CoCalc Shared Fileswww / papers / motive_visibility / motive_visibility_v8.tex
Author: William A. Stein
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3% motive_visibility.tex
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5% 25 August 2002
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7% Project of William Stein, Neil Dummigan, Mark Watkins
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10
11\documentclass{amsart}
12\usepackage{amssymb}
13\usepackage{amsmath}
14\usepackage{amscd}
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21\newtheorem{lem}[prop]{Lemma}
22\newtheorem{thm}[prop]{Theorem}
23\newtheorem{cor}[prop]{Corollary}
24\newtheorem{examp}[prop]{Example}
25\newtheorem{remar}[prop]{Remark}
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76
77% ---- SHA ----
78\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
79  \newcommand{\textcyr}[1]{%
80    {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
81     \selectfont #1}}
82\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
83
84\newcommand{\HH}{{\mathfrak H}}
85\newcommand{\aaa}{{\mathfrak a}}
86\newcommand{\bb}{{\mathfrak b}}
87\newcommand{\dd}{{\mathfrak d}}
88\newcommand{\ee}{{\mathbf e}}
89\newcommand{\Fbar}{\overline{F}}
90\newcommand{\CH}{\mathrm {CH}}
91
92\begin{document}
93\title[Shafarevich-Tate Groups of Modular Motives]{Constructing Elements in Shafarevich-Tate Groups of Modular Motives}
94\author{Neil Dummigan}
95\author{William Stein}
96\author{Mark Watkins}
97\date{August 24th, 2002}
98\subjclass{11F33, 11F67, 11G40.}
99
100\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
101Shafarevich-Tate group.}
102
103\address{University of Sheffield\\ Department of Pure
104Mathematics\\
105Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
106U.K.}
108One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
110University Park\\State College, PA 16802\\ U.S.A.}
111
112\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
113\email{watkins@math.psu.edu}
114
115
116\begin{abstract}
117
118We study Shafarevich-Tate groups of motives attached to modular forms
119on $\Gamma_0(N)$ of weight bigger than~$2$.  We deduce a criterion for
120the existence of nontrivial elements of these Shafarevich-Tate groups,
121and give $16$ examples in which the Beilinson-Bloch conjecture implies
122the existence of such elements.  We also use modular symbols and
123observations about Tamagawa numbers to compute nontrivial conjectural
124lower bounds on the orders of the Shafarevich-Tate groups of modular
125motives of low level and weight at most $12$.  Our methods build upon
126Mazur's idea of visibility, but in the context of motives instead of
127abelian varieties.
128\end{abstract}
129
130\maketitle
131
132\section{Introduction}
133Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
134be the associated $L$-function. The conjecture of Birch and
135Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$
136at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
137also gives an interpretation of the leading term in the Taylor
138expansion in terms of various quantities, including the order of
139the Shafarevich-Tate group of $E$.
140
141Cremona and Mazur \cite{CM} look, among all strong Weil elliptic
142curves over $\QQ$ of conductor $N\leq 5500$, at those with
143non-trivial Shafarevich-Tate group (according to the Birch and
144Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
145group has predicted elements of prime order $m$. In most cases
146they find another elliptic curve, often of the same conductor,
147whose $m$-torsion is Galois-isomorphic to that of the first one,
148and which has positive rank. The rational points on the second elliptic
149curve produce classes in the common $H^1(\QQ,E[m])$. They show
150\cite{CM2} that these lie in the Shafarevich-Tate group of the
151first curve, so rational points on one curve explain elements of
152the Shafarevich-Tate group of the other curve.
153
154The Bloch-Kato conjecture \cite{BK} is the generalisation to
155arbitrary motives of the leading term part of the Birch and
156Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
157\cite{B} generalises the part about the order of vanishing at the
158central point, identifying it with the rank of a certain Chow
159group.
160
161This paper is a partial generalisation of \cite{CM} and \cite{AS}
162from abelian varieties over $\QQ$ associated to modular forms of
163weight~$2$ to the motives attached to modular forms of higher weight.
164It also does for congruences between modular forms of equal weight
165what \cite{Du2} did for congruences between modular forms of different
166weights.
167
168We consider the situation where two newforms~$f$ and~$g$, both of
169even weight $k>2$ and level~$N$, are congruent modulo a maximal ideal
170$\qq$ of odd residue characteristic, and $L(g,k/2)=0$ but
171$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish
172to order at least $2$ at $s=k/2$.  In Section~\ref{sec:examples},
173we give sixteen
174examples (all with $k=4$ and $k=6$), and in each $\qq$ divides the
175numerator of the algebraic number $L(f,k/2)/\vol_{\infty}$, where
176$\vol_{\infty}$ is a certain canonical period.
177
178In fact, we show how this divisibility may be deduced from the
179vanishing of $L(g,k/2)$ using recent work of Vatsal \cite{V}. The
180point is, the congruence between$f$ and~$g$ leads to a congruence
181between suitable algebraic parts'' of the special values
182$L(f,k/2)$ and $L(g,k/2)$. In slightly more detail, a multiplicity
183one result of Faltings and Jordan shows that the congruence of
184Fourier expansions leads to a congruence of certain associated
185cohomology classes. These are then identified with the modular
186symbols which give rise to the algebraic parts of special values.
187If $L(g,k/2)$ vanishes then the congruence implies that
188$L(f,k/2)/\vol_{\infty}$ must be divisible by $\qq$.
189
190The Bloch-Kato conjecture sometimes then implies that the
191Shafarevich-Tate group attached to~$f$ has nonzero $\qq$-torsion.
192Under certain hypotheses and assumptions, the most substantial of
193which is the Beilinson-Bloch conjecture relating the vanishing of
194$L(g,k/2)$ to the existence of algebraic cycles, we are able to
195construct some of the predicted elements of~$\Sha$ using the
196Galois-theoretic interpretation of the congruences to transfer
197elements from a Selmer group for~$g$ to a Selmer group for~$f$.
198One might say that algebraic cycles for one motive explain
199elements of~$\Sha$ for the other, or that we use congruences to
200link the Beilinson-Bloch conjecture for one motive with the
201Bloch-Kato conjecture for the other.
202%In proving the local
203%conditions at primes dividing the level, and also in examining the
204%local Tamagawa factors at these primes, we make use of a higher weight
205%level-lowering result due to Jordan and Livn\'e \cite{JL}.
206
207We also compute data which, assuming the Bloch-Kato conjecture,
208provides lower bounds for the orders of numerous Shafarevich-Tate
209groups (see Section~\ref{sec:invis}).
