))\quad\text{and}\quad 827 h^1(D'(k/2))\simeq H^1_f(\QQ_q,T'_{\qq}(k/2)).$$
828The exact sequence in the middle of page 366 of \cite{BK} gives us a
829commutative diagram.
830$$\hspace{3.5em}\begin{CD} 831h^1(D'(k/2))@>\pi >>h^1(D'(k/2))@>>>h^1(D'(k/2)/\qq D'(k/2))\\ 832@VVV@VVV@VVV\\ 833H^1(\QQ_q,T'_{\qq}(k/2))@>\pi 834>>H^1(\QQ_q,T'_{\qq}(k/2))@>>>H^1(\QQ_q,A'[\qq](k/2)). 835\end{CD}$$
836The vertical arrows are all inclusions and we know that the image
837of $h^1(D'(k/2))$ in $H^1(\QQ_q,T'_{\qq}(k/2))$ is exactly
838$H^1_f(\QQ_q,T'_{\qq}(k/2))$. The top right horizontal map is
839surjective since $h^2(D'(k/2))=0$.
840
841The class $\res_q(c)\in H^1(\QQ_q,A'[\qq](k/2))$ is in the image
842of $H^1_f(\QQ_q,T'_{\qq}(k/2))$, by construction, and therefore is
843in the image of $h^1(D'(k/2)/\qq D'(k/2))$. By the fullness and
844exactness of the Fontaine-Lafaille functor \cite{FL} (see Theorem
8454.3 of \cite{BK}), $D'(k/2)/\qq D'(k/2)$ is isomorphic to
846$D(k/2)/\qq D(k/2)$.
847
848It follows that the class $\res_q(c)\in H^1(\QQ_q,A[\qq](k/2))$ is
849in the image of $h^1(D(k/2)/\qq D(k/2))$ by the vertical map in
850the exact sequence analogous to the above. Since the map from
851$h^1(D(k/2))$ to $h^1(D(k/2)/\qq D(k/2))$ is surjective,
852$\res_q(c)$ lies in the image of $H^1_f(\QQ_q,T_{\qq}(k/2))$. From
853this it follows that $\res_q(\gamma)\in 854H^1_f(\QQ_q,A_{\qq}(k/2))$, as desired.
855\end{enumerate}
856\end{proof}
857
858Theorem~2.7 of \cite{AS} is concerned with verifying local
859conditions in the case $k=2$, where~$f$ and~$g$ are associated
860with abelian varieties~$A$ and~$B$. (Their theorem also applies to
861abelian varieties over number fields.) Our restriction outlawing
862congruences modulo $\qq$ with cusp forms of lower level is
863analogous to theirs forbidding~$q$ from dividing Tamagawa factors
864$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic
865curve with $\ord_l(j(A))<0$, consideration of a Tate
866parametrisation shows that if $q\mid c_{A,l}$, i.e., if
867$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
868at~$l$.)
869
870In this paper we have encountered two technical problems which we
871dealt with in quite similar ways:
872\begin{enumerate}
873\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
874\item proving local conditions at primes $p\mid N$, for an element
875of $\qq$-torsion.
876\end{enumerate}
877If our only interest was in testing the Bloch-Kato conjecture at
878$\qq$, we could have made these problems cancel out, as in Lemma
8798.11 of \cite{DFG}, by weakening the local conditions. However, we
880have chosen not to do so, since we are also interested in the
881Shafarevich-Tate group, and since the hypotheses we had to assume
882are not particularly strong. Note that, since $A[\qq]$ is
883irreducible, the $\qq$-part of $\Sha$ does not depend on the
884choice of $T_{\qq}$.
885
886\section{Examples and Experiments}
887\label{sec:examples} This section contains tables and numerical
888examples that illustrate the main themes of this paper.  In
889Section~\ref{sec:vistable}, we explain Table~\ref{tab:newforms},
890which contains~$16$ examples of pairs $f,g$ such that the strong
891Beilinson-Bloch conjecture and Theorem~\ref{local} together imply
892the existence of nontrivial elements of the Shafarevich-Tate group
893of the motive attached to~$f$. Section~\ref{sec:howdone} outlines
894the higher-weight modular symbol computations that were used in
895making Table~\ref{tab:newforms}. Section~\ref{sec:invis} discusses
896Table~\ref{tab:invisforms}, which summarizes the results of an
897extensive computation of conjectural orders of Shafarevich-Tate
898groups for modular motives of low level and weight.
