CoCalc Shared Fileswww / papers / motive_visibility / dsw_4.tex
Author: William A. Stein
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77% ---- SHA ----
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90\newcommand{\CH}{\mathrm {CH}}
91
92\begin{document}
93\title{Constructing Elements in
94Shafarevich-Tate Groups of Modular Motive}
95\author{Neil Dummigan}
96\author{William Stein}
97\author{Mark Watkins}
98\date{May 29th, 2002}
99\subjclass{11F33, 11F67, 11G40.}
100
101\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
102Shafarevich-Tate group.}
103
104\address{University of Sheffield\\ Department of Pure
105Mathematics\\
106Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
107U.K.}
109One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
111University Park\\State College, PA 16802\\ U.S.A.}
112
113\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
114\email{watkins@math.psu.edu}
115
116\maketitle {\bf Not for distribution}
117\section{Introduction}
118Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
119be the associated $L$-function. The conjecture of Birch and
120Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$
121at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
122also gives an interpretation of the leading term in the Taylor
123expansion in terms of various quantities, including the order of
124the Shafarevich-Tate group of $E$.
125
126Cremona and Mazur [2000] look, among all strong Weil elliptic
127curves over $\QQ$ of conductor $N\leq 5500$, at those with
128non-trivial Shafarevich-Tate group (according to the Birch and
129Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
130group has predicted elements of prime order $m$. In most cases
131they find another elliptic curve, often of the same conductor,
132whose $m$-torsion is Galois-isomorphic to that of the first one,
133and which has positive rank. The rational points on the second elliptic
134curve produce classes in the common $H^1(\QQ,E[m])$. They show
135\cite{CM2} that these lie in the Shafarevich-Tate group of the
136first curve, so rational points on one curve explain elements of
137the Shafarevich-Tate group of the other curve.
138
139The Bloch-Kato conjecture \cite{BK} is the generalisation to
140arbitrary motives of the leading term part of the Birch and
141Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
142\cite{B} generalises the part about the order of vanishing at the
143central point, identifying it with the rank of a certain Chow
144group.
145
146The present work may be considered as a partial generalisation of
147the work of Cremona and Mazur, from elliptic curves over $\QQ$
148(which are associated to modular forms of weight $2$) to the
149motives attached to modular forms of higher weight. (See \cite{AS}
150for a different generalisation, to modular abelian varieties of
151higher dimension.) It may also be regarded as doing, for
152congruences between modular forms of equal weight, what \cite{Du2}
153did for congruences between modular forms of different weights.
154
155We consider the situation where two newforms~$f$ and~$g$, both of
156weight $k>2$ and level~$N$, are congruent modulo a maximal ideal
157$\qq$ of odd residue characteristic, and $L(g,k/2)=0$ but
158$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish
159to order at least $2$ at $s=k/2$. We are able to find sixteen
160examples (all with $k=4$ and $k=6$), and in each $\qq$
161divides the numerator of the algebraic number
162$L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$ is a certain
163canonical period. In fact, we show how this divisibility may be
164deduced from the vanishing of $L(g,k/2)$ using recent work of
165Vatsal \cite{V}. The point is the congruence between$f$ and~$g$
166leads to a congruence between suitable algebraic parts'' of the
167special values $L(f,k/2)$ and $L(g,k/2)$. If one vanishes then the
168other is divisible by $\qq$. Under certain hypotheses, the
169Bloch-Kato conjecture then implies that the Shafarevich-Tate group
170attached to~$f$ has non-zero $\qq$-torsion. Under certain
171hypotheses and assumptions, the most substantial of which is the
172Beilinson-Bloch conjecture relating the vanishing of $L(g,k/2)$ to
173the existence of algebraic cycles, we are able to construct the
174predicted elements of~$\Sha$, using the Galois-theoretic
175interpretation of the congruences to transfer elements from a
176Selmer group for~$g$ to a Selmer group for~$f$. In proving the
177local conditions at primes dividing the level, and also in
178examining the local Tamagawa factors at these primes, we make use
179of a higher weight level-lowering result due to Jordan and Livn\'e
180\cite{JL}.
181
182One might say that algebraic cycles for one motive explain
183elements of~$\Sha$ for the other. A main point of \cite{CM} was to
184observe the frequency with which those elements of~$\Sha$
185predicted to exist for one elliptic curve may be explained by
186finding a congruence with another elliptic curve containing points
187of infinite order. One shortcoming of our work, compared to the
188elliptic curve case, is that, due to difficulties with local
189factors in the Bloch-Kato conjecture, we are unable to compute the
190exact order of~$\Sha$ predicted by the Bloch-Kato conjecture. We
192congruence. However, Vatsal's work allows us to explain how the
193vanishing of one $L$-function leads, via the congruence, to the
194divisibility by~$\qq$ of (an algebraic part of) another,
195independent of observations of computational data. The
196computational data does however show that there exist examples to
197which our results apply. Moreover, it displays factors of $\qq^2$,
198whose existence we do not prove theoretically, but which are
199predicted by Bloch-Kato.
200
201\section{Motives and Galois representations}
202Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
203$\Gamma_0(N)$, with coefficients in an algebraic number field~$E$,
204which is necessarily totally real. A theorem of Deligne \cite{De1}
205implies the existence, for each (finite) prime~$\lambda$ of~$E$,
206of a two-dimensional vector space $V_{\lambda}$ over
207$E_{\lambda}$, and a continuous representation
208$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
209such that
210\begin{enumerate}
211\item $\rho_{\lambda}$ is unramified at $p$ for all primes $p$ not dividing
212$lN$ (where $\lambda \mid l$);
213\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
214characteristic polynomial of $\Frob_p^{-1}$ acting on
215$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
216\end{enumerate}
217
218Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
219the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
220There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
221both $2$-dimensional $E$-vector spaces. For details of the
222construction see \cite{Sc}. The de Rham realisation has a Hodge
223filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset 224F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
225cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
226cohomology. There are natural isomorphisms $V_B\otimes 227E_{\lambda}\simeq V_{\lambda}$. We may choose a
228$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
229each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
230Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
231There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
232which amounts to multiplying the action of $\Frob_p$ by $p^j$.