210%Our data is consistent
211%with the fact \cite{Fl2} that the part of $\#\Sha$ coprime to the
212%congruence modulus is necessarily a perfect square (assuming that~$\Sha$
213%is finite).
214
215\section{Motives and Galois representations}
216This section and the next provide definitions of some of the
217quantities appearing later in the Bloch-Kato conjecture. Let
218$f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
219$\Gamma_0(N)$, with coefficients in an algebraic number field~$E$,
220which is necessarily totally real. Let~$\lambda$ be any finite
221prime of~$E$, and let~$\ell$ denote its residue characteristic.  A
222theorem of Deligne \cite{De1} implies the existence of a
223two-dimensional vector space $V_{\lambda}$ over $E_{\lambda}$, and
224a continuous representation
225$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
226such that
227\begin{enumerate}
228\item $\rho_{\lambda}$ is unramified at~$p$ for all primes~$p$ not dividing
229$lN$, and
230\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
231characteristic polynomial of $\Frob_p^{-1}$ acting on
232$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
233\end{enumerate}
234
235Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
236the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
237There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
238both $2$-dimensional $E$-vector spaces. For details of the
239construction see \cite{Sc}. The de Rham realisation has a Hodge
240filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset 241F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
242cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
243cohomology.
244For each prime $\lambda$, there is a natural isomorphism
245$V_B\otimes E_{\lambda}\simeq V_{\lambda}$. We may choose a
246$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
247each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
248Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
249There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
250which amounts to multiplying the action of $\Frob_p$ by $p^j$.
251
252Following \cite{BK} (Section 3), for $p\neq l$ (including
253$p=\infty$) let
254$$255H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow 256H^1(I_p,V_{\lambda}(j))). 257$$
258The subscript~$f$ stands for finite
259part'', $D_p$ is a decomposition subgroup at a prime above~$p$,
260$I_p$ is the inertia subgroup, and the cohomology is for
261continuous cocycles and coboundaries. For $p=l$ let
262$$263H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow 264H^1(D_l,V_{\lambda}(j)\otimes B_{\cris})) 265$$
266(see Section 1 of
267\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
268$B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
269elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
270in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes~$p$.
271
272There is a natural exact sequence
273$$274 \begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}. 275$$
276Let
277$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
278Define the $\lambda$-Selmer group
279$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
280$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
281$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes~$p$. Note that the
282condition at $p=\infty$ is superfluous unless $l=2$. Define the
283Shafarevich-Tate group
284$$285 \Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/ 286 \pi_*H^1_f(\QQ,V_{\lambda}(j)). 287$$
288Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any
289prime ideal~$\lambda$ is the length of the $\lambda$-component of
290$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,
291and write~$\Sha$ for~$\Sha(j)$.
292
293Define the group of global torsion points
294$$295 \Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)). 296$$
297This is analogous to the group of rational torsion points on an
298elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in
299which the exponent of any prime ideal~$\lambda$ is the length of
300the $\lambda$-component of $\Gamma_{\QQ}$.
301
302\section{Canonical periods}
303We assume from now on for convenience that $N\geq 3$. We need to
304choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
305and de Rham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
306in a way such that $T_B$ and $T_{\dR}\otimes_{O_E}O_E[1/Nk!]$
307agree with (respectively) the $O_E$-lattice $\mathfrak{M}_{f,B}$
308and the $O_E[1/Nk!]$-lattice $\mathfrak{M}_{f,\dR}$ defined in
309\cite{DFG} using cohomology, with non-constant coefficients, of
310modular curves. (In \cite{DFG}, see especially Sections 2.2 and
3115.4, and the paragraph preceding Lemma 2.3.)
312
313For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
314module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
315$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes 316E_{\lambda}\simeq V_{\lambda}$. Then the $O_{\lambda}$ module
317$T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable.
318
319Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
320generalised elliptic curves with full level-$N$ structure. Let
321$\mathfrak{E}$ be the universal generalised elliptic curve over
322$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product
323of $\mathfrak{E}$ over $M(N)$. (The motive $M_f$ is constructed
324using a projector on the cohomology of a desingularisation of
325$\mathfrak{E}^{k-2}$. Realising $M(N)(\CC)$ as the disjoint union
326of $\phi(N)$ copies of the quotient
327$\Gamma(N)\backslash\mathfrak{H}^*$ (where $\mathfrak{H}^*$ is the
328completed upper half plane), and letting $\tau$ be a variable on
329$\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is isomorphic to
330the elliptic curve with period lattice generated by $1$ and
331$\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a variable on
332the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the fibre product.
333Then $2\pi i f(\tau)\,d\tau\wedge dz_1\wedge\ldots\wedge dz_{k-2}$
334is a well-defined differential form on (a desingularisation of)
335$\mathfrak{E}^{k-2}$ and naturally represents a generating element
336of $F^{k-1}T_{\dR}$. (At least we can make our choices locally at
337primes dividing $Nk!$ so that this is the case.) We shall call
338this element $e(f)$.
339
340Under the de Rham isomorphism between $V_{\dR}\otimes\CC$ and
341$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
342a natural action of complex conjugation on $V_B$, breaking it up
343into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
344Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
345to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
346$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
347are rank one $O_E$-modules, but not necessarily free, since the
348class number of $O_E$ may be greater than one. Choose nonzero
349elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
350the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
351$\Omega_f^{\pm}$ by $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$.
352
353\section{The Bloch-Kato conjecture}\label{sec:bkconj}
354In this section we extract from the Bloch-Kato conjecture for
355$L(f,k/2)$ a prediction about the order of the Shafarevich-Tate
356group, by analysing the other terms in the formula.
357
358Let $L(f,s)$ be the $L$-function attached to~$f$. For
359$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with
360Euler product
361$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but
362there is an analytic continuation given by an integral, as
363described in the next section. Suppose that $L(f,k/2)\neq 0$. The
364Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the
365following equality of fractional ideals of~$E$:
366$$367 \frac{L(f,k/2)}{\vol_{\infty}}= 368 \left(\prod_pc_p(k/2)\right) 369 \frac{\#\Sha}{\aaa^{\pm}(\#\Gamma_{\QQ})^2}. 370$$
371(Strictly speaking, the conjecture in \cite{BK}
372is only given for $E=\QQ$.) Here, $\pm$
373represents the parity of $(k/2)-1$, and $\vol_{\infty}$ is equal
374to $(2\pi i)^{k/2}$ multiplied by the determinant of the
375isomorphism $V_B^{\pm}\otimes\CC\simeq 376(V_{\dR}/F^{k/2})\otimes\CC$, calculated with respect to the
377lattices $O_E\delta_f^{\pm}$ and the image of $T_{\dR}$. For
378$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be
379\begin{align*}
380\length&\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-
381  \ord_{\lambda}(P_p(p^{-j}))\\
382=&\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).
