CoCalc Shared Fileswww / papers / motive_visibility / motive_visibility_v4.tex
Author: William A. Stein
1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2%
3% motive_visibility_v4.tex
4%
5% Project of William Stein, Neil Dummigan, Mark Watkins
6%
7%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
8
9\documentclass{amsart}
10\usepackage{amssymb}
11\usepackage{amsmath}
12\usepackage{amscd}
13
14\newcommand{\edit}[1]{\footnote{#1}\marginpar{\hfill {\sf\thefootnote}}}
15
16\newtheorem{prop}{Proposition}[section]
17\newtheorem{defi}[prop]{Definition}
18\newtheorem{conj}[prop]{Conjecture}
19\newtheorem{lem}[prop]{Lemma}
20\newtheorem{thm}[prop]{Theorem}
21\newtheorem{cor}[prop]{Corollary}
22\newtheorem{examp}[prop]{Example}
23\newtheorem{remar}[prop]{Remark}
24\newcommand{\Ker}{\mathrm {Ker}}
25\newcommand{\Aut}{{\mathrm {Aut}}}
26\def\id{\mathop{\mathrm{ id}}\nolimits}
27\renewcommand{\Im}{{\mathrm {Im}}}
28\newcommand{\ord}{{\mathrm {ord}}}
29\newcommand{\End}{{\mathrm {End}}}
30\newcommand{\Hom}{{\mathrm {Hom}}}
31\newcommand{\Mor}{{\mathrm {Mor}}}
32\newcommand{\Norm}{{\mathrm {Norm}}}
33\newcommand{\Nm}{{\mathrm {Nm}}}
34\newcommand{\tr}{{\mathrm {tr}}}
35\newcommand{\Tor}{{\mathrm {Tor}}}
36\newcommand{\Sym}{{\mathrm {Sym}}}
37\newcommand{\Hol}{{\mathrm {Hol}}}
38\newcommand{\vol}{{\mathrm {vol}}}
39\newcommand{\tors}{{\mathrm {tors}}}
40\newcommand{\cris}{{\mathrm {cris}}}
41\newcommand{\length}{{\mathrm {length}}}
42\newcommand{\dR}{{\mathrm {dR}}}
43\newcommand{\lcm}{{\mathrm {lcm}}}
44\newcommand{\Frob}{{\mathrm {Frob}}}
45\def\rank{\mathop{\mathrm{ rank}}\nolimits}
46\newcommand{\Gal}{\mathrm {Gal}}
47\newcommand{\Spec}{{\mathrm {Spec}}}
48\newcommand{\Ext}{{\mathrm {Ext}}}
49\newcommand{\res}{{\mathrm {res}}}
50\newcommand{\Cor}{{\mathrm {Cor}}}
51\newcommand{\AAA}{{\mathbb A}}
52\newcommand{\CC}{{\mathbb C}}
53\newcommand{\RR}{{\mathbb R}}
54\newcommand{\QQ}{{\mathbb Q}}
55\newcommand{\ZZ}{{\mathbb Z}}
56\newcommand{\NN}{{\mathbb N}}
57\newcommand{\EE}{{\mathbb E}}
58\newcommand{\HHH}{{\mathbb H}}
59\newcommand{\pp}{{\mathfrak p}}
60\newcommand{\qq}{{\mathfrak q}}
61\newcommand{\FF}{{\mathbb F}}
62\newcommand{\KK}{{\mathbb K}}
63\newcommand{\GL}{\mathrm {GL}}
64\newcommand{\SL}{\mathrm {SL}}
65\newcommand{\Sp}{\mathrm {Sp}}
66\newcommand{\Br}{\mathrm {Br}}
67\newcommand{\Qbar}{\overline{\mathbb Q}}
68%\newcommand{\Sha}{\underline{III}}
69%\newcommand{\Sha}{\amalg\kern-0.575em\amalg}
70% ---- SHA ----
71\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
72  \newcommand{\textcyr}[1]{%
73    {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
74     \selectfont #1}}
75\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
76
77\newcommand{\HH}{{\mathfrak H}}
78\newcommand{\aaa}{{\mathfrak a}}
79\newcommand{\bb}{{\mathfrak b}}
80\newcommand{\dd}{{\mathfrak d}}
81\newcommand{\ee}{{\mathbf e}}
82\newcommand{\Fbar}{\overline{F}}
83\newcommand{\CH}{\mathrm {CH}}
84
85\begin{document}
86\title{Constructing elements in
87Shafarevich-Tate groups of modular motives\\
88{\sc (NOT FOR DISTRIBUTION!)}}
89\author{Neil Dummigan}
90\author{William Stein}
91\author{Mark Watkins}
92\date{September 12, 2001}
93\subjclass{11F33, 11F67, 11G40.}
94
95\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
96Shafarevich-Tate group.}
97
98\address{University of Sheffield\\ Department of Pure
99Mathematics\\
100Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
101U.K.}
102\address{Harvard University\\Department of Mathematics\\
103One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
104\address{Penn State Mathematics Department\\
105University Park\\State College, PA 16802\\ U.S.A.}
106
107\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
108\email{watkins@math.psu.edu}
109
110\maketitle
111\section{Introduction}
112Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
113be the associated $L$-function. The conjecture of Birch and
114Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$
115at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
116also gives an interpretation of the leading term in the Taylor
117expansion in terms of various quantities, including the order of
118the Shafarevich-Tate group.