899Section~\ref{sec:other_ex} gives specific examples in which
900various hypotheses fail. Note that in \S 7 modular symbol'' has
901a different meaning from in \S 5, being related to homology rather
902than cohomology. For precise definitions see \cite{SV}.
903
904\subsection{Visible $\Sha$ Table~\ref{tab:newforms}}\label{sec:vistable}
905\begin{table}
906\caption{\label{tab:newforms}Visible $\Sha$}\vspace{-3ex}
907
908$$909\begin{array}{|c|c|c|c|c|}\hline 910 g & \deg(g) & f & \deg(f) & q\text{'}s \\\hline 911\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\ 912\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\ 913\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\ 914\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\ 915\vspace{-2ex} & & & & \\ 916\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\ 917\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\ 918\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\ 919\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\ 920\vspace{-2ex} & & & & \\ 921\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\ 922\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\ 923\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\ 924\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\ 925\vspace{-2ex} & & & & \\ 926\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\ 927\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\ 928\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\ 929\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\ 930\hline 931\end{array} 932$$
933\end{table}
934
935
936Table~\ref{tab:newforms} on page~\pageref{tab:newforms} lists
937sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
938along with at least one prime~$q$ such that there is a prime
939$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
940$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
941The notation is as follows.
942The first column contains a label whose structure is
943\begin{center}
944{\bf [Level]k[Weight][GaloisOrbit]}
945\end{center}
946This label determines a newform $g=\sum a_n q^n$, up to Galois
947conjugacy. For example, \nf{127k4C} denotes a newform in the third
948Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
949orbits are ordered first by the degree of $\QQ(\ldots, a_n, 950\ldots)$, then by the sequence of absolute values $|\mbox{\rm 951Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
952being first in the event that the two absolute values are equal,
953and the first Galois orbit is denoted {\bf A}, the second {\bf B},
954and so on. The second column contains the degree of the field
955$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
956contain~$f$ and its degree, respectively. The fifth column
957contains at least one prime~$q$ such that there is a prime
958$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
959hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
960satisfied for~$f$,~$g$, and~$\qq$.
961
962For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
963prime $q$ appears in the $q$-column, meaning $q^2$ divides the
964order of the group $S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp})$, defined
965at the end of 7.3 below.
966
967
968We describe the first line of Table~\ref{tab:newforms}
969in more detail.  See the next section for further details
970on how the computations were performed.
971
972Using modular symbols, we find that there is a newform
973$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots 974\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
975the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
976also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
977coefficients generate a number field~$K$ of degree~$17$, and by
978computing the image of the modular symbol $XY\{0,\infty\}$ under
979the period mapping, we find that $L(f,2)\neq 0$.  The newforms~$f$
980and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
981characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
982both equal to
983$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 984+ \cdots\in \FF_{43}[[q]].$$
985
986There is no form in the Eisenstein subspaces of
987$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
988$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
989$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
990prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
991level~$1$ form of weight~$4$. Thus we have checked the hypotheses
992of Theorem~\ref{local}, so if $r$ is the dimension of
993$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
994$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
995
996Recall that since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that
997$r\geq 2$. Then, since $L(f,k/2)\neq 0$, we expect that the
998$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to
999the $\qq$-torsion subgroup of $\Sha$. Admitting these assumptions,
1000we have constructed the $\qq$-torsion in $\Sha$ predicted by the
1001Bloch-Kato conjecture.
1002
1003For particular examples of elliptic curves one can often find and
1004write down rational points predicted by the Birch and
1005Swinnerton-Dyer conjecture. It would be nice if likewise one could
1006explicitly produce algebraic cycles predicted by the
1007Beilinson-Bloch conjecture in the above examples. Since
1008$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
10090.3.2 of \cite{Z}), so ought to be trivial in
1010$\CH_0^{k/2}(M_g)\otimes\QQ$.