383\end{align*}
384
385We omit the definition of $\ord_{\lambda}(c_p(j))$ for
386$\lambda\mid p$, which requires one to assume Fontaine's de Rham
387conjecture (\cite{Fo}, Appendix A6), and depends on the choices of
388$T_{\dR}$ and $T_B$, locally at~$\lambda$. (We shall mainly be
389concerned with the $q$-part of the Bloch-Kato conjecture, where~$q$
390is a prime of good reduction. For such primes, the de Rham
391conjecture follows from Theorem 5.6 of \cite{Fa1}.)
392
393\begin{lem}\label{vol}
394$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and
395$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.
396\end{lem}
397\begin{proof}
398$\vol_{\infty}$ is also equal to the determinant of the period map
399from $F^{k/2}V_{\dR}\otimes\CC$ to $V_B^{\pm}\otimes\CC$, with
400respect to lattices dual to those we used above in the definition
401of $\vol_{\infty}$ (c.f. the last paragraph of 1.7 of \cite{De2}).
402We are using here natural pairings. Recall that the index of
403$O_E\delta_f^{\pm}$ in $T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then
404the proof is completed by noting that, locally away from primes
405dividing $Nk!$, the index of $T_{\dR}$ in its dual is equal to the
406index of $T_B$ in its dual, both being equal to the ideal
407denoted~$\eta$ in \cite{DFG2}.
408\end{proof}
409\begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer.
410Then the fractional ideal $c_p(j)$ is supported at most on
411divisors of~$p$.
412\end{lem}
413\begin{proof}
414As on p.~30 of \cite{Fl1}, for odd $l\neq p$,
415$\ord_{\lambda}(c_p(j))$ is the length of the finite
416$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
417where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
418trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
419torsion free.
420\end{proof}
421
422\begin{lem}\label{local1}
423Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
424is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
425$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
426that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not congruent
427modulo $\qq$ to any newform of weight~$k$, trivial character and
428level dividing $N/p$. Then any integer~$j$,
429$\ord_{\qq}(c_p(j))=0$.
430\end{lem}
431\begin{proof}
432It suffices to show that
433$$434 \dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)), 435$$
436since this ensures that
437$H^0(I_p,A_{\qq}(j))=V_{\qq}^{I_p}/T_{\qq}^{I_p}$, hence that
438$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}^{I_p}/T_{\qq}^{I_p})$. If
439the dimensions differ then, given that $f$ is not congruent modulo
440$\qq$ to a newform of level dividing $N/p$, Proposition 2.2 of
441\cite{L} shows that we are in the situation covered by one of the
442three cases in Proposition 2.3 of \cite{L}. Since $p\not\equiv 443-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is
444unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's
445result that $N$ is the prime-to-$q$ part of the conductor of
446$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses
447the condition $q>k$) implies the existence of a newform of weight
448$k$, trivial character and level dividing $N/p$, congruent to~$g$
449modulo $\qq$. This contradicts our hypotheses.
450\end{proof}
451
452\begin{remar}
453For an example of what can be done when~$f$ is congruent to
454a form of lower level, see the first example in Section~\ref{sec:other_ex}
455below.
456\end{remar}
457
458\begin{lem}\label{at q}
459If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then
460$\ord_{\qq}(c_q)=0$.
461\end{lem}
462\begin{proof}
463It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an
464application, at the end of Section~2.2, of the results of
465\cite{Fa1}) that $T_{\qq}$ is the
466$O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered
467module $T_{\dR}\otimes O_{\qq}$ by the functor they call
468$\mathbb{V}$. (This property is part of the definition of an
469$S$-integral premotivic structure given in Section~1.2 of
470\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
471of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
472in Theorem~4.1 of \cite{BK} follows from the first paragraph of
4732(h) of \cite{Fa1}.)
474\end{proof}
475
476\begin{lem}
477If $A[\lambda]$ is an
478irreducible representation of $\Gal(\Qbar/\QQ)$,
479then
480$$\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0.$$
481\end{lem}
482This follows trivially from the definition.
483
484Putting together the above lemmas we arrive at the following:
485\begin{prop}\label{sha}
486Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that $A[\qq]$
487is an irreducible representation of $\Gal(\Qbar/\QQ)$, where $\qq\mid q$.
488Assume the same hypotheses as in Lemma \ref{local1},
489for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$
490are as in the previous section. If
491$$492 \ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty})>0 493$$
494(with numerator nonzero) then the Bloch-Kato conjecture
495predicts that
496$$497 \ord_{\qq}(\#\Sha)>0. 498$$
499\end{prop}
500
501\section{Congruences of special values}
502Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
503weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
504large enough to contain all the coefficients $a_n$ and $b_n$.
505Suppose that $\qq\mid q$ is a prime of~$E$ such that $f\equiv 506g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$, and assume
507that $q\nmid N\phi(N)k!$.
508Choose $\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
509$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates
510$T_B^{\pm}$ locally at $\qq$.
511Make two further assumptions:
512$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$
513
514\begin{prop} \label{div}
515With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
516\end{prop}
517\begin{proof} This is based on some of the ideas used in Section 1 of
518\cite{V}.  Note the apparent typo in Theorem~1.13 of \cite{V},
519which presumably should refer to Condition 2''. Since
520$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
521$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm 5221=(-1)^{(k/2)-1}$. It is well known, and easy to prove, that
523$$524 \int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s). 525$$
526Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
527$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
528where the integral is taken along the positive imaginary axis,
529then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
530Thus we are reduced
531to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
532
533Let $\mathcal{D}_0$ be the group of divisors of degree zero
534supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
535integer $r\geq 0$, let $P_r(R)$ be the additive group of
536homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
537groups have a natural action of $\Gamma_1(N)$. Let
538$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
539be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
540
541Via the isomorphism (8) in Section~1.5 of \cite{V}, combined with
542the argument in 1.7 of \cite{V}, the cohomology class
543$\omega_f^{\pm}$ corresponds to a modular symbol $\Phi_f^{\pm}\in 544S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
545element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_{E,\qq})$. We are
546now dealing with cohomology over $X_1(N)$ rather than $M(N)$,
547which is why we insist that $q\nmid \phi(N)$. It follows from the
548last line of Section~4.2 of \cite{St} that, up to some small
549factorials which do not matter locally at $\qq$,
550$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv 551(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since
552$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
553$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv 554(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
555coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
556show is divisible by $\qq$.