119
120Cremona and Mazur [2000] look, among all strong Weil elliptic
121curves over $\QQ$ of conductor $N\leq 5500$, at those with
122non-trivial Shafarevich-Tate group (according to the Birch and
123Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
124group has predicted elements of order $m$. In most cases they find
125another elliptic curve, often of the same conductor, whose
126$m$-torsion is Galois-isomorphic to that of the first one, and
127which has rank two. The rational points on the second elliptic
128curve produce classes in the common $H^1(\QQ,E[m])$. They expect
129that these lie in the Shafarevich-Tate group of the first curve,
130so rational points on one curve explain elements of the
131Shafarevich-Tate group of the other curve.
132
133The Bloch-Kato conjecture \cite{BK} is the generalisation to
134arbitrary motives of the leading term part of the Birch and
135Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
136\cite{B} generalises the part about the order of vanishing at the
137central point, identifying it with the rank of a certain Chow
138group.
139
140The present work may be considered as a partial generalisation of
141the work of Cremona and Mazur, from elliptic curves over $\QQ$
142(with which are associated modular forms of weight $2$) to the
143motives attached to modular forms of higher weight. (See \cite{AS}
144for a different generalisation, to modular abelian varieties of
145higher dimension.) It may also be regarded as doing, for
146congruences between modular forms of equal weight, what \cite{Du2}
147did for congruences between modular forms of different weights.
148
149We consider the situation where two newforms $f$ and $g$, both of
150weight $k>2$ and level $N$, are congruent modulo some $\qq$, and
151$L(g,s)$ vanishes to order at least $2$ at $s=k/2$. We are able to
152find eleven examples (all with $k=4$ and $k=6$), and in each case
153$\qq$ appears in the numerator of the algebraic number
154$L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$ is a certain
155canonical period. In fact, we show how this divisibility may be
156proved from the vanishing of $L(g,k/2)$ using recent work of
157Vatsal \cite{V}. The point is, the congruence between $f$ and $g$
158leads to a congruence between suitable algebraic parts'' of the
159special values $L(f,k/2)$ and $L(g,k/2)$. If one vanishes then the
160other is divisible by $\qq$. Under certain hypotheses, the
161Bloch-Kato conjecture then implies that the Shafarevich-Tate group
162attached to $f$ has non-zero $\qq$-torsion. Under certain
163hypotheses and assumptions, the most substantial of which is the
164Beilinson-Bloch conjecture relating the vanishing of $L(g,k/2)$ to
165the existence of algebraic cycles, we are able to construct the
166predicted elements of $\Sha$, using the Galois-theoretic
167interpretation of the congruences to transfer elements from a
168Selmer group for $g$ to a Selmer group for $f$. In proving the
169local conditions at primes dividing the level, and also in
170examining the local Tamagawa factors at these primes, we make use
171of a higher weight level-lowering result due to Jordan and Livn\'e
172\cite{JL}.
173
174One might say that algebraic cycles for one motive explain
175elements of $\Sha$ for the other. A main point of \cite{CM} was to
176observe the frequency with which those elements of $\Sha$
177predicted to exist for one elliptic curve may be explained by
178finding a congruence with another elliptic curve containing points
179of infinite order. One shortcoming of our work, compared to the
180elliptic curve case, is that, due to difficulties with local
181factors in the Bloch-Kato conjecture, we are unable to predict the
182exact order of $\Sha$. We have to start with modular forms between
183which there exists a congruence. However, Vatsal's work allows us
184to explain how the vanishing of one $L$-function leads, via the
185congruence, to the divisibility by $\qq$ of (an algebraic part of)
186another, independent of observations of computational data. The
187computational data does however show that there exist examples to
188which our results apply.