233
234Following \cite{BK} (Section 3), for $p\neq l$ (including
235$p=\infty$) let
236$$237H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow 238H^1(I_p,V_{\lambda}(j))). 239$$
240The subscript~$f$ stands for finite
241part'', $D_p$ is a decomposition subgroup at a prime above~$p$,
242$I_p$ is the inertia subgroup, and the cohomology is for
243continuous cocycles and coboundaries. For $p=l$ let
244$$245H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow 246H^1(D_l,V_{\lambda}(j)\otimes B_{\cris})) 247$$
248(see Section 1 of
249\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
250$B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
251elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
252in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes~$p$.
253
254There is a natural exact sequence
255$$256 \begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}. 257$$
258Let
259$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
260Define the $\lambda$-Selmer group
261$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
262$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
263$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes~$p$. Note that the
264condition at $p=\infty$ is superfluous unless $l=2$. Define the
265Shafarevich-Tate group
266$$267 \Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/ 268 \pi_*H^1_f(\QQ,V_{\lambda}(j)). 269$$
270Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any
271prime ideal~$\lambda$ is the length of the $\lambda$-component of
272$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,
273and write~$\Sha$ for~$\Sha(j)$.
274
275Define the group of global torsion points
276$$277 \Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)). 278$$
279This is analogous to the group of rational torsion points on an
280elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in
281which the exponent of any prime ideal~$\lambda$ is the length of
282the $\lambda$-component of $\Gamma_{\QQ}$.
283
284\section{Canonical periods}
285We assume from now on for convenience that $N\geq 3$. We need to
286choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
287and deRham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
288in a way such that $T_B\otimes_{O_E}O_E[1/Nk!]$ and
289$T_{\dR}\otimes_{O_E}O_E[1/Nk!]$ agree with the
290$O_E[1/Nk!]$-lattices $\mathfrak{M}_{f,B}$ and
291$\mathfrak{M}_{f,\dR}$ defined in \cite{DFG}. (See especially
292Sections 2.2 and 5.4 of \cite{DFG}.)
293
294For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
295module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
296$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes 297E_{\lambda}\simeq V_{\lambda}$. Then for $\lambda\nmid Nk!$, the
298$O_{\lambda}$ module $T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable,
299since it comes from $\ell$-adic cohomology with $O_{\lambda}$
300coefficients. We may assume that $T_{\lambda}$ is
301$\Gal(\Qbar/\QQ)$-stable for all finite $\lambda$, by adjusting
302$T_B$ locally at primes $\lambda\mid Nk!$ if necessary.
303
304Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
305generalised elliptic curves with full level-$N$ structure. Let
306$\mathfrak{E}$ be the universal generalised elliptic curve over
307$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product
308of $\mathfrak{E}$ over $M(N)$. Realising $M(N)(\CC)$ as the
309disjoint union of $\phi(N)$ copies of the quotient
310$\Gamma(N)\backslash\mathfrak{H}^*$, and letting $\tau$ be a
311variable on $\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is
312isomorphic to the elliptic curve with period lattice generated by
313$1$ and $\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a
314variable on the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the
315fibre product. Then $2\pi i f(\tau)\,d\tau\wedge 316dz_1\wedge\ldots\wedge dz_{k-2}$ is a well-defined differential
317form on (a desingularisation of) $\mathfrak{E}^{k-2}$ and
318naturally represents a generating element of $F^{k-1}T_{\dR}$. (At
319least, we can make our choices locally at primes dividing $Nk!$ so
320that this is the case.) We shall call this element $e(f)$.
321
322Under the deRham isomorphism between $V_{\dR}\otimes\CC$ and
323$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
324a natural action of complex conjugation on $V_B$, breaking it up
325into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
326Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
327to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
328$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
329are rank one $O_E$-modules, but not necessarily free, since the
330class number of $O_E$ may be greater than one. Choose non-zero
331elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
332the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
333$\Omega_f^{\pm}$ by $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$.
334
335\section{The Bloch-Kato conjecture}\label{sec:bkconj}
336Let $L(f,s)$ be the $L$-function attached to~$f$. For
337$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with
338Euler product
339$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but
340there is an analytic continuation given by an integral, as
341described in the next section. Suppose that $L(f,k/2)\neq 0$. The
342Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the
343following equality of fractional ideals of $E$:
344$$345 {L(f,k/2)\over \vol_{\infty}}= 346 {\left(\prod_pc_p(k/2)\right)\#\Sha\over 347 \aaa^{\pm}(\#\Gamma_{\QQ})^2}. 348$$
349(Strictly speaking, the conjecture in \cite{BK}
350is only given for $E=\QQ$.) Here, $\pm$
351represents the parity of $(k/2)-1$, and $\vol_{\infty}$ is equal
352to $(2\pi i)^{k/2}$ multiplied by the determinant of the
353isomorphism $V_B^{\pm}\otimes\CC\simeq 354(V_{\dR}/F^{k/2})\otimes\CC$, calculated with respect to the
355lattices $O_E\delta_f^{\pm}$ and the image of $T_{\dR}$. For
356$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be
357\begin{align*}
358\length&\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-
359  \ord_{\lambda}(P_p(p^{-j}))\\
360=&\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).