557Similarly
558$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv 559(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of
560$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
561Therefore it would suffice to show that, for some $\mu\in O_E$,
562the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
563$\qq$ in $S_{\Gamma_1(N)}(k,O_{E,\qq})$. It suffices to show that,
564for some $\mu\in O_E$, the element
565$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
566considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
567non-constant coefficients. This would be the case if
568$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
569one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
570a consequence of Theorem 2.1(1) of \cite{FJ}.
571\end{proof}
572\begin{remar}\label{sign}
573The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are
574equal. They are determined by the eigenvalue of the involution $W_N$,
575which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and
576$b_N$ are each $N^{k/2-1}$ times this sign and~$\qq$ has residue
577characteristic coprime to $2N$.  The common sign in the functional
578equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of
579$W_N$ acting on~$f$ and~$g$.
580\end{remar}
581
582This is analogous to the remark at the end of Section~3 of \cite{CM},
583which shows that if~$\qq$ has odd residue characteristic and
584$L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then $L(g,s)$ must vanish to order
585at least two at $s=k/2$.  Note that Maeda's conjecture
586implies that there are no examples of~$g$ of
588$L(g,k/2)=0$ (see \cite{CF}).
589
590\section{Constructing elements of the Shafarevich-Tate group}
591Let~$f$ and~$g$ be as in the first paragraph of the previous
592section. In the previous section we showed how the congruence
593between $f$ and $g$ relates the vanishing of $L(g,k/2)$ to the
594divisibility by $\qq$ of an algebraic part'' of $L(f,k/2)$.
595Conjecturally the former is associated with the existence of
596certain algebraic cycles (for $M_g$) while the latter is
597associated with the existence of certain elements of the
598Shafarevich-Tate group (for $M_f$, as we saw in \S 4). In this
599section we show how the congruence, interpreted in terms of Galois
600representations, provides a direct link between algebraic cycles
601and the Shafarevich-Tate group.
602
603For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
604$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
605$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
606is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
607the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
608irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
609
610Suppose that $L(g,k/2)=0$. If the sign in the functional equation
611is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
612\ref{sign}), this implies that the order of vanishing of $L(g,s)$
613at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
614conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
615is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational
616rational equivalence classes of null-homologous,
617algebraic cycles  of codimension $k/2$
618on the motive $M_g$. (This generalises the part
619of the Birch--Swinnerton-Dyer conjecture which says that for an
620elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at
621$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)
622
623Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
624to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
625subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
626If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
627get (assuming also the Beilinson-Bloch conjecture) a subspace of
628$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
629vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
630conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
631equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
632follow from the conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
633Sections~1 and~6.5 of \cite{Fo2}.
634
635Similarly, if $L(f,k/2)\neq 0$ then we expect that
636$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
637coincides with the $\qq$-part of $\Sha$.
638\begin{thm}\label{local}
639Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
640$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
641$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$ and
642that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a
643newform of weight~$k$, trivial character and level dividing $N/p$.
644Suppose that, for all primes $p\mid N$, $\,p\not\equiv 645-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here
646$w_p$ is the common eigenvalue of the Atkin-Lehner involution
647$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of
648$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
649\end{thm}
650
651\begin{proof}
652Take a nonzero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
653continuity and rescaling we may assume that it lies in
654$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq 655H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction (note that
656$T'_{\qq}(k/2)/\qq T'_{\qq}(k/2)\simeq A'[\qq](k/2)$) we get a
657nonzero class $c\in H^1(\QQ,A'[\qq](k/2))\simeq 658H^1(\QQ,A[\qq](k/2))$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$
659is trivial. It follows that $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
660$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
661we get a nonzero, $\qq$-torsion class $\gamma\in 662H^1(\QQ,A_{\qq}(k/2))$.
663
664Our aim is to show that $\res_p(\gamma)\in 665H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
666consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
667
668\begin{enumerate}
669\item {\bf $p\nmid qN$. }
670
671Consider the $I_p$-cohomology of the short exact sequence
672$$673 \begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD}, 674$$
675where~$\pi$ is multiplication by a uniformising element of
676$O_{\qq}$. Since in this case $A'_{\qq}(k/2)$ is unramified at
677$p$, $H^0(I_p, A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is
678$\qq$-divisible. Therefore $H^1(I_p,A'[\qq](k/2))$ (which,
679remember, is the same as $H^1(I_p,A[\qq](k/2))$) injects into
680$H^1(I_p,A'_{\qq}(k/2))$. It follows from the fact that $d\in 681H^1_f(\QQ,V'_{\qq}(k/2))$ that the image in
682$H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is zero, hence
683that the restriction of~$c$ (to $H^1(I_p,A'[\qq](k/2))\simeq 684H^1(I_p,A[\qq](k/2))$) is zero. Hence the restriction of $\gamma$
685to $H^1(I_p,A_{\qq}(k/2))$ is also zero. By line~3 of p.~125 of
686\cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
687contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
688to $H^1(I_p,A_{\qq}(k/2))$,  so we have shown that
689$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
690
691\item {\bf $p\mid N$. }
692
693First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
694It suffices to show that
695$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
696since then the natural map from
697$H^0(I_p,V'_{\qq}(k/2))$ to $H^0(I_p, A'_{\qq}(k/2))$ is
698surjective; this may be done as in the proof of Lemma
699\ref{local1}. It follows as above that the image of $c\in 700H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is zero. Then
701$\res_p(c)$ comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by
702inflation-restriction. The order of this group is the same as the
703order of the group $H^0(\QQ_p,A[\qq](k/2))$, which we claim is
704trivial. By the work of Carayol \cite{Ca1}, $p\mid N$ implies that
705$V_{\qq}(k/2)$ is ramified at $p$, so $\dim 706H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim 707H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
708consider the case where this common dimension is $1$. The
709(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha 710p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication by~$\alpha$
711on the one-dimensional space $H^0(I_p,V_{\qq}(k/2))$. It
712follows from Theor\'eme A of \cite{Ca1} that this is the same as
713the Euler factor at $p$ of $L(f,s)$. By the work of Atkin and
714Lehner \cite{AL}, it then follows that $p^2\nmid N$ and
715$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
716$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
717$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$ as
718$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
719$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
720$\res_p(\gamma)=0$ and certainly lies in
721$H^1_f(\QQ_p,A_{\qq}(k/2))$.