189
190\section{Motives and Galois representations}
191Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
192$\Gamma_0(N)$, with coefficients in an algebraic number field $E$.
193A theorem of Deligne \cite{De1} implies the existence, for each
194(finite) prime $\lambda$ of $E$, of a two-dimensional vector space
195$V_{\lambda}$ over $E_{\lambda}$, and a continuous representation
196$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
197such that
198\begin{enumerate}
199\item $\rho_{\lambda}$ is unramified at $p$ for all primes $p$ not dividing
200$lN$ (where $\lambda \mid l$);
201\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
202characteristic polynomial of $\Frob_p^{-1}$ acting on
203$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
204\end{enumerate}
205
206Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
207the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
208There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
209both $2$-dimensional $E$-vector spaces. For details of the
210construction see \cite{Sc}. The de Rham realisation has a Hodge
211filtration $V_{\dR}=F^0\supset F^1=\ldots =F^{k-1}\supset 212F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
213cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
214cohomology. There are natural isomorphisms $V_B\otimes 215E_{\lambda}\simeq V_{\lambda}$. We may choose a
216$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
217each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
218There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
219which amounts to multiplying the action of $\Frob_p$ by $p^j$.
220
221Following \cite{BK} (Section 3), for $p\neq l$ (including
222$p=\infty$) let
223$$H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow 224H^1(I_p,V_{\lambda}(j))).$$ The subscript $f$ stands for finite
225part''. $D_p$ is a decomposition subgroup at a prime above $p$,
226$I_p$ is the inertia subgroup, and the cohomology is for
227continuous cocycles and coboundaries. For $p=l$ let
228$$H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow 229H^1(D_l,V_{\lambda}(j)\otimes B_{\cris}))$$ (see Section 1 of
230\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
231$B_{dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
232elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
233in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes $p$.
234
235There is a natural exact sequence
236$$\begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.$$
237
238Let
239$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
240Define the $\lambda$-Selmer group \newline
241$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
242$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
243$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes $p$. Note that the
244condition at $p=\infty$ is superfluous unless $l=2$. Define the
245Shafarevich-Tate group
246$$\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/\pi_*H^1_f(\QQ,V_{\lambda}(j)).$$
247The length of its $\lambda$-component may be taken for the
248exponent of $\lambda$ in an ideal of $O_E$, which we call
249$\#\Sha(j)$. We shall only concern ourselves with the case
250$j=k/2$, and write $\Sha$ for $\Sha(j)$.
251
252Define the set of global points
253$$\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).$$
254This is analogous to the group of rational torsion points on an
255elliptic curve. The length of its $\lambda$-component may be taken
256for the exponent of $\lambda$ in an ideal of $O_E$, which we call
257$\#\Gamma_{\QQ}$.
258\section{Vatsal's work on congruences between special values}
259\section{The Bloch-Kato conjecture}
260Suppose that $L(f,k/2)\neq 0$. The Bloch-Kato conjecture for the
261motive $M_f(k/2)$ predicts that
262$${L(f,k/2)\over \vol_{\infty}}={\left(\prod_pc_p(k/2)\right)\#\Sha\over (\#\Gamma_{\QQ})^2}.$$
263The local factors $\vol_{\infty}$ and $c_p(k/2)$ depend on the
264choice of a basis for $V_{\dR}$, but their product does not. For
265$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be
266$$\length H^1_f(\QQ_p,T_{\lambda}(j))_{\tors.}-\ord_{\lambda}((1-a_pp^{-j}+p^{k-1-2j})).$$
267We omit the definition of $\ord_{\lambda}(c_p(j))$ for
268$\lambda\mid p$, which requires one to assume Fontaine's de Rham
269conjecture (\cite{Fo}, Appendix A6). (We shall mainly be concerned
270with the $q$-part of the Bloch-Kato conjecture, where $q$ is a
271prime of good reduction. For such primes, the de Rham conjecture
272follows from Theorem 5.6 of \cite{Fa1}.) The above formula is to
273be interpreted as an equality of fractional ideals of $E$.
274(Strictly speaking, the conjecture in \cite{BK} is only given for
275$E=\QQ$.)
276
277\begin{lem} Let $p\nmid N$ be a prime, $j$ an integer.