1011
1012\subsection{How the computation was performed}\label{sec:howdone}
1013We give a brief summary of how the computation was performed.  The
1014algorithms that we used were implemented by the second author, and
1015most are a standard part of MAGMA (see \cite{magma}).
1016
1017Let~$g$,~$f$, and~$q$ be some data from a line of
1018Table~\ref{tab:newforms} and let~$N$ denote the level of~$g$. We
1019verified the existence of a congruence modulo~$q$, that
1020$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq 10210$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
1022not arise from any $S_k(\Gamma_0(N/p))$, as follows:
1023
1024To prove there is a congruence, we showed that the corresponding
1025{\em integral} spaces of modular symbols satisfy an appropriate
1026congruence, which forces the existence of a congruence on the
1027level of Fourier expansions.  We showed that $\rho_{g,\qq}$ is
1028irreducible by computing a set that contains all possible residue
1029characteristics of congruences between~$g$ and any Eisenstein
1030series of level dividing~$N$, where by congruence, we mean a
1031congruence for all Fourier coefficients of index~$n$ with
1032$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
1033form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
1034listing a basis of such~$h$ and finding the possible congruences,
1035where again we disregard the Fourier coefficients of index not
1036coprime to~$N$.
1037
1038To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
1039modular symbol ${\mathbf e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$
1040under a map with the same kernel as the period mapping, and found that the
1041image was~$0$.  The period mapping sends the modular
1042symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
1043so that ${\mathbf e}$ maps to~$0$ implies that
1044$L(g,\frac{k}{2})=0$. In a similar way, we verified that
1045$L(f,\frac{k}{2})\neq 0$.  Next, we checked that $W_N(g) 1046=(-1)^{k/2} g$ which, because of the functional equation, implies
1047that $L'(g,\frac{k}{2})=0$. Table~\ref{tab:newforms} is of
1048independent interest because it includes examples of modular forms
1049of even weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
1050the functional equation.  We found no such examples of weights
1051$\geq 8$.
1052
1053\subsection{Conjecturally nontrivial $\Sha$}\label{sec:invis}
1054In this section we apply some of the results of
1055Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders
1056of Shafarevich-Tate groups of many modular motives. The results of
1057this section suggest that~$\Sha$ of a modular motive is usually not
1058visible at level~$N$'', i.e., explained by congruences at level~$N$,
1059which agrees with the observations of \cite{CM} and \cite{AS}. For
1060example, when $k>6$ we find many examples of conjecturally
1061nontrivial~$\Sha$ but no examples of nontrivial visible~$\Sha$.
1062
1063For any newform~$f$, let $L(M_f/\QQ,s) = \prod_{i=1}^{d} 1064L(f^{(i)},s)$ where $f^{(i)}$ runs over the
1065$\Gal(\Qbar/\QQ)$-conjugates of~$f$. Let~$T$ be the complex torus
1066$\CC^d/(2\pi i)^{k/2}\mathcal{L}$, where the lattice $\mathcal{L}$
1067is defined by integrating integral cuspidal modular symbols (for
1068$\Gamma_0(N)$) against the conjugates of~$f$.  Let
1069$\Omega_{M_f/\QQ}$ denote the volume of the $(-1)^{(k/2)-1}$
1070eigenspace $T^{\pm}=\{z \in T : \overline{z}=(-1)^{(k/2)-1}z\}$
1071for complex conjugation on~$T$.