361\end{align*}
362
363We omit the definition of $\ord_{\lambda}(c_p(j))$ for
364$\lambda\mid p$, which requires one to assume Fontaine's de Rham
365conjecture (\cite{Fo}, Appendix A6), and depends on the choices of
366$T_{\dR}$ and $T_B$, locally at~$\lambda$. (We shall mainly be
367concerned with the $q$-part of the Bloch-Kato conjecture, where
368$q$ is a prime of good reduction. For such primes, the de Rham
369conjecture follows from Theorem 5.6 of \cite{Fa1}.)
370
371\begin{lem}\label{vol}
372$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and
373$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.
374\end{lem}
375\begin{proof}
376$\vol_{\infty}$ is also equal to the determinant
377of the period map from $F^{k/2}V_{\dR}\otimes\CC$ to
378$V_B^{\pm}\otimes\CC$, with respect to lattices dual to those we
379used above in the definition of $\vol_{\infty}$ (c.f. the last
380paragraph of 1.7 of \cite{De2}). We are using here the natural
381pairings. Recall that the index of $O_E\delta_f^{\pm}$ in
382$T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then the proof is completed
383by noting that, locally away from primes dividing $Nk!$, the index
384of $T_{\dR}$ in its dual is equal to the index of $T_B$ in its
385dual, both being equal to the ideal denoted~$\eta$ in \cite{DFG2}.
386\end{proof}
387\begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer.
388Then the fractional ideal $c_p(j)$ is supported at most on
389divisors of~$p$.
390\end{lem}
391\begin{proof}
392As on p.~30 of \cite{Fl1}, for odd $l\neq p$,
393$\ord_{\lambda}(c_p(j))$ is the length of the finite
394$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
395where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
396trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
397torsion free.
398\end{proof}
399
400\begin{lem}\label{local1}
401Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
402is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
403$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
404that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not congruent
405modulo $\qq$ to any newform of weight~$k$, trivial character and
406level dividing $N/p$. Then for~$j$ any integer,
407$\ord_{\qq}(c_p(j))=0$.
408\end{lem}
409\begin{proof}
410It suffices to show that
411$$412 \dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)), 413$$
414since this ensures that
415$H^0(I_p,A_{\qq}(j))=V_{\qq}^{I_p}/T_{\qq}^{I_p}$, hence that
416$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}^{I_p}/T_{\qq}^{I_p})$. If
417the dimensions differ then, given that $f$ is not congruent modulo
418$\qq$ to a newform of level dividing $N/p$, Proposition 2.2 of
419\cite{L} shows that we are in the situation covered by one of the
420three cases in Proposition 2.3 of \cite{L}. Since $p\not\equiv 421-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is
422unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's
423result that $N$ is the prime-to-$q$ part of the conductor of
424$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses
425the condition $q>k$) implies the existence of a newform of weight
426$k$, trivial character and level dividing $N/p$, congruent to~$g$
427modulo $\qq$. This contradicts our hypotheses.
428\end{proof}
429
430\begin{remar}
431For an example of what can be done when~$f$ {\em is } congruent to
432a form of lower level, see the first example in Section~\ref{sec:other_ex}
433below.
434\end{remar}
435
436\begin{lem}\label{at q}
437If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then
438$\ord_{\qq}(c_q)=0$.
439\end{lem}
440\begin{proof}
441It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an
442application, at the end of Section~2.2, of the results of
443\cite{Fa1}) that $T_{\qq}$ is
444the $O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
445filtered module $T_{\dR}\otimes O_{\qq}$ by the functor they call
446$\mathbb{V}$. (This property is part of the definition of an
447$S$-integral premotivic structure given in Section~1.2 of
448\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
449of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
450in Theorem~4.1 of \cite{BK} follows from the first paragraph of
4512(h) of \cite{Fa1}.)
452\end{proof}
453
454\begin{lem}
455If $A[\lambda]$ is an
456irreducible representation of $\Gal(\Qbar/\QQ)$,
457then
458$\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0$.
459\end{lem}
460This follows trivially from the definition.
461
462Putting together the above lemmas we arrive at the following:
463\begin{prop}\label{sha}
464Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that $A[\qq]$
465is an irreducible representation of $\Gal(\Qbar/\QQ)$, where $\qq\mid q$.
466Assume the same hypotheses as in Lemma \ref{local1},
467for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$
468are as in the previous section. If
469$$470 \ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty})>0 471$$
472(with numerator non-zero) then the Bloch-Kato conjecture
473predicts that
474$$475 \ord_{\qq}(\#\Sha)>0. 476$$
477\end{prop}
478
479\section{Congruences of special values}
480Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
481weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
482large enough to contain all the coefficients $a_n$ and $b_n$.
483Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv 484g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Suppose
485that $q\nmid N\phi(N)k!$ It is easy to see that we may choose the
486$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
487$\ord_{\qq}(\aaa^{\pm})=0$, i.e. $\delta_f^{\pm}$ generates
488$T_B^{\pm}$ locally at $\qq$. Let us suppose that such a choice
490
491We shall now make two further assumptions:
492\begin{enumerate}
493\item $L(f,k/2)\neq 0$;
494\item $L(g,k/2)=0$.
495\end{enumerate}
496\begin{prop} \label{div}
497With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
498\end{prop}
499\begin{proof} This is based on some of the ideas used in Section 1 of
500\cite{V}.  Note the apparent typo in Theorem~1.13 of \cite{V},
501which presumably should refer to Condition 2''. Since
502$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
503$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm 5041=(-1)^{(k/2)-1}$. It is well-known, and easy to prove, that
505$$\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).$$
506Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
507$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
508where the integral is taken along the positive imaginary axis,
509then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
510Thus we are reduced
511to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
512
513Let $\mathcal{D}_0$ be the group of divisors of degree zero
514supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
515integer $r\geq 0$, let $P_r(R)$ be the additive group of
516homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
517groups have a natural action of $\Gamma_1(N)$. Let
518$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
519be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
520
521Via the isomorphism (8) in Section~1.5 of \cite{V},
522$\omega_f^{\pm}$ corresponds to an element $\Phi_f^{\pm}\in 523S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
524element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$.