278Then the fractional ideal $c_p(j)$ is supported at most on
279divisors of $p$.
280\end{lem}
281\begin{proof}
282As on p. 30 of \cite{Fl1}, for odd $l\neq p$,
283$\ord_{\lambda}(c_p(j))$ is the length of the finite
284$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
285where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
286trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
287torsion-free.
288\end{proof}
289\begin{lem}\label{local1}
290Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
291is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
292$\qq\mid q$, and that, for all primes $p\mid N$, $\,p\neq \pm 2931\pmod{q}$. Suppose also that $f$ is not congruent modulo $\qq$ to
294any newform of weight~$k$, trivial character and level dividing
295$N/p$, with~$p$ any prime that exactly divides~$N$. Then for any
296$p\mid N$, and $j$ an integer, $\ord_{\qq}(c_p(j))=0$.
297\end{lem}
298\begin{proof} Bearing in mind that
299$$\ord_{\qq}((1-a_pp^{-j}+p^{k-1-2j}))=\#H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}),$$
300and also the long exact sequence in $I_p$-cohomology arising from
301$$\begin{CD}0@>>>T_{\qq}(j)@>>>V_{\qq}(j)@>\pi>>A_{\qq}(j)@>>>0\end{CD},$$
302it suffices to show that
303$$\dim H^0(I_p,A[\qq](j))=\dim H^0(I_p,V_{\qq}(j)).$$
304If the dimensions differ then, given that $f$ is not congruent
305modulo $\qq$ to a newform of level strictly dividing $N$, and
306since $p\neq \pm 1\pmod{q}$, Propositions 3.1 and 2.3 of \cite{L}
307tell us that $A[\qq](j)$ is unramified at $p$ and that
308$\ord_p(N)=1$. (Here we are using Carayol's result that $N$ is the
309prime-to-$q$ part of the conductor of $V_{\qq}$ \cite{Ca1}.) But
310then Theorem 1 of \cite{JL} (which uses the condition $q>k$)
311implies the existence of a newform of weight $k$, trivial
312character and level dividing $N/p$, congruent to $g$ modulo $\qq$.
313This contradicts our hypotheses.
314\end{proof}
315\begin{lem} Let $\qq\mid q$ be a prime of $E$ such that $q\nmid 316N$. Choosing bases for $V_{\dR}$ and $V_B$ which locally at $\qq$
317are as in the previous section, we have $\ord_{\qq}(c_q)=0$.
318\end{lem}
319\begin{proof}
320\end{proof}
321
322\begin{lem} $\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0$ if $A[\lambda]$ is an
323irreducible representation of $\Gal(\Qbar/\QQ)$.
324\end{lem}
325This follows trivially from the definition.
326
327Putting together the above lemmas we arrive at the following:
328\begin{prop}\label{sha} Let $q\nmid N$ be an odd prime such that $q>k$.
329Let $\qq\mid q$ be a prime of $E$ such that $A[\qq]$ is an
330irreducible representation of $\Gal(\Qbar/\QQ)$. Choose bases for
331$V_{\dR}$ and $V_B$ which locally at $\qq$ are as in the previous
332section. If
333$$\ord_{\qq}(L(f,k/2)/\vol_{\infty}>0$$
334(with numerator non-zero) then the Bloch-Kato conjecture predicts
335that $\ord_{\qq}(\#\Sha)>0$.
336\end{prop}
337
338\section{Constructing elements of the Shafarevich-Tate group}
339Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
340weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
341large enough to contain all the coefficients $a_n$ and $b_n$.
342Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv 343g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. For $f$
344we have defined $V_{\lambda}$, $T_{\lambda}$ and $A_{\lambda}$.
345Let $V'_{\lambda}$, $T'_{\lambda}$ and $A'_{\lambda}$ be the
346corresponding objects for $g$. Let $A[\lambda]$ denote the
347$\lambda$-torsion in $A_{\lambda}$. Since $a_p$ is the trace of
348$\Frob_p^{-1}$ on $V_{\lambda}$, it follows from the Cebotarev
349Density Theorem that $A[\qq]$ and $A'[\qq]$, if irreducible, are
350isomorphic as $\Gal(\Qbar/\QQ)$-modules.