722
723\item {\bf $p=q$. }
724
725Since $q\nmid N$ is a prime of good reduction for the motive
726$M_g$, $\,V'_{\qq}$ is a crystalline representation of
727$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
728$V'_{\qq}$ have the same dimension, where
729$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q} 730B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
731As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
732the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
733filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we
734may now prove, in the same manner as Proposition 9.2 of
735\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.
736\end{enumerate}
737\end{proof}
738
739Theorem~2.7 of \cite{AS} is concerned with verifying local
740conditions in the case $k=2$, where~$f$ and~$g$ are associated
741with abelian varieties~$A$ and~$B$. (Their theorem also applies to
742abelian varieties over number fields.) Our restriction outlawing
743congruences modulo $\qq$ with cusp forms of lower level is
744analogous to theirs forbidding~$q$ from dividing Tamagawa factors
745$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic
746curve with $\ord_l(j(A))<0$, consideration of a Tate
747parametrisation shows that if $q\mid c_{A,l}$, i.e., if
748$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
749at~$l$.)
750
751In this paper we have encountered two technical problems which we
752dealt with in quite similar ways:
753\begin{enumerate}
754\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
755\item proving local conditions at primes $p\mid N$, for an element
756of $\qq$-torsion.
757\end{enumerate}
758If our only interest was in testing the Bloch-Kato conjecture at
759$\qq$, we could have made these problems cancel out, as in Lemma
7608.11 of \cite{DFG}, by weakening the local conditions. However, we
761have chosen not to do so, since we are also interested in the
762Shafarevich-Tate group, and since the hypotheses we had to assume
763are not particularly strong. Note that, since $A[\qq]$ is
764irreducible, the $\qq$-part of $\Sha$ does not depend on the
765choice of $T_{\qq}$.
766
767\section{Examples and Experiments}
768\label{sec:examples} This section contains tables and numerical
769examples that illustrate the main themes of this paper.  In
770Section~\ref{sec:vistable}, we explain Table~\ref{tab:newforms},
771which contains~$16$ examples of pairs $f,g$ such that the
772Beilinson-Bloch conjecture and Theorem~\ref{local} together imply
773the existence of nontrivial elements of the Shafarevich-Tate group
774of the motive attached to~$f$. Section~\ref{sec:howdone} outlines
775the higher-weight modular symbol computations that were used in
776making Table~\ref{tab:newforms}. Section~\ref{sec:invis} discusses
777Table~\ref{tab:invisforms}, which summarizes the results of an
778extensive computation of conjectural orders of Shafarevich-Tate
779groups for modular motives of low level and weight.
780Section~\ref{sec:other_ex} gives specific examples in which
781various hypotheses fail. Note that in this section modular
782symbol'' has a different meaning from in \S 5, being related to
783homology rather than cohomology. For precise definitions see
784\cite{SV}.
785
786\subsection{Visible $\Sha$ Table~\ref{tab:newforms}}\label{sec:vistable}
787\begin{table}
788\caption{\label{tab:newforms}Visible $\Sha$}\vspace{-3ex}
789
790$$791\begin{array}{|c|c|c|c|c|}\hline 792 g & \deg(g) & f & \deg(f) & q's \\\hline 793\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\ 794\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\ 795\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\ 796\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\ 797\vspace{-2ex} & & & & \\ 798\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\ 799\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\ 800\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\ 801\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\ 802\vspace{-2ex} & & & & \\ 803\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\ 804\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\ 805\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\ 806\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\ 807\vspace{-2ex} & & & & \\ 808\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\ 809\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\ 810\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\ 811\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\ 812\hline 813\end{array} 814$$
815\end{table}
816
817
818Table~\ref{tab:newforms} on page~\pageref{tab:newforms} lists
819sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
820along with at least one prime~$q$ such that there is a prime
821$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
822$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
823The notation is as follows.
824The first column contains a label whose structure is
825\begin{center}
826{\bf [Level]k[Weight][GaloisOrbit]}
827\end{center}
828This label determines a newform $g=\sum a_n q^n$, up to Galois
829conjugacy. For example, \nf{127k4C} denotes a newform in the third
830Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
831orbits are ordered first by the degree of $\QQ(\ldots, a_n, 832\ldots)$, then by the sequence of absolute values $|\mbox{\rm 833Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
834being first in the event that the two absolute values are equal,
835and the first Galois orbit is denoted {\bf A}, the second {\bf B},
836and so on. The second column contains the degree of the field
837$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
838contain~$f$ and its degree, respectively. The fifth column
839contains at least one prime~$q$ such that there is a prime
840$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
841hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
842satisfied for~$f$,~$g$, and~$\qq$.
843
844For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
845prime $q$ appears in the $q$-column, meaning $q^2$ divides the
846order of the group $S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp})$, defined
847at the end of 7.3 below.
848
849
850We describe the first line of Table~\ref{tab:newforms}
851in more detail.  See the next section for further details
852on how the computations were performed.
853
854Using modular symbols, we find that there is a newform
855$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots 856\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
857the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
858also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
859coefficients generate a number field~$K$ of degree~$17$, and by
860computing the image of the modular symbol $XY\{0,\infty\}$ under
861the period mapping, we find that $L(f,2)\neq 0$.  The newforms~$f$
862and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
863characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
864both equal to
865$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 866+ \cdots\in \FF_{43}[[q]].$$
867
868There is no form in the Eisenstein subspaces of
869$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
870$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
871$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
872prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
873level~$1$ form of weight~$4$. Thus we have checked the hypotheses
874of Theorem~\ref{local}, so if $r$ is the dimension of
875$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
876$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
877
878Recall that since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that
879$r\geq 2$. Then, since $L(f,k/2)\neq 0$, we expect that the
880$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to
881the $\qq$-torsion subgroup of $\Sha$. Admitting these assumptions,
882we have constructed the $\qq$-torsion in $\Sha$ predicted by the
883Bloch-Kato conjecture.
884
885For particular examples of elliptic curves one can often find and
886write down rational points predicted by the Birch and
887Swinnerton-Dyer conjecture. It would be nice if likewise one could
888explicitly produce algebraic cycles predicted by the
889Beilinson-Bloch conjecture in the above examples. Since
890$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
8910.3.2 of \cite{Z}), so ought to be trivial in
892$\CH_0^{k/2}(M_g)\otimes\QQ$.
893
894\subsection{How the computation was performed}\label{sec:howdone}
895We give a brief summary of how the computation was performed.  The
896algorithms that we used were implemented by the second author, and
897most are a standard part of MAGMA (see \cite{magma}).