1072
1073
1074{\begin{table}
1075\vspace{-2ex}
1076\caption{\label{tab:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}
1077\vspace{-4ex}
1078
1079$$1080\begin{array}{|c|c|c|c|}\hline 1081f & \deg(f) & B\,\, (\text{\Sha bound})& \text{all odd congruence primes}\\\hline 1082\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\ 1083\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\ 1084\nf{263k4B} & 39 & 41^{2} & 263 \\ 1085\nf{269k4C} & 39 & 23^{2} & 269 \\ 1086\nf{271k4B} & 39 & 29^{2} & 271 \\ 1087\nf{281k4B} & 40 & 29^{2} & 281 \\ 1088\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\ 1089\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\ 1090%\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\ 1091% 319k4C removed since Lemma not satisfied. 1092\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\ 1093\hline 1094\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\ 1095\nf{101k6B} & 24 & 17^{2} & 101 \\ 1096\nf{103k6B} & 24 & 23^{2} & 103 \\ 1097\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\ 1098\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\ 1099\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\ 1100\nf{157k6B} & 34 & 251^{2} & 157 \\ 1101\nf{167k6B} & 40 & 41^{2} & 167 \\ 1102\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\ 1103\nf{173k6B} & 39 & 71^{2} & 173 \\ 1104\nf{181k6B} & 40 & 107^{2} & 181 \\ 1105\nf{191k6B} & 46 & 85091^{2} & 191 \\ 1106\nf{193k6B} & 41 & 31^{2} & 193 \\ 1107\nf{199k6B} & 46 & 200329^2 & 199 \\ 1108\hline 1109\nf{47k8B} & 16 & 19^{2} & 47 \\ 1110\nf{59k8B} & 20 & 29^{2} & 59 \\ 1111\nf{67k8B} & 20 & 29^{2} & 67 \\ 1112\nf{71k8B} & 24 & 379^{2} & 71 \\ 1113\nf{73k8B} & 22 & 197^{2} & 73 \\ 1114\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\ 1115\nf{79k8B} & 25 & 307^{2} & 79 \\ 1116\nf{83k8B} & 27 & 1019^{2} & 83 \\ 1117\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\ 1118\nf{89k8B} & 29 & 44491^{2} & 89 \\ 1119\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\ 1120\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\ 1121\nf{103k8B} & 32 & 75367^{2} & 103 \\ 1122\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\ 1123\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\ 1124\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\ 1125\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\ 1126\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\ 1127\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\ 1128\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\ 1129\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\ 1130 \hline 1131\end{array} 1132$$
1133\end{table}
1134\begin{table}
1135$$1136\begin{array}{|c|c|c|c|}\hline 1137f & \deg(f) & B\,\, (\text{\Sha bound})& \text{all odd congruence primes}\\\hline 1138\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\ 1139\nf{121k8G} & 12 & 13^{2} & 3, 11 \\ 1140\nf{121k8H} & 12 & 19^{2} & 5, 11 \\ 1141\nf{125k8D} & 16 & 179^{2} & 5 \\ 1142\nf{127k8B} & 39 & 59^{2} & 127 \\ 1143\nf{128k8F} & 4 & 11^{2} & 1 \\ 1144\nf{131k8B} & 43 & 241^{2} \!\cdot\! 817838201^{2}&131\\ 1145\nf{134k8C} & 11 & 61^{2} & 11, 17, 41, 67, 71, 421, 2356138931854759 \\ 1146\nf{137k8B} & 42 & 71^{2} \!\cdot\! 749093^{2} & 137 \\ 1147\nf{139k8B} & 43 & 47^{2} \!\cdot\! 89^{2} \!\cdot\! 1021^{2} & 139 \\ 1148\nf{141k8C} & 14 & 13^{2} & 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\ 1149\nf{142k8B} & 10 & 11^{2} & 3, 53, 71, 56377, 1965431024315921873 \\ 1150\nf{143k8C} & 19 & 307^{2} & 3, 11, 13, 89, 199, 409, 178397, 1151639259, 17440535 115297287 \\ 1153\nf{143k8D} & 21 & 109^{2} & 3, 7, 11, 13, 61, 79, 103, 173, 241, 1154769, 36583 1155\\ 1156\nf{145k8C} & 17 & 29587^{2} & 5, 11, 29, 107, 251623, 393577, 1157518737, 9837145 1158699 \\ 1159\nf{146k8C} & 12 & 3691^{2} & 11, 73, 269, 503, 1673540153, 11374452082219 \\ 1160\nf{148k8B} & 11 & 19^{2} & 3, 37 \\ 1161\nf{149k8B} & 47 & 11^{4} \!\cdot\! 40996789^{2} & 149\\ 1162 1163\hline 1164 1165\nf{43k10B} & 17 & 449^{2} & 43 \\ 1166\nf{47k10B} & 20 & 2213^{2} & 47 \\ 1167\nf{53k10B} & 21 & 673^{2} & 53 \\ 1168\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\ 1169\nf{59k10B} & 25 & 37^{2} & 59 \\ 1170\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\ 1171\nf{64k10K} & 2 & 19^{2} & 3 \\ 1172\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\ 1173\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\ 1174\nf{71k10B} & 30 & 1103^{2} & 71 \\ 1175 1176\hline 1177\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\ 1178\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\ 1179\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\ 1180\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\ 1181\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\ 1182\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\ 1183\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\ 1184\nf{49k12H} & 12 & 271^{2} & 7 \\ 1185\hline 1186\end{array} 1187$$
1188\end{table}
1189
1190\begin{lem}\label{lem:lrat}
1191If $p\nmid Nk!$ is such that $f$ is not congruent to any of its
1192Galois conjugates modulo a prime dividing $p$ then the $p$-parts
1193of
1194$$1195 \frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}\qquad\text{and}\qquad 1196 \Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right) 1197$$
1198are equal, where $\vol_\infty$ is as in Section~\ref{sec:bkconj}.