351
352Suppose that $L(g,k/2)=0$. If the sign in the functional equation
353is positive, this implies that the order of vanishing at $s=k/2$
354is at least $2$. According to the Beilinson-Bloch conjecture
355\cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$ is the
356rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational,
357null-homologous, codimension $k/2$ algebraic cycles on the motive
358$M_g$, modulo rational equivalence. (This generalises the part of
359the Birch-Swinnerton-Dyer conjecture which says that for an
360elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at
361$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)
362
363Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
364to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
365subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
366If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
367get (assuming also the Beilinson-Bloch conjecture) a subspace of
368$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
369vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
370conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
371equal to the order of vanishing of $L(g,s)$ at $s=k/2$, but it
372seems a shame not to mention the cycles.
373
374Similarly, for $j\geq k/2$, the rank of $H^1_f(\QQ,V_{\qq}(j))$ is
375conjectured to be the order of vanishing of $L(f,s)$ at $s=k-j$.
376If $L(f,k/2)\neq 0$ then we expect that
377$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
378coincides with the $\qq$-part of $\Sha$.
379\begin{thm}\label{local}
380Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
381$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
382$A[\qq]$ and $A'[\qq]$ are irreducible representations of
383$\Gal(\Qbar/\QQ)$ and that, for all primes $p\mid N$, $\,p\neq \pm 3841\pmod{q}$. Suppose also that neither $f$ nor $g$ is congruent
385modulo $\qq$ to any newform of weight~$k$, trivial character and
386level dividing $N/p$, with~$p$ any prime that exactly divides~$N$.
387Then the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
388$\FF_{\qq}$-rank at least $r-1$.
389\end{thm}
390\begin{proof}
391Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
392continuity and rescaling we may assume that it lies in
393$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq 394H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction we get a non-zero
395class $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$. By
396irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial, so
397$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
398we get a non-zero, $\qq$-torsion class $\gamma\in 399H^1(\QQ,A_{\qq}(k/2))$.
400
401Our aim is to show that $\res_p(\gamma)\in 402H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
403consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
404The case $p=q$ does not quite work, which is the reason for the
405$r-1$'' in the statement of the theorem.
406
407\begin{enumerate}
408\item {\bf $p\nmid qN$. }
409
410Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p, 411A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
412$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
413$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
414follows from $d\in H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of
415$\gamma$ in $H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p. 125
416of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
417contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
418to $H^1(I_p,A_{\qq}(k/2))$,  so we have shown that
419$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
420
421\item {\bf $p\mid N$. }
422
423First we must show that $H^0(I_p, A'_{\qq}(k/2))$ is
424$\qq$-divisible. It suffices to show that
425$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
426but this may be done as in the proof of Lemma\ref{local1}. It
427follows as above that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in
428$H^1(I_p,A[\qq](k/2))$ is zero. $\res_p(c)$ then comes from
429$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
430order of this group is the same as the order of the group
431$H^0(\QQ_p,A[\qq](k/2))$, which we claim is trivial. By the work
432of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is
433ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As
434above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim 435H^0(I_p,A[\qq](k/2))$, so we need only consider the case where
436this common dimension is $1$. The (motivic) Euler factor at $p$
437for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts
438as multiplication by $\alpha$ on the one-dimensional space
439$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of
440\cite{Ca1} that this is the same as the Euler factor at $p$ of
441$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then
442follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where
443$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,
444$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on
445$H^0(I_p,A[\qq](k/2))$ as $\pm p^{-1}$. Since $p\neq \pm 4461\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence
447$\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in
448$H^1_f(\QQ_p,A_{\qq}(k/2))$.
449
450\item {\bf $p=q$. }
451
452Since $q\nmid N$ is a prime of good reduction for the motive
453$M_g$, $\,V'_{\qq}$ is a crystalline representation of
454$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
455$V'_{\qq}$ have the same dimension, where
456$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q} 457B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
458It follows from Theorem 4.1(ii) of \cite{BK} that
459$$D_{\dR}(V'_{\qq})/(F^{k/2}D_{\dR}(V'_{\qq}))\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)),$$
460via their exponential map.
461
462Since $p$ is a prime of good reduction, the de Rham conjecture is
463a consequence of the crystalline conjecture, which follows from
464Theorem 5.6 of \cite{Fa1}. Hence
465$$(V'_{\dR}\otimes_E E_{\qq})/(F^{k/2}V'_{\dR}\otimes_E E_{\qq})\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)).$$
466Observe that the dimension of the left-hand-side is $1$. Hence
467there is a subspace of $H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension at
468least $r-1$, mapping to zero in $H^1(\QQ_q,V'_{\qq}(k/2))$. It
469follows that our $r$-dimensional $\FF_{\qq}$-subspace of the
470$\qq$-torsion in $H^1(\QQ,A_{\qq}(k/2))$ has at least an
471$(r-1)$-dimensional subspace landing in
472$H^1_f(\QQ_{\qq},A_{\qq}(k/2))$ (in fact, mapping to zero).