898
899Let~$g$,~$f$, and~$q$ be some data from a line of
900Table~\ref{tab:newforms} and let~$N$ denote the level of~$g$. We
901verified the existence of a congruence modulo~$q$, that
902$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq 9030$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
904not arise from any $S_k(\Gamma_0(N/p))$, as follows:
905
906To prove there is a congruence, we showed that the corresponding
907{\em integral} spaces of modular symbols satisfy an appropriate
908congruence, which forces the existence of a congruence on the
909level of Fourier expansions.  We showed that $\rho_{g,\qq}$ is
910irreducible by computing a set that contains all possible residue
911characteristics of congruences between~$g$ and any Eisenstein
912series of level dividing~$N$, where by congruence, we mean a
913congruence for all Fourier coefficients of index~$n$ with
914$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
915form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
916listing a basis of such~$h$ and finding the possible congruences,
917where again we disregard the Fourier coefficients of index not
918coprime to~$N$.
919
920To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
921modular symbol ${\mathbf 922e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$ under a map
923with the same kernel as the period mapping, and found that the
924image was~$0$.  The period mapping sends the modular
925symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
926so that ${\mathbf e}$ maps to~$0$ implies that
927$L(g,\frac{k}{2})=0$. In a similar way, we verified that
928$L(f,\frac{k}{2})\neq 0$.  Next, we checked that $W_N(g) 929=(-1)^{k/2} g$ which, because of the functional equation, implies
930that $L'(g,\frac{k}{2})=0$. Table~\ref{tab:newforms} is of
931independent interest because it includes examples of modular forms
932of even weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
933the functional equation.  We found no such examples of weights
934$\geq 8$.
935
936\subsection{Conjecturally nontrivial $\Sha$}\label{sec:invis}
937In this section we apply some of the results of
938Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders
939of Shafarevich-Tate groups of many modular motives. The results of
940this section suggest that~$\Sha$ of a modular motive is usually not
941visible at level~$N$'', i.e., explained by congruences at level~$N$,
942which agrees with the observations of \cite{CM} and \cite{AS}. For
943example, when $k>6$ we find many examples of conjecturally
944nontrivial~$\Sha$ but no examples of nontrivial visible~$\Sha$.
945
946For any newform~$f$, let $L(M_f/\QQ,s) = \prod_{i=1}^{d} 947L(f^{(i)},s)$ where $f^{(i)}$ runs over the
948$\Gal(\Qbar/\QQ)$-conjugates of~$f$. Let~$T$ be the complex torus
949$\CC^d/\mathcal{L}$, where the lattice $\mathcal{L}$ is defined by
950integrating integral cuspidal modular symbols against the
951conjugates of~$f$.  Let $\Omega_{M_f/\QQ}$ denote the volume of
952the $-1$ eigenspace $T^{-}=\{z \in T : \overline{z}=-z\}$ for
953complex conjugation on~$T$.
954
955
956{\begin{table}
957\vspace{-2ex}
958\caption{\label{tab:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}
959\vspace{-4ex}
960
961$$962\begin{array}{|c|c|c|c|}\hline 963f & \deg(f) & B\,\, (\text{\Sha bound})& \text{all odd congruence primes}\\\hline 964\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\ 965\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\ 966\nf{263k4B} & 39 & 41^{2} & 263 \\ 967\nf{269k4C} & 39 & 23^{2} & 269 \\ 968\nf{271k4B} & 39 & 29^{2} & 271 \\ 969\nf{281k4B} & 40 & 29^{2} & 281 \\ 970\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\ 971\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\ 972\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\ 973\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\ 974\hline 975\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\ 976\nf{101k6B} & 24 & 17^{2} & 101 \\ 977\nf{103k6B} & 24 & 23^{2} & 103 \\ 978\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\ 979\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\ 980\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\ 981\nf{157k6B} & 34 & 251^{2} & 157 \\ 982\nf{167k6B} & 40 & 41^{2} & 167 \\ 983\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\ 984\nf{173k6B} & 39 & 71^{2} & 173 \\ 985\nf{181k6B} & 40 & 107^{2} & 181 \\ 986\nf{191k6B} & 46 & 85091^{2} & 191 \\ 987\nf{193k6B} & 41 & 31^{2} & 193 \\ 988\nf{199k6B} & 46 & 200329^2 & 199 \\ 989\hline 990\nf{47k8B} & 16 & 19^{2} & 47 \\ 991\nf{59k8B} & 20 & 29^{2} & 59 \\ 992\nf{67k8B} & 20 & 29^{2} & 67 \\ 993\nf{71k8B} & 24 & 379^{2} & 71 \\ 994\nf{73k8B} & 22 & 197^{2} & 73 \\ 995\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\ 996\nf{79k8B} & 25 & 307^{2} & 79 \\ 997\nf{83k8B} & 27 & 1019^{2} & 83 \\ 998\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\ 999\nf{89k8B} & 29 & 44491^{2} & 89 \\ 1000\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\ 1001\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\ 1002\nf{103k8B} & 32 & 75367^{2} & 103 \\ 1003\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\ 1004\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\ 1005\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\ 1006\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\ 1007\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\ 1008\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\ 1009\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\ 1010\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\ 1011 \hline 1012\end{array} 1013$$
1014\end{table}
1015\begin{table}
1016$$1017\begin{array}{|c|c|c|c|}\hline 1018f & \deg(f) & B\,\, (\text{\Sha bound})& \text{all odd congruence primes}\\\hline 1019\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\ 1020\nf{121k8G} & 12 & 13^{2} & 3, 11 \\ 1021\nf{121k8H} & 12 & 19^{2} & 5, 11 \\ 1022\nf{125k8D} & 16 & 179^{2} & 5 \\ 1023\nf{127k8B} & 39 & 59^{2} & 127 \\ 1024\nf{128k8F} & 4 & 11^{2} & 1 \\ 1025\nf{131k8B} & 43 & 241^{2} \!\cdot\! 817838201^{2}&131\\ 1026\nf{134k8C} & 11 & 61^{2} & 11, 17, 41, 67, 71, 421, 2356138931854759 \\ 1027\nf{137k8B} & 42 & 71^{2} \!\cdot\! 749093^{2} & 137 \\ 1028\nf{139k8B} & 43 & 47^{2} \!\cdot\! 89^{2} \!\cdot\! 1021^{2} & 139 \\ 1029\nf{141k8C} & 14 & 13^{2} & 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\ 1030\nf{142k8B} & 10 & 11^{2} & 3, 53, 71, 56377, 1965431024315921873 \\ 1031\nf{143k8C} & 19 & 307^{2} & 3, 11, 13, 89, 199, 409, 178397, 1032639259, 17440535 103397287 \\ 1034\nf{143k8D} & 21 & 109^{2} & 3, 7, 11, 13, 61, 79, 103, 173, 241, 1035769, 36583 1036\\ 1037\nf{145k8C} & 17 & 29587^{2} & 5, 11, 29, 107, 251623, 393577, 1038518737, 9837145 1039699 \\ 1040\nf{146k8C} & 12 & 3691^{2} & 11, 73, 269, 503, 1673540153, 11374452082219 \\ 1041\nf{148k8B} & 11 & 19^{2} & 3, 37 \\ 1042\nf{149k8B} & 47 & 11^{4} \!\cdot\! 40996789^{2} & 149\\ 1043 1044\hline 1045 1046\nf{43k10B} & 17 & 449^{2} & 43 \\ 1047\nf{47k10B} & 20 & 2213^{2} & 47 \\ 1048\nf{53k10B} & 21 & 673^{2} & 53 \\ 1049\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\ 1050\nf{59k10B} & 25 & 37^{2} & 59 \\ 1051\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\ 1052\nf{64k10K} & 2 & 19^{2} & 3 \\ 1053\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\ 1054\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\ 1055\nf{71k10B} & 30 & 1103^{2} & 71 \\ 1056 1057\hline 1058\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\ 1059\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\ 1060\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\ 1061\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\ 1062\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\ 1063\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\ 1064\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\ 1065\nf{49k12H} & 12 & 271^{2} & 7 \\ 1066\hline 1067\end{array} 1068$$
1069\end{table}
1070
1071The following lemma may be proved by tensoring the space of
1072integral modular symbols with $O_E$, then decomposing it in such a
1073way that $\Omega_{M_f/\QQ}$ becomes the determinant of a diagonal
1074matrix with $d$ non-zero entries. We omit the somewhat awkward
1075details.