1199\end{lem}
1200
1201\begin{proof} (Sketch.) Let $H$ be the $\ZZ$-module of all
1202integral cuspidal modular symbols for $\Gamma_0(N)$. Let $I$ be
1203the image of $H$ under the projection into $H\otimes\QQ$
1204corresponding to $f$ and its Galois conjugates. $I$ is not
1205necessarily contained in $H$ since we will have inverted the
1206residue characteristics of any primes of congruence between $f$
1207and cuspforms for $\Gamma_0(N)$ which are not Galois conjugate to
1208$f$.
1209
1210Now $\mathcal{L}$ is (up to divisors of $Nk!$) the lattice
1211obtained by pairing the cohomology modular symbols
1212$\Phi_{f^{(i)}}^{\pm}$ (as in \S 5) with the homology modular
1213symbols in $H$, or equivalently in $I$. For $1\leq i\leq d$ let
1214$I_i$ be the $O_E$-module generated by the image of the projection
1215of $I$ into $I\otimes E$ corresponding to $f^{(i)}$. The finite
1216index of $I\otimes O_E$ in $\oplus_{i=1}^d I_i$ is divisible only
1217by primes of congruence between $f$ and its Galois conjugates. Up
1218to these primes, $\Omega_{M_f/\QQ}/(2\pi i)^{((k/2)-1)d}$ is then
1219a product of the $d$ complex numbers obtained by pairing
1220$\Phi_{f^{(i)}}^{\pm}$ with $I_i$, for $1\leq i\leq d$. Bearing in
1221mind the last line of \S 3, and ignoring divisors of $\aaa^{\pm}$,
1222which are clearly of no importance, we see that these complex
1223numbers are the $\Omega^{\pm}_{f^{(i)}}$, up to divisors of $Nk!$.
1224We have then a factorisation of the left hand side which shows it
1225to be equal to the right hand side, to the extent claimed by the
1226lemma.
1227\end{proof}
1228
1229\begin{remar}
1230The newform $f=\nf{319k4C}$ is congruent to one of its Galois conjugates
1231modulo~$17$ and $17\mid \frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}$ so the lemma
1232and our computations
1233say nothing about whether or not $17$ divides
1234$\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right)$.
1235\end{remar}
1236
1237
1238Let~$\mathcal{S}$ be the set of newforms with~level $N$ and
1239weight~$k$ satisfying either $k=4$ and $N\leq 321$, or $k=6$ and
1240$N\leq 199$, or $k=8$ and $N\leq 149$, or $k=10$ and $N\leq 72$,
1241or $k=12$ and $N\leq 49$.  Given $f\in \mathcal{S}$, let~$B$ be
1242defined as follows:
1243\begin{enumerate}
1244\item Let $L_1$ be the numerator of the
1245      rational number $L(M_f/\QQ,k/2)/\Omega_{M_f/\QQ}$.
1246      If $L_1=0$ let $B=1$ and terminate.
1247\item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$.
1248\item Let $L_3$ be the part of $L_2$ that is coprime to
1249      $p+1$ for every prime~$p$ such that $p^2\mid N$.