473
474\end{enumerate}
475\end{proof}
476
477Theorem 2.7 of \cite{AS} is concerned with verifying local
478conditions in the case $k=2$, where $f$ and $g$ are associated
479with abelian varieties $A$ and $B$. (Their theorem also applies to
480abelian varieties over number fields.) Our restriction outlawing
481congruences modulo $\qq$ with cusp forms of lower level is
482analogous to theirs forbidding $q$ from dividing Tamagawa factors
483$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
484curve with $\ord_l(j(A))<0$, consideration of a Tate
485parametrisation shows that if $q\mid c_{A,l}$, i.e. if
486$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
487at $l$.)
488
489\section{Eleven examples}
490\newcommand{\nf}[1]{\mbox{\bf #1}}
491\begin{figure}
492\caption{\label{fig:newforms}Newforms Relevant to
493Theorem~\ref{local}}
494$$495\begin{array}{|ccccc|}\hline 496 g & \deg(g) & f & \deg(f) & q's \\\hline 497\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\ 498\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\ 499\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\ 500\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\ 501\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\ 502\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\ 503\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\ 504\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\ 505\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19 \\ 506\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\ 507\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\ 508\hline 509\end{array} 510$$
511\end{figure}
512
513\edit{Is it possible to stop this table appearing part way through
514the proof of the previous theorem?} Table~\ref{fig:newforms} on
515page~\pageref{fig:newforms} lists eleven pairs of newforms~$f$
516and~$g$ (of equal weights and levels) along with at least one
517prime~$q$ such that there is a prime $\qq\mid q$ with $f\equiv 518g\pmod{\qq}$. In each case, $\ord_{s=k/2}L(g,k/2)\geq 2$ while
519$L(f,k/2)\neq 0$.
520\subsection{Notation}
521Table~\ref{fig:newforms} is laid out as follows.
522The first column contains a label whose structure is
523\begin{center}
524{\bf [Level]k[Weight][GaloisOrbit]}
525\end{center}
526This label determines a newform $g=\sum a_n q^n$, up to Galois
527conjugacy. For example, \nf{127k4C} denotes a newform in the third
528Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
529orbits are ordered first by the degree of $\QQ(\ldots, a_n, 530\ldots)$, then by $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not dividing
531the level, with positive trace being first in the event that the
532two absolute values are equal, and the first Galois orbit is
533denoted {\bf A}, the second {\bf B}, and so on. The second column
534contains the degree of the field $\QQ(\ldots, a_n, \ldots)$. The
535third and fourth columns contain~$f$ and its degree, respectively.
536The fifth column contains at least one prime~$q$ such that there
537is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that
538the hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
539satisfied for~$f$,~$g$, and~$\qq$.
540
541\subsection{The first example in detail}
542\newcommand{\fbar}{\overline{f}}
543We describe the first line of Table~\ref{fig:newforms}
544in more detail.  See the next section for further details
545on how the computations were performed.
546
547Using modular symbols, we find that there is a newform
548$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots 549\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
550the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
551also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
552coefficients generate a number field~$K$ of degree~$17$, and by
553computing the image of the modular symbol $XY\{0,\infty\}$ under
554the period mapping, we find that $L(f,2)\neq 0$.  The newforms~$f$
555and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
556characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
557both equal to
558$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots\in \FF_{43}[[q]].$$
559
560There is no form in the Eisenstein subspaces of
561$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
562$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
563$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
564prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
565level~$1$ form of weight~$4$. Thus we have checked the hypotheses
566of Theorem~\ref{local}, so if $r$ is the dimension of
567$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
568$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r-1$.
569
570Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,
571since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup
572of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion
573subgroup of $\Sha$. Admitting these assumptions, we have
574constructed the $\qq$-torsion in $\Sha$ predicted by the
575Bloch-Kato conjecture.
576
577For particular examples of elliptic curves one can often find and
578write down rational points predicted by the Birch and
579Swinnerton-Dyer conjecture. It would be nice if likewise one could
580explicitly produce algebraic cycles predicted by the
581Beilinson-Bloch conjecture in the above examples. Since
582$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
5830.3.2 of \cite{Z}), so ought to be trivial in
584$\CH_0^{k/2}(M_g)\otimes\QQ$.