1076\begin{lem}\label{lem:lrat}
1077If $p\nmid Nk!$ is a non-congruence prime for $f$ then the
1078$p$-parts of
1079$$1080 \frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}\qquad\text{and}\qquad 1081 \Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\right) 1082$$
1083are equal, where $\vol_\infty$ is as in Section~\ref{sec:bkconj}.
1084\end{lem}
1085For the rest of this section, {\em we officially assume the
1086Bloch-Kato conjecture.}
1087
1088Let~$\mathcal{S}$ be the set of newforms with~level $N$ and weight~$k$
1089satisfying either $k=4$ and $N\leq 321$, or $k=6$ and $N\leq 199$, or
1090$k=8$ and $N\leq 133$, or $k=10$ and $N\leq 72$, or $k=12$ and $N\leq 109149$.  Given $f\in \mathcal{S}$, let~$B$ be the lower bound on $\#\Sha$ defined
1092as follows:
1093\begin{enumerate}
1094\item Let $L_1$ be the rational number $L(M_f,k/2)/\Omega_{M_f}$.
1095      If $L_1=0$ let $B=1$ and terminate.
1096\item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$.
1097\item Let $L_3$ be the part of $L_2$ that is coprime to
1098      $p+1$ for every prime~$p$ such that $p^2\mid N$.
1099\item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic
1100      of any prime of
1101      congruence between~$f$ and a form of weight~$k$ and
1102      lower level. (By congruence here, we mean a congruence for coefficients
1103      $a_n$ with $n$ coprime to the level of~$f$.)
1104\item Let $B$ be the part of $L_4$ coprime to the residue characteristic
1105      of any prime of congruence
1106      between~$f$ and an Eisenstein series.  (This eliminates
1107      residue characteristics of reducible representations.)
1108\end{enumerate}
1109Proposition~\ref{sha} and Lemma~\ref{lem:lrat} imply that if
1110$\ord_p(B) 1111> 0$, and $p$ is not a congruence prime for $f$, then
1112$\ord_p(\#\Sha) > 0$. We have left the congruence primes in $B$ in
1113the starred examples since the squares are still suggestive.
1114
1115We computed~$B$ for every newform in~$\mathcal{S}$.  There are
1116many examples in which $L_3$ is large, but~$B$ is not, and this is
1117because of Tamagawa factors. For example, {\bf 39k4C} has
1118$L_3=19$, but $B=1$ because of a $19$-congruence with a form of
1119level~$13$; in this case we must have $19\mid c_{13}(2)$, where
1120$c_{13}(2)$ is as in Section~\ref{sec:bkconj}.  See
1121Section~\ref{sec:other_ex} for more details.  Also note that in
1122every example~$B$ is a perfect square, which is consistent with
1123the fact \cite{Fl2} that the order of $\Sha$ (if finite) is
1124necessarily a perfect square.  That our computed value of~$B$
1125should be a square is not a priori obvious.
1126
1128of rings of integers, so our definition of~$B$ is overly conservative.
1129For example,~$5$ occurs in row~$2$ of Table~\ref{tab:newforms} but not
1130in Table~\ref{tab:invisforms}, because \nf{159k4E} is Eisenstein at
1131some prime above~$5$, but the prime of congruences of
1132characteristic~$5$ between \nf{159k4B} and \nf{159k4E} is not
1133Eisenstein.
1134
1135
1136The newforms for which $B>1$ are given in
1137Table~\ref{tab:invisforms}. The second column of the table records
1138the degree of the field generated by the Fourier coefficients
1139of~$f$.  The third contains~$B$.  Let~$W$ be the intersection of
1140the span of all conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and
1141$W^{\perp}$ the Petersson orthogonal complement of~$W$ in
1142$S_k(\Gamma_0(N),\ZZ)$. Then the fourth column contains the odd
1143prime divisors of $\#(S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp}))$, which
1144are exactly the possible primes of congruence for~$f$. We place a
1145$*$ next to the four entries of Table~\ref{tab:invisforms} that
1146also occur in Table~\ref{tab:newforms}.
1147
1148\subsection{Examples in which hypotheses fail}\label{sec:other_ex}
1149We have some other examples where forms of
1150different levels are congruent.
1151However, Remark~\ref{sign} does not
1152apply, so that one of the forms could have an odd functional
1153equation, and the other could have an even functional equation.
1154For instance, we have a $19$-congruence between the
1155newforms $g=\nf{13k4A}$ and $f=\nf{39k4C}$ of Fourier
1156coefficients coprime to $39$.
1157Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$
1158has {\it odd} functional equation.