525(See also Section~4.2 of \cite{St}.) Inverting $N\phi(N)$ takes
526into account the fact that we are now dealing with $X_1(N)$ rather
527that $M(N)$. Up to some small factorials which do not matter
528locally at $\qq$,
529$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv 530(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since
531$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
532$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv 533(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
534coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
535show is divisible by $\qq$.
536Similarly
537$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv 538(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of
539$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
540Therefore it would suffice to show that, for some $\mu\in O_E$,
541the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
542$\qq$ in $S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$. It suffices to show
543that, for some $\mu\in O_E$, the element
544$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
545considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
546non-constant coefficients. This would be the case if
547$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
548one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
549a consequence of Theorem 2.1(1) of \cite{FJ}.
550\end{proof}
551\begin{remar}
552By Proposition~\ref{sha} (assuming, for all $p\mid N$ the same
553hypotheses as in Lemma~\ref{local1}, together with
554$q\nmid\phi(N)$), the Bloch-Kato conjecture now predicts that
555$\ord_{\qq}(\#\Sha)>0$. The next section provides a conditional
556construction of the required elements of $\Sha$.
557\end{remar}
558\begin{remar}\label{sign}
559The signs in the functional equations of $L(f,s)$ and $L(g,s)$
560have to be equal, since they are determined by the action of the
561involution $W_N$ on the common subspace generated by the
562reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.
563Specifically, the sign is $(-1)^{k/2}w_N$, where $w_N$ is the
564common eigenvalue of $W_N$ acting on~$f$ and~$g$.
565\end{remar}
566This is analogous to the remark at the end of Section~3 of
567\cite{CM}, which shows that if $L(f,k/2)\neq 0$ but $L(g,k/2)=0$
568then $L(g,s)$ must vanish to order at least two, as in all the
569examples below. It is worth pointing out that there are no
570examples of $g$ of level one, and positive sign in the functional
571equation, such that $L(g,k/2)=0$, unless Maeda's conjecture (that
572all the normalised cuspidal eigenforms of weight~$k$ and level one
573are Galois conjugate) is false. See \cite{CF}.
574
575\section{Constructing elements of the Shafarevich-Tate group}
576Let~$f$ and~$g$ be as in the first paragraph of the previous
577section. For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
578$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
579$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
580is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
581the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
582irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
583
584Suppose that $L(g,k/2)=0$. If the sign in the functional equation
585is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
586\ref{sign}), this implies that the order of vanishing of $L(g,s)$
587at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
588conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
589is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational
590rational equivalence classes of null-homologous, codimension $k/2$
591algebraic cycles on the motive $M_g$. (This generalises the part
592of the Birch--Swinnerton-Dyer conjecture which says that for an
593elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at
594$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)
595
596Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
597to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
598subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
599If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
600get (assuming also the Beilinson-Bloch conjecture) a subspace of
601$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
602vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
603conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
604equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
605follow from the conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
606Sections~1 and~6.5 of \cite{Fo2}.
607
608Similarly, if $L(f,k/2)\neq 0$ then we expect that
609$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
610coincides with the $\qq$-part of $\Sha$.
611\begin{thm}\label{local}
612Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
613$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
614$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$ and
615that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a
616newform of weight~$k$, trivial character and level dividing $N/p$.
617Suppose that, for all primes $p\mid N$, $\,p\not\equiv 618-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here
619$w_p$ is the common eigenvalue of the Atkin-Lehner involution
620$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of
621$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
622\end{thm}
623
624\begin{proof}
625Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
626continuity and rescaling we may assume that it lies in
627$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq 628H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction (note that
629$T'_{\qq}(k/2)/\qq T'_{\qq}(k/2)\simeq A'[\qq](k/2)$) we get a
630non-zero class $c\in H^1(\QQ,A'[\qq](k/2))\simeq 631H^1(\QQ,A[\qq](k/2))$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$
632is trivial. It follows that $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
633$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
634we get a non-zero, $\qq$-torsion class $\gamma\in 635H^1(\QQ,A_{\qq}(k/2))$.
636
637Our aim is to show that $\res_p(\gamma)\in 638H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
639consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
640
641\begin{enumerate}
642\item {\bf $p\nmid qN$. }
643
644Consider the $I_p$-cohomology of the short exact sequence
645$$646 \begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD}, 647$$
648where~$\pi$ is multiplication by a uniformising element of
649$O_{\qq}$. Since in this case $A'_{\qq}(k/2)$ is unramified at
650$p$, $H^0(I_p, A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is
651$\qq$-divisible. Therefore $H^1(I_p,A'[\qq](k/2))$ (which,
652remember, is the same as $H^1(I_p,A[\qq](k/2))$) injects into
653$H^1(I_p,A'_{\qq}(k/2))$. It follows from the fact that $d\in 654H^1_f(\QQ,V'_{\qq}(k/2))$ that the image in
655$H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is zero, hence
656that the restriction of~$c$ (to $H^1(I_p,A'[\qq](k/2))\simeq 657H^1(I_p,A[\qq](k/2))$) is zero. Hence the restriction of $\gamma$
658to $H^1(I_p,A_{\qq}(k/2))$ is also zero. By line~3 of p.~125 of
659\cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
660contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
661to $H^1(I_p,A_{\qq}(k/2))$,  so we have shown that
662$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
663
664\item {\bf $p\mid N$. }
665
666First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
667It suffices to show that
668$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
669since this would imply that the natural map from
670$H^0(I_p,V'_{\qq}(k/2))$ to $H^0(I_p, A'_{\qq}(k/2))$ is
671surjective, but this may be done as in the proof of Lemma
672\ref{local1}. It follows as above that the image of $c\in 673H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is zero. Then
674$\res_p(c)$ comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by
675inflation-restriction. The order of this group is the same as the
676order of the group $H^0(\QQ_p,A[\qq](k/2))$, which we claim is
677trivial. By the work of Carayol \cite{Ca1}, $p\mid N$ implies that
678$V_{\qq}(k/2)$ is ramified at $p$, so $\dim 679H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim 680H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
681consider the case where this common dimension is $1$. The
682(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha 683p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication by
684$\alpha$ on the one-dimensional space $H^0(I_p,V_{\qq}(k/2))$. It
685follows from Theor\'eme A of \cite{Ca1} that this is the same as
686the Euler factor at $p$ of $L(f,s)$. By the work of Atkin and
687Lehner \cite{AL}, it then follows that $p^2\nmid N$ and
688$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
689$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
690$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$ as
691$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
692$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
693$\res_p(\gamma)=0$ and certainly lies in
694$H^1_f(\QQ_p,A_{\qq}(k/2))$.