1250\item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic
1251      of any prime of
1252      congruence between~$f$ and a form of weight~$k$ and
1253      lower level. (By congruence here, we mean a congruence for coefficients
1254      $a_n$ with $n$ coprime to the level of~$f$.)
1255\item Let $L_5$ be the part of $L_4$ coprime to the residue characteristic
1256      of any prime of congruence
1257      between~$f$ and an Eisenstein series.  (This eliminates
1258      residue characteristics of reducible representations.)
1259\item Let $B$ be the part of $L_5$ coprime to the residue characteristic
1260      of any prime of congruence between $f$ and any one of its Galois
1261      conjugates.
1262\end{enumerate}
1263Proposition~\ref{sha} and Lemma~\ref{lem:lrat} imply that if
1264$\ord_p(B) 1265> 0$ then, according
1266to the Bloch-Kato conjecture, $\ord_p(\#\Sha)=\ord_p(B) > 0$. We
1267have left the congruence primes in $B$ in the starred examples
1268since the squares are still suggestive.
1269
1270We computed~$B$ for every newform in~$\mathcal{S}$.  There are
1271many examples in which $L_3$ is large, but~$B$ is not, and this is
1272because of Tamagawa factors. For example, {\bf 39k4C} has
1273$L_3=19$, but $B=1$ because of a $19$-congruence with a form of
1274level~$13$; in this case we must have $19\mid c_{3}(2)$, where
1275$c_{3}(2)$ is as in Section~\ref{sec:bkconj}.  See
1276Section~\ref{sec:other_ex} for more details.  Also note that in
1277every example~$B$ is a perfect square, which is as predicted by
1278the existence of Flach's generalised Cassels-Tate pairing
1279\cite{Fl2}. (Note that for $\lambda\mid l$, a non-congruence prime
1280for $f$, the lattice $T_{\lambda}$ is self-dual, so the pairing
1281shows that the order of the $\lambda$-part of $\Sha$, if finite,
1282is a square.) That our computed value of~$B$ should be a square is
1283not {\it a priori} obvious.
1284
1286of rings of integers, so our definition of~$B$ is overly conservative.
1287For example,~$5$ occurs in row~$2$ of Table~\ref{tab:newforms} but not
1288in Table~\ref{tab:invisforms}, because \nf{159k4E} is Eisenstein at
1289some prime above~$5$, but the prime of congruences of
1290characteristic~$5$ between \nf{159k4B} and \nf{159k4E} is not
1291Eisenstein.
1292
1293
1294The newforms for which $B>1$ are given in
1295Table~\ref{tab:invisforms}. The second column of the table records
1296the degree of the field generated by the Fourier coefficients
1297of~$f$.  The third contains~$B$.  Let~$W$ be the intersection of
1298the span of all conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and
1299$W^{\perp}$ the Petersson orthogonal complement of~$W$ in
1300$S_k(\Gamma_0(N),\ZZ)$. The fourth column contains the odd
1301prime divisors of $\#(S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp}))$, which
1302are exactly the possible primes of congruence for~$f$. We place a
1303$*$ next to the four entries of Table~\ref{tab:invisforms} that
1304also occur in Table~\ref{tab:newforms}.
1305
1306\subsection{Examples in which hypotheses fail}\label{sec:other_ex}
1307We have some other examples where forms of
1308different levels are congruent.
1309However, Remark~\ref{sign} does not
1310apply, so that one of the forms could have an odd functional
1311equation, and the other could have an even functional equation.
1312For instance, we have a $19$-congruence between the
1313newforms $g=\nf{13k4A}$ and $f=\nf{39k4C}$ of Fourier
1314coefficients coprime to $39$.
1315Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$
1316has {\it odd} functional equation.
1317Here~$f$ fails the condition about not being congruent
1318to a form of lower level, so in Lemma~\ref{local1} it is possible that
1319$\ord_{\qq}(c_{3}(2))>0$. In fact this does happen. Because
1320$V'_{\qq}$ (attached to~$g$ of level $13$) is unramified at $p=3$,
1321$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
1322two-dimensional. As in (2) of the proof of Theorem~\ref{local},
1323one of the eigenvalues of $\Frob_p^{-1}$ acting on this
1324two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
1325$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
1326$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
1327$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
1328$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
1329Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
1330our example here with $p=3$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
1331nontrivial when $w_p=-1$, so (2) of the proof of Theorem~\ref{local}
1332does not work. This is just as well, since had it
1333worked we would have expected
1334$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
1335shows not to be the case.