1159Here~$f$ fails the condition about not being congruent
1160to a form of lower level, so in Lemma~\ref{local1} it is possible that
1161$\ord_{\qq}(c_{19}(2))>0$. In fact this does happen. Because
1162$V'_{\qq}$ (attached to~$g$ of level $81$) is unramified at $p=19$,
1163$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
1164two-dimensional. As in (2) of the proof of Theorem~\ref{local},
1165one of the eigenvalues of $\Frob_p^{-1}$ acting on this
1166two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
1167$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
1168$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
1169$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
1170$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
1171Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
1172our example here with $p=7$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
1173non-trivial when $w_p=-1$, so (2) of the proof of Theorem~\ref{local}
1174does not work. This is just as well, since had it
1175worked we would have expected
1176$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
1177shows not to be the case.
1178
1179In the following example, the divisibility between the levels is the
1180other way round.  There is a $7$-congruence between $g=\nf{122k6A}$
1181and $f=\nf{61k6B}$, both $L$-functions have even functional equation,
1182and $L(g,3)=0$. In the proof of Theorem~\ref{local},
1183there is a problem with the local condition at $p=2$. The map from
1184$H^1(I_2,A'[\qq](3))$ to $H^1(I_2,A'_{\qq}(3))$ is not necessarily
1185injective, but its kernel is at most one dimensional, so we still get
1186the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
1187$\FF_{\qq}$-rank at least~$1$ (assuming $r\geq 2$), and thus get
1188(probable) elements of $\Sha$ for \nf{61k6B}. In particular, these
1189elements of $\Sha$ are {\it invisible} at level 61. When the levels
1190are different we are no longer able to apply Theorem 2.1 of
1191\cite{FJ}. However, we still have the congruences of integral modular
1192symbols required to make the proof of Proposition \ref{div} go
1193through. Indeed, as noted above, the congruences of modular forms were
1194found by producing congruences of modular symbols. Despite these
1195congruences of modular symbols, Remark 5.3 does not apply, since there
1196is no reason to suppose that $w_N=w_{N'}$, where $N$ and $N'$ are the
1197distinct levels.
1198
1199Finally, there are two examples where we have a form $g$ with even
1200functional equation such that $L(g,k/2)=0$, and a congruent form
1201$f$ which has odd functional equation; these are a 23-congruence
1202between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
1203between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
1204$\ord_{s=2}L(f,s)=1$, it ought to be the case that
1205$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
1206similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
1207the appropriate modification of Theorem \ref{local} does not
1208necessarily provide us with non-trivial $\qq$-torsion in $\Sha$.
1209It only tells us that the $\qq$-torsion subgroup of
1210$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It
1211could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$
1212appears in the conjectural formula for the first derivative of the
1213complex $L$ function, evaluated at $s=k/2$, but in combination
1214with a regulator that we have no way of calculating.
1215
1216Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
1217associated with $f$ and $g$ by the construction of Mazur, Tate and
1218Teitelbaum \cite{MTT}, each divided by a suitable canonical
1219period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
1220quite clear what to make of this. This divisibility may be proved
1221as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
1222times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
1223normalised) are congruent $\bmod{\,\qq}$, as a result of the
1224congruence between the modular symbols out of which they are
1225constructed. Integrating an appropriate function against these
1226measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
1227to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
1228since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
1229where the signs in the functional equations of $L(g,s)$ and
1230$L_q(g,s)$ are the same, positive in this instance. (According to
1231the proposition in Section 18 of \cite{MTT}, the signs differ
1232precisely when $L_q(g,s)$ has a trivial zero'' at $s=k/2$.)
1233
1234We also found some examples for which the conditions of Theorem~\ref{local}
1235were not met.  For example, we have a $7$-congruence between
1236\nf{639k4B} and \nf{639k4H}, but $w_{71}=-1$, so that $71\equiv 1237-w_{71}\pmod{7}$. There is a similar problem with a $7$-congruence
1238between \nf{260k6A} and \nf{260k6E} --- here $w_{13}=1$ so that
1239$13\equiv -w_{13}\pmod{7}$. Finally, there is a $5$-congruence between
1240\nf{116k6A} and \nf{116k6D}, but here the prime~$5$ is less than the
1241weight~$6$.
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1335
1336
1337\end{document}
1338127k4A    43  127k4C   17   [43]
1339159k4A  5,23  159k4E    8   [5]x[23]
1340365k4B    29  365k4E   18   [29]x[5] (extra factor of 5 divides the level)
1341369k4A  5,13  369k4J    9   [5]x[13]x[2]
1342453k4A  5,17  453k4E   23   [5]x[17]
1343453k4A    23  151k4A   30   Odd func eq for g
1344465k4A    11  465k4H    7   [11]x[5]x[2]
1345477k4A    73  477k4M   12   [73]x[2]
1346567k4A    23  567k4I    8   [23]x[3]
134781k4A     13  567k4L   12   Odd func eq for f, Theorem 4.1 gives nothing.
1348581k4A 19,19  581k4E   34   [19^2]x[4] (possible BIG Sha, as 19^2 divides MD)
1349639k4A     7  639k4H   12   [7]
1350657k4A     5  657k4C    7   [5]x[3]x[2] (see next note)
1351657k4A     5  657k4G   12   [5]x[4] (does 657k4A make both these visible?)
1352681k4A    43  227k4A   23   Odd func eq for g
1353681k4A    59  681k4D   30   [59]x[3]x[2]
1354684k4C   7,7  684k4K    4   [7^2]x[2] (see note to 581k4A)
135595k6A  31,59   95k6D    9   [31]x[59]
1356116k6A     5  116k6D    6   [5]x[29]x[2]
1357122k6A     7  61k6B    14   7^2 appears in L(61k6B,3)
1358122k6A    73  122k6C    6   [73]x[3] (guess that 3 is a bad prime now)
1359260k6A  7,17  260k6E    4   [7]x[17]x[4] <-- Did not compute MD or LROP
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372\nf{263k4B} &  & 41^2 & \\
1373\nf{269k4C} &  & 23^2 & \\
1374\nf{271k4B} &  & 29^2 &\\
1375\nf{281k4B} &  & 29^2\\
1376\hline
1377\nf{101k6B} &  & 17^2 & 101\\
1378\nf{103k6B} &  & 23^2\\
1379\nf{111k6C} &  & 11^2\\
1380\nf{153k6G} &  & 7^2\\
1381\nf{157k6B} &  & 252^2\\
1382\nf{167k6B} &  & 41^2\\
1383\nf{172k6B} &  & 7^2\\
1384\nf{173k6B} &  & 71^2\\
1385\nf{181k6B} &  & 107^2\\
1386\hline
1387\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571\\
1388