695
696\item {\bf $p=q$. }
697
698Since $q\nmid N$ is a prime of good reduction for the motive
699$M_g$, $\,V'_{\qq}$ is a crystalline representation of
700$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
701$V'_{\qq}$ have the same dimension, where
702$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q} 703B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
704As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
705the $O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
706filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we
707may now prove, in the same manner as Proposition 9.2 of
708\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.
709\end{enumerate}
710\end{proof}
711
712Theorem~2.7 of \cite{AS} is concerned with verifying local
713conditions in the case $k=2$, where~$f$ and~$g$ are associated
714with abelian varieties~$A$ and~$B$. (Their theorem also applies to
715abelian varieties over number fields.) Our restriction outlawing
716congruences modulo $\qq$ with cusp forms of lower level is
717analogous to theirs forbidding~$q$ from dividing Tamagawa factors
718$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic
719curve with $\ord_l(j(A))<0$, consideration of a Tate
720parametrisation shows that if $q\mid c_{A,l}$, i.e., if
721$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
722at~$l$.)
723
724In this paper we have encountered two technical problems which we
725dealt with in quite similar ways:
726\begin{enumerate}
727\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
728\item proving local conditions at primes $p\mid N$, for an element
729of $\qq$-torsion.
730\end{enumerate}
731If our only interest was in testing the Bloch-Kato conjecture at
732$\qq$, we could have made these problems cancel out, as in Lemma
7338.11 of \cite{DFG}, by weakening the local conditions. However, we
734have chosen not to do so, since we are also interested in the
735Shafarevich-Tate group, and since the hypotheses we had to assume
736are not particularly strong.
737
738\section{Examples and Experiments}
739
740\subsection{Visible $\Sha$ Table~\ref{fig:newforms}}
741\begin{figure}
742\caption{\label{fig:newforms}Visible $\Sha$}
743$$744\begin{array}{|c|c|c|c|c|}\hline 745 g & \deg(g) & f & \deg(f) & q's \\\hline 746\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\ 747\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\ 748\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\ 749\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\ 750\vspace{-2ex} & & & & \\ 751\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\ 752\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\ 753\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\ 754\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\ 755\vspace{-2ex} & & & & \\ 756\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\ 757\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\ 758\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\ 759\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\ 760\vspace{-2ex} & & & & \\ 761\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\ 762\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\ 763\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\ 764\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\ 765\hline 766\end{array} 767$$
768\end{figure}
769
770
771Table~\ref{fig:newforms} on page~\pageref{fig:newforms} lists
772sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
773along with at least one prime~$q$ such that there is a prime
774$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
775$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
776The notation is as follows.
777The first column contains a label whose structure is
778\begin{center}
779{\bf [Level]k[Weight][GaloisOrbit]}
780\end{center}
781This label determines a newform $g=\sum a_n q^n$, up to Galois
782conjugacy. For example, \nf{127k4C} denotes a newform in the third
783Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
784orbits are ordered first by the degree of $\QQ(\ldots, a_n, 785\ldots)$, then by the sequence of absolute values $|\mbox{\rm 786Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
787being first in the event that the two absolute values are equal,
788and the first Galois orbit is denoted {\bf A}, the second {\bf B},
789and so on. The second column contains the degree of the field
790$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
791contain~$f$ and its degree, respectively. The fifth column
792contains at least one prime~$q$ such that there is a prime
793$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
794hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
795satisfied for~$f$,~$g$, and~$\qq$.
796
797
798We describe the first line of Table~\ref{fig:newforms}
799in more detail.  See the next section for further details
800on how the computations were performed.
801
802Using modular symbols, we find that there is a newform
803$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots 804\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
805the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
806also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
807coefficients generate a number field~$K$ of degree~$17$, and by
808computing the image of the modular symbol $XY\{0,\infty\}$ under
809the period mapping, we find that $L(f,2)\neq 0$.  The newforms~$f$
810and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
811characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
812both equal to
813$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 814+ \cdots\in \FF_{43}[[q]].$$
815
816There is no form in the Eisenstein subspaces of
817$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
818$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
819$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
820prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
821level~$1$ form of weight~$4$. Thus we have checked the hypotheses
822of Theorem~\ref{local}, so if $r$ is the dimension of
823$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
824$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
825
826Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,
827since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup
828of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion
829subgroup of $\Sha$. Admitting these assumptions, we have
830constructed the $\qq$-torsion in $\Sha$ predicted by the
831Bloch-Kato conjecture.