585
586\subsection{Some remarks on how the computation was performed}
587We give a brief summary of how the computation was performed.  The
588algorithms that we used were implemented by the second author, and
589most are a standard part of the MAGMA V2.8 (see \cite{magma}).
590
591Let~$g$,~$f$, and~$q$ be some data from a line of
592Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We
593verified the existence of a congruence modulo~$q$, that
594$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq 5950$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
596not arise from any $S_k(\Gamma_0(N/p))$, as follows:
597
598To prove there is a congruence, we showed that the corresponding
599{\em integral} spaces of modular symbols satisfy an appropriate
600congruence, which forces the existence of a congruence on the
601level of Fourier expansions.  We showed that $\rho_{g,\qq}$ is
602irreducible by computing a set that contains all possible residue
603characteristics of congruences between~$g$ and any Eisenstein
604series of level dividing~$N$, where by congruence, we mean a
605congruence for all Fourier coefficients of index~$n$ with
606$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
607form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
608listing a basis of such~$h$ and finding the possible congruences,
609where again we disregard the Fourier coefficients of index not
610coprime to~$N$.
611
612To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
613modular symbol ${\mathbf 614e}=X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}\{0,\infty\}$ under a
615map with the same kernel as the period mapping, and found that the
616image was~$0$.  The period mapping sends the modular
617symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
618so that ${\mathbf e}$ maps to~$0$ implies that
619$L(g,\frac{k}{2})=0$. In a similar way, we verified that
620$L(f,\frac{k}{2})\neq 0$.  Next, we checked that $W_N(g) = g$
621which, because of the functional equation, implies that
622$L'(g,\frac{k}{2})=0$.
623
624In the course of our search, we found~$17$ plausible examples and
625had to discard~$6$ of them because either the representation
626$\rho_{f,\qq}$ arose from lower level, was reducible, or $p\equiv 627\pm 1\pmod{q}$ for some  $p\mid N$.  Table~\ref{fig:newforms} is
628of independent interest because it includes examples of modular
629forms of prime level and weight $>2$ with a zero at $\frac{k}{2}$
630that is not forced by the functional equation.  We found no such
631examples of weights $\geq 8$.
632
633
634
635
636\section{Some Remarks on Visibility}
637It would be nice to add a section on visibility here.  I.e.,
638systematically list examples where the rational part of the central
639value has a big square factor, which {\em probably} should be
640nontrivial $\Sha$, and say whether or not the theory of this paper
641predicts that it is visible.  Answer the question: Does it look like
642all Sha be visible for motives?'' probably no''.
643
644
645\begin{thebibliography}{AL}
646\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
647$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
648\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
649Shafarevich-Tate groups of abelian varieties: evidence for the
650Birch and Swinnerton-Dyer conjecture, in preparation.
651\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
652{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
653\bibitem[BCP]{magma}
654W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
655  {T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
656  235--265, Computational algebra and number theory (London, 1993).
657\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
658of motives, The Grothendieck Festschrift Volume I, 333--400,
659Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
660\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques
661associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
662\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
663\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
664Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em
665Duke Math. J. }{\bf 59 }(1989), 785--801.
666\bibitem[CM]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
667Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
66813--28.
669\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
670$\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math.
671{\bf 179, } 139--172, Springer, 1969.
672\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
673d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
674part 2, 313--346.
675\bibitem[DFG]{DFG} F. Diamond, M. Flach, L. Guo, Adjoint motives
676of modular forms and the Tamagawa number conjecture, preprint.
677{{\sf
678http://www.andromeda.rutgers.edu/\~{\mbox{}}krm/liguo/lgpapers.html}}
679\bibitem[Du1]{Du1} N. Dummigan, Period ratios of modular forms, {\em
680Math. Ann. }{\bf 318 }(2000), 621--636.
681\bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and
682Selmer groups, {\em Math. Res. Lett. }{\bf 8 }(2001), 479--494.
683\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
684Galois representations, {\em in }Algebraic analysis, geometry and
685number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
686Press, Baltimore, 1989.
687\bibitem[Fl2]{Fl2} M. Flach, A generalisation of the Cassels-Tate
688pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.\edit{I
689switched these two Flach references, because they were out of chrono
690order. --was}
691\bibitem[Fl1]{Fl1} M. Flach, On the degree of modular parametrisations,
692S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
693ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
694Boston Berlin, 1993.