1336
1337In the following example, the divisibility between the levels is
1338the other way round.  There is a $7$-congruence between
1339$g=\nf{122k6A}$ and $f=\nf{61k6B}$, both $L$-functions have even
1340functional equation, and $L(g,3)=0$. In the proof of
1341Theorem~\ref{local}, there is a problem with the local condition
1342at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to
1343$H^1(I_2,A'_{\qq}(3))$ is not necessarily injective, but its
1344kernel is at most one dimensional, so we still get the
1345$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
1346$\FF_{\qq}$-rank at least~$1$ (assuming $r\geq 2$), and thus get
1347elements of $\Sha$ for \nf{61k6B} (assuming all along the strong
1348Beilinson-Bloch conjecture). In particular, these elements of
1349$\Sha$ are {\it invisible} at level 61. When the levels are
1350different we are no longer able to apply Theorem 2.1 of \cite{FJ}.
1351However, we still have the congruences of integral modular symbols
1352required to make the proof of Proposition \ref{div} go through.
1353Indeed, as noted above, the congruences of modular forms were
1354found by producing congruences of modular symbols. Despite these
1355congruences of modular symbols, Remark~\ref{sign} does not apply,
1356since there is no reason to suppose that $w_N=w_{N'}$, where $N$
1357and $N'$ are the distinct levels.
1358
1359Finally, there are two examples where we have a form $g$ with even
1360functional equation such that $L(g,k/2)=0$, and a congruent form
1361$f$ which has odd functional equation; these are a 23-congruence
1362between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
1363between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
1364$\ord_{s=2}L(f,s)=1$, it ought to be the case that
1365$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
1366similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
1367the appropriate modification of Theorem \ref{local} (with strong
1368Beilinson-Bloch conjecture) does not necessarily provide us with
1369nontrivial $\qq$-torsion in $\Sha$. It only tells us that the
1370$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ has
1371$\FF_{\qq}$-rank at least $1$. It could all be in the image of
1372$H^1_f(\QQ,V_{\qq}(2))$. $\Sha$ appears in the conjectural formula
1373for the first derivative of the complex $L$ function, evaluated at
1374$s=k/2$, but in combination with a regulator that we have no way
1375of calculating.
1376
1377Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
1378associated with $f$ and $g$ by the construction of Mazur, Tate and
1379Teitelbaum \cite{MTT}, each divided by a suitable canonical
1380period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
1381quite clear what to make of this. This divisibility may be proved
1382as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
1383times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
1384normalised) are congruent $\bmod{\,\qq}$, as a result of the
1385congruence between the modular symbols out of which they are
1386constructed. Integrating an appropriate function against these
1387measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
1388to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
1389since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
1390where the signs in the functional equations of $L(g,s)$ and
1391$L_q(g,s)$ are the same, positive in this instance. (According to
1392the proposition in Section 18 of \cite{MTT}, the signs differ
1393precisely when $L_q(g,s)$ has a trivial zero'' at $s=k/2$.)
1394
1395We also found some examples for which the conditions of
1396Theorem~\ref{local} were not met.  For example, we have a
1397$7$-congruence between \nf{639k4B} and \nf{639k4H}, but
1398$w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a
1399similar problem with a $7$-congruence between \nf{260k6A} and
1400\nf{260k6E} --- here $w_{13}=1$ so that $13\equiv 1401-w_{13}\pmod{7}$. According to Propositions \ref{div} and
1402\ref{sha}, Bloch-Kato still predicts that the $\qq$-part of $\Sha$
1403is non-trivial in these examples. Finally, there is a
1404$5$-congruence between \nf{116k6A} and \nf{116k6D}, but here the
1405prime~$5$ is less than the weight~$6$ so Propositions \ref{div}
1406and \ref{sha} do not even apply.
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1515