832
833For particular examples of elliptic curves one can often find and
834write down rational points predicted by the Birch and
835Swinnerton-Dyer conjecture. It would be nice if likewise one could
836explicitly produce algebraic cycles predicted by the
837Beilinson-Bloch conjecture in the above examples. Since
838$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
8390.3.2 of \cite{Z}), so ought to be trivial in
840$\CH_0^{k/2}(M_g)\otimes\QQ$.
841
842\subsection{How the computation was performed}
843We give a brief summary of how the computation was performed.  The
844algorithms that we used were implemented by the second author, and
845most are a standard part of MAGMA (see \cite{magma}).
846
847Let~$g$,~$f$, and~$q$ be some data from a line of
848Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We
849verified the existence of a congruence modulo~$q$, that
850$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq 8510$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
852not arise from any $S_k(\Gamma_0(N/p))$, as follows:
853
854To prove there is a congruence, we showed that the corresponding
855{\em integral} spaces of modular symbols satisfy an appropriate
856congruence, which forces the existence of a congruence on the
857level of Fourier expansions.  We showed that $\rho_{g,\qq}$ is
858irreducible by computing a set that contains all possible residue
859characteristics of congruences between~$g$ and any Eisenstein
860series of level dividing~$N$, where by congruence, we mean a
861congruence for all Fourier coefficients of index~$n$ with
862$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
863form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
864listing a basis of such~$h$ and finding the possible congruences,
865where again we disregard the Fourier coefficients of index not
866coprime to~$N$.
867
868To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
869modular symbol ${\mathbf 870e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$ under a map
871with the same kernel as the period mapping, and found that the
872image was~$0$.  The period mapping sends the modular
873symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
874so that ${\mathbf e}$ maps to~$0$ implies that
875$L(g,\frac{k}{2})=0$. In a similar way, we verified that
876$L(f,\frac{k}{2})\neq 0$.  Next, we checked that $W_N(g) 877=(-1)^{k/2} g$ which, because of the functional equation, implies
878that $L'(g,\frac{k}{2})=0$. Table~\ref{fig:newforms} is of
879independent interest because it includes examples of modular forms
880of weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
881the functional equation.  We found no such examples of weights
882$\geq 8$.
883
884For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
885prime appears in the $q$-column, meaning $f$ and $g$ are congruent
886$\bmod{\,\qq^2}$ for some $\qq\mid q$. In these cases, a
887modification of Theorem \ref{local} will show that the
888$\qq^2$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
889$O_E/\qq^2$-rank at least $r$.
890
891\subsection{Conjecturally nontrivial $\Sha$}
892In this section we apply some of the results of
893Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders
894of Shafarevich-Tate group of many modular motives.
895The results of this section suggest that~$\Sha$ of modular motives
896is usually not explained by congruences at level~$N$, which agrees
897with the observations of \cite{CM} and \cite{AS}.
898For example, when $k>6$ we find many examples of nontrivial~$\Sha$
899but no examples of nontrivial visible~$\Sha$.
900
901For any newform~$f$, let $L(M_f,s) = \prod_{\sigma}L(f^\sigma,s)$
902where $f^\sigma$ runs over the $\Gal(\Qbar/\QQ)$-conjugates of~$f$.
903Let~$T$ be the complex torus $\CC^d/\mathcal{L}$, where the
904lattice $\mathcal{L}$ is defined by integrating integral modular
905symbols against the conjugates of~$f$.  Let $\Omega_{M_f}$ denote the
906volume of the $-1$ eigenspace $T^{-}=\{z \in T : \overline{z}=-z\}$ for complex conjugation on~$T$.
907
908
909{\begin{figure}
910\vspace{-2ex}
911\caption{\label{fig:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}
912\vspace{-2ex}
913$$914\begin{array}{|c|c|c|c|}\hline 915f & \deg(f) & B\,\, (\text{\Sha bound})& \text{all odd congruence primes}\\\hline 916\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\ 917\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\ 918\nf{263k4B} & 39 & 41^{2} & 263 \\ 919\nf{269k4C} & 39 & 23^{2} & 269 \\ 920\nf{271k4B} & 39 & 29^{2} & 271 \\ 921\nf{281k4B} & 40 & 29^{2} & 281 \\ 922\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\ 923\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\ 924\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\ 925\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\ 926\hline 927\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\ 928\nf{101k6B} & 24 & 17^{2} & 101 \\ 929\nf{103k6B} & 24 & 23^{2} & 103 \\ 930\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\ 931\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\ 932\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\ 933\nf{157k6B} & 34 & 251^{2} & 157 \\ 934\nf{167k6B} & 40 & 41^{2} & 167 \\ 935\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\ 936\nf{173k6B} & 39 & 71^{2} & 173 \\ 937\nf{181k6B} & 40 & 107^{2} & 181 \\ 938\nf{191k6B} & 46 & 85091^{2} & 191 \\ 939\nf{193k6B} & 41 & 31^{2} & 193 \\ 940\nf{199k6B} & 46 & 200329^2 & 199 \\ 941\hline 942\nf{47k8B} & 16 & 19^{2} & 47 \\ 943\nf{59k8B} & 20 & 29^{2} & 59 \\ 944\nf{67k8B} & 20 & 29^{2} & 67 \\ 945\nf{71k8B} & 24 & 379^{2} & 71 \\ 946\nf{73k8B} & 22 & 197^{2} & 73 \\ 947\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\ 948\nf{79k8B} & 25 & 307^{2} & 79 \\ 949\nf{83k8B} & 27 & 1019^{2} & 83 \\ 950\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\ 951\nf{89k8B} & 29 & 44491^{2} & 89 \\ 952\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\ 953\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\ 954\nf{103k8B} & 32 & 75367^{2} & 103 \\ 955\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\ 956\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\ 957\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\ 958\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\ 959\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\ 960\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\ 961\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\ 962\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\ 963\hline 964\end{array} 965$$
966\end{figure}
967\begin{figure}
968$$969\begin{array}{|c|c|c|c|}\hline 970f & \deg(f) & B\,\, (\text{\Sha bound})& \text{all odd congruence primes}\\\hline 971\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\ 972\nf{121k8G} & 12 & 13^{2} & 3, 11 \\ 973\nf{121k8H} & 12 & 19^{2} & 5, 11 \\ 974\nf{125k8D} & 16 & 179^{2} & 5 \\ 975\hline 976 977\nf{43k10B} & 17 & 449^{2} & 43 \\ 978\nf{47k10B} & 20 & 2213^{2} & 47 \\ 979\nf{53k10B} & 21 & 673^{2} & 53 \\ 980\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\ 981\nf{59k10B} & 25 & 37^{2} & 59 \\ 982\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\ 983\nf{64k10K} & 2 & 19^{2} & 3 \\ 984\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\ 985\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\ 986\nf{71k10B} & 30 & 1103^{2} & 71 \\ 987 988\hline 989\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\ 990\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\ 991\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\ 992\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\ 993\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\ 994\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\ 995\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\ 996\nf{49k12H} & 12 & 271^{2} & 7 \\ 997\hline 998\end{array} 999$$
1000\end{figure}
1001
1002The following conjecture is probably not difficult to prove, but we
1003haven't given a proof, so we formally state it as a conjecture, then
1004assume it.