695\bibitem[Fo]{Fo} J.-M. Fontaine, Sur certains types de
696repr\'esentations $p$-adiques du groupe de Galois d'un corps
697local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
698}{\bf 115 }(1982), 529--577.
699\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture epsilon''
700for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
70151--56.
702\bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois
703representations coming from modular forms, {\em J. Number Theory
704}{\bf 31 }(1989), 133--141.
705
706\bibitem[Ne1]{Ne1} J. Nekov\'ar, Kolyvagin's method for Chow groups of
707Kuga-Sato varieties, {\em Invent. Math. }{\bf 107 }(1992),
70899--125.
709\bibitem[Ne2]{Ne2} J. Nekov\'ar, $p$-adic Abel-Jacobi maps and $p$-adic
710heights. The arithmetic and geometry of algebraic cycles (Banff,
711AB, 1998), 367--379, CRM Proc. Lecture Notes, 24, Amer. Math.
712Soc., Providence, RI, 2000.
713\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
714{\em Invent. Math. }{\bf 100 }(1990), 419--430.
715\bibitem[SwD]{SwD} H. P. F. Swinnerton-Dyer, On $\ell$-adic representations and
716congruences for coefficients of modular forms,{\em Modular
717functions of one variable} III, Lect. Notes Math. {\bf 350, }
718Springer, 1973.
719\bibitem[V]{V} V. Vatsal, Canonical periods and congruence
720formulae, {\em Duke Math. J. }{\bf 98 }(1999), 397--419.
721\bibitem[Z]{Z} S. Zhang, Heights of Heegner cycles and derivatives of $L$-series,
722{\em Invent. Math. }{\bf 130 }(1997), 99--152.
723\end{thebibliography}
724
725
726\end{document}
727\subsection{Plan for finishing the computation}
728
729What I (William) have to do with the below stuff:\\
730\begin{enumerate}
731\item Fill in the missing blanks at these levels:
732$$453, 639, 657, 681, 684, 95, 116, 122, 260$$
733\item Make a MAGMA program that has a table of forms to try.
734\item First, unfortunately, for some reason the
735labels are sometimes wrong, e.g., \nf{159k4A} should
736have been \nf{159k4B}, so I have to check all the
737rational forms to see which has $L(1)=0$.
738\item Try each level using the MAGMA program (this will take a long time to run).
739\item Finish the Table of examples'' above.
740\end{enumerate}
741
742
743\begin{verbatim}
744
745
746(This stuff below is being integrated into the above, as I do
747the required (rather time consuming) computations.)
748**************************************************
749FROM Mark Watkins:
750
751Here's a list of stuff; the left-hand (sinister) column
752contains forms with a double zero at the central point,
753whilst the right-hand (dexter) column contains forms which
754have a large square factor in LRatio at 2. The middle column
755is the large prime factor in ModularDegree of the LHS,
756and/or the lpf in the LRatio at 2 of the RHS.
757
758It seems that 567k4L has invisible Sha possibly.
759The ModularDegree for 639k4B has no large factors.
760
761127k4A  43  127k4C
762159k4A  23  159k4E
763365k4A  29  365k4E
764369k4A  13  369k4I
765453k4A  17  453k4E
766453k4A  23
767465k4A  11  465k4H
768477k4A  73  477k4L
769567k4A  23  567k4G
770        13  567k4L
771581k4A  19  581k4E
772639k4B  --
773
774
775Forms with spurious zeros to do:
776
777657k4A
778681k4A
779684k4B
78095k6A   31,59
781116k6A  --
782122k6A  73
783260k6A
784
785If we allow 5 and 7 to be small primes, then we get more
786visibility info.
787
788159k4A   5  159k4E
789369k4A   5  369k4I
790453k4A   5  453k4E
791639k4B   7  639k4H
792
793Maybe the general theorem does not apply to primes which are so small,
794but we might be able to show that they are OK in these specific cases.
795
796I checked the first three weight 6 examples with AbelianIntersection,
797but actually computing the LRatio bogs down too much. Will need
798new version.
799
80095k6A   31     95k6D
80195k6A   59     95k6D
802116k6A   5    116k6D
803122k6A  73    122k6D
804
805Incidentally, the LRatio for 581k4E has a power of 19^4.
806Perhaps not surprising, as the ModularDegree of 581k4A
807has a factor of 19^2. But maybe it means a bigger Sha.
808\end{verbatim}
809