1005\begin{conj}\label{conj:lrat}
1006If $p\nmid Nk$ is a prime and
1007$p\mid \frac{L(M_f,k/2)}{\Omega_{k/2}}$, then
1008$p\mid \Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\right)$.
1009\end{conj}
1010For the rest of this section, {\em we officially assume the Bloch-Kato
1011conjecture and Conjecture~\ref{conj:lrat}.}
1012
1013Let~$S$ be the set of newforms with~level $N$ and weight~$k$
1014satisfying either $k=4$ and $N\leq 321$, or $k=6$ and $N\leq 199$, or
1015$k=8$ and $N\leq 125$, or $k=10$ and $N\leq 72$, or $k=12$ and
1016$N\leq 49$. Compute a bound~$B$ on $\#\Sha$, for each
1017newform $f\in S$, as follows:
1018\begin{enumerate}
1019\item Let $L_1$ be the rational number $L(M_f,k/2)/\Omega_{M_f}$.
1020      If $L_1=0$ let $B=1$ and terminate.
1021\item Let $L_2$ be the part of $L_1$ that is coprime to $N\cdot k!$.
1022\item Let $L_3$ be the part of $L_2$ that is coprime to
1023      $p+1$ for every prime~$p$ such that $p^2\mid N$.
1024\item Let $L_4$ be the part of $L_3$ coprime to any prime of
1025      congruence between~$f$ and a form of weight~$k$ and
1026      lower level.
1027\item Let $B$ be the part of $L_4$ coprime to any prime of congruence
1028      between~$f$ and an Eisenstein series.  (This eliminates
1029      residue characteristic of reducible representations.)
1030\end{enumerate}
1031Proposition~\ref{sha} implies that if
1032$\ord_p(B) > 0$, then $\ord_p(\#\Sha) > 0$.
1033
1034We computed~$B$ for every newform in~$S$.  Because of Tamagawa
1035numbers, there are many examples in which $L_3$ is
1036large, but~$B$ is not.
1037For example, {\bf 39k4C} has $L_3=19$, but $B=1$ because
1038of a $19$-congruence with a form of level~$13$;
1039in this case we must have $19\mid c_{13}$.
1040
1041
1042The newforms for which $B>1$ are given in Table~\ref{fig:invisforms}.
1043The second column of the table records the degree of the field
1044generated by the Fourier coefficients of~$f$.  This third
1045contains~$B$.  Let~$W$ be the intersection of the span of all
1046conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and $W^{\perp}$ the
1047Petersson orthogonal complement of~$W$ in $S_k(\Gamma_0(N),\ZZ)$.
1048Then the fourth column contains the odd prime divisors of
1049$\#(S_k(\Gamma_0(N),\ZZ))/(W+W^{\perp})$, which are exactly the
1050possible primes of congruence for~$f$.
1051We place a $*$ next to the four entries of Table~\ref{fig:invisforms}
1052that also occur in Table~\ref{fig:newforms}.
1053
1054\subsection{Other examples}\label{sec:other_ex}
1055We have some other examples where forms of
1056different levels are congruent.
1057However, Remark~\ref{sign} does not
1058apply, so that one of the forms could have an odd functional
1059equation, and the other could have an even functional equation.
1060For instance, we have a $13$-congruence between $g=\nf{81k4A}$ and
1061$f=\nf{567k4L}$; here $L(\nf{567k4L},2)\neq 0$, while
1062$L(\nf{81k4},2)=0$ since it has {\it odd} functional equation.
1063Here $f$ obviously fails the condition about not being congruent
1064to a form of lower level, so in Lemma~\ref{local1} it is possible that
1065$\ord_{\qq}(c_7(2))>0$. In fact this does happen. Because
1066$V'_{\qq}$ (attached to~$g$ of level $81$) is unramified at $p=7$,
1067$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
1068two-dimensional. As in (2) of the proof of Theorem~\ref{local},
1069one of the eigenvalues of $\Frob_p^{-1}$ acting on this
1070two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
1071$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
1072$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
1073$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
1074$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
1075Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
1076our example here with $p=7$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
1077non-trivial when $w_p=-1$, so (2) of the proof of Theorem~\ref{local}
1078does not work. This is just as well, since had it
1079worked we would have expected
1080$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
1081shows not to be the case.
1082
1083Here is an example where the divisibility between the levels is
1084the other way round: a $7$-congruence between $g=\nf{122k6A}$ and
1085$f=\nf{61k6B}$. In this case both $L$-functions have even
1086functional equation, and we have $L(\nf{122k6A},3)=0$. In the
1087proof of Theorem~\ref{local}, we find a problem with the local condition
1088at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to
1089$H^1(I_2,A'_{\qq}(3))$ is not necessarily injective, but its
1090kernel is at most one-dimensional, so we still get the
1091$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
1092$\FF_{\qq}$-rank at least $1$ (assuming $r\geq 2$), and thus get
1093(probable) elements of $\Sha$ for \nf{61k6B}. In particular, these
1094elements of $\Sha$ are {\it invisible} at level 61. When the
1095levels are different we are no longer able to apply Theorem 2.1 of
1096\cite{FJ}. However, we still have the congruences of integral
1097modular symbols required to make the proof of Proposition
1098\ref{div} go through. Indeed, as noted above, the congruences of
1099modular forms were found by producing congruences of modular
1100symbols. Despite these congruences of modular symbols, Remark 5.3
1101does not apply, since there is no reason to suppose that
1102$w_N=w_{N'}$, where $N$ and $N'$ are the distinct levels.
1103
1104Finally, there are two examples where we have a form $g$ with even
1105functional equation such that $L(g,k/2)=0$, and a congruent form
1106$f$ which has odd functional equation; these are a 23-congruence
1107between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
1108between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
1109$\ord_{s=2}L(f,s)=1$, it ought to be the case that
1110$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
1111similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
1112the appropriate modification of Theorem \ref{local} does not
1113necessarily provide us with non-trivial $\qq$-torsion in $\Sha$.
1114It only tells us that the $\qq$-torsion subgroup of
1115$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It
1116could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$
1117appears in the conjectural formula for the first derivative of the
1118complex $L$ function, evaluated at $s=k/2$, but in combination
1119with a regulator that we have no way of calculating.
1120
1121Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
1122associated with $f$ and $g$ by the construction of Mazur, Tate and
1123Teitelbaum \cite{MTT}, each divided by a suitable canonical
1124period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
1125quite clear what to make of this. This divisibility may be proved
1126as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
1127times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
1128normalised) are congruent $\bmod{\,\qq}$, as a result of the
1129congruence between the modular symbols out of which they are
1130constructed. Integrating an appropriate function against these
1131measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
1132to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
1133since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
1134where the signs in the functional equations of $L(g,s)$ and
1135$L_q(g,s)$ are the same, positive in this instance. (According to
1136the proposition in Section 18 of \cite{MTT}, the signs differ
1137precisely when $L_q(g,s)$ has a trivial zero'' at $s=k/2$.)
1138
1139
1140
1141\subsection{Excluded data}
1142We also found some examples for which the conditions of Theorem \ref{local}
1143were not met. We have a 7-congruence between \nf{639k4B} and \nf{639k4H},
1144but $w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a similar
1145problem with a 7-congruence between \nf{260k6A} and \nf{260k6E} --- here
1146$w_{13}=1$ so that $13\equiv -w_{13}\pmod{7}$. Finally, there is a
11475-congruence between \nf{116k6A} and \nf{116k6D}, but here the prime 5
1148is less than the weight 6.
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1247
1248\end{document}
1249127k4A    43  127k4C   17   [43]
1250159k4A  5,23  159k4E    8   [5]x[23]
1251365k4B    29  365k4E   18   [29]x[5] (extra factor of 5 divides the level)
1252369k4A  5,13  369k4J    9   [5]x[13]x[2]
1253453k4A  5,17  453k4E   23   [5]x[17]
1254453k4A    23  151k4A   30   Odd func eq for g
1255465k4A    11  465k4H    7   [11]x[5]x[2]
1256477k4A    73  477k4M   12   [73]x[2]
1257567k4A    23  567k4I    8   [23]x[3]
125881k4A     13  567k4L   12   Odd func eq for f, Theorem 4.1 gives nothing.
1259581k4A 19,19  581k4E   34   [19^2]x[4] (possible BIG Sha, as 19^2 divides MD)
1260639k4A     7  639k4H   12   [7]
1261657k4A     5  657k4C    7   [5]x[3]x[2] (see next note)
1262657k4A     5  657k4G   12   [5]x[4] (does 657k4A make both these visible?)
1263681k4A    43  227k4A   23   Odd func eq for g
1264681k4A    59  681k4D   30   [59]x[3]x[2]
1265684k4C   7,7  684k4K    4   [7^2]x[2] (see note to 581k4A)
126695k6A  31,59   95k6D    9   [31]x[59]
1267116k6A     5  116k6D    6   [5]x[29]x[2]
1268122k6A     7  61k6B    14   7^2 appears in L(61k6B,3)
1269122k6A    73  122k6C    6   [73]x[3] (guess that 3 is a bad prime now)
1270260k6A  7,17  260k6E    4   [7]x[17]x[4] <-- Did not compute MD or LROP
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283\nf{263k4B} &  & 41^2 & \\
1284\nf{269k4C} &  & 23^2 & \\
1285\nf{271k4B} &  & 29^2 &\\
1286\nf{281k4B} &  & 29^2\\
1287\hline
1288\nf{101k6B} &  & 17^2 & 101\\
1289\nf{103k6B} &  & 23^2\\
1290\nf{111k6C} &  & 11^2\\
1291\nf{153k6G} &  & 7^2\\
1292\nf{157k6B} &  & 252^2\\
1293\nf{167k6B} &  & 41^2\\
1294\nf{172k6B} &  & 7^2\\
1295\nf{173k6B} &  & 71^2\\
1296\nf{181k6B} &  & 107^2\\
1297\hline
1298\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571\\
1299
1300