CoCalc Shared Fileswww / papers / motive_visibility / motive_visibility_v2.tex
Author: William A. Stein
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21\newtheorem{lem}[prop]{Lemma}
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70%\newcommand{\Sha}{\underline{III}}
71%\newcommand{\Sha}{\amalg\kern-0.575em\amalg}
72% ---- SHA ----
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86
87\begin{document}
88\title{Constructing elements in
89Shafarevich-Tate groups of modular motives\\
90{\sc (NOT FOR DISTRIBUTION!)}}
91\author{Neil Dummigan}
92\author{William Stein}
93\author{Mark Watkins}
94\date{\today}
95\subjclass{11F33, 11F67, 11G40.}
96
97\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
98Shafarevich-Tate group.}
99
100\address{University of Sheffield\\ Department of Pure
101Mathematics\\
102Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
103U.K.}
105One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
107University Park\\State College, PA 16802\\ U.S.A.}
108
109\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
110\email{watkins@math.psu.edu}
111
112\begin{abstract}
113\end{abstract}
114
115\maketitle
116\section{Introduction}
117Mention \cite{CM}.
118
119\section{Motives and Galois representations}
120Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
121$\Gamma_0(N)$, with coefficients in an algebraic number field $E$.
122A theorem of Deligne \cite{De1} implies the existence, for each
123(finite) prime $\lambda$ of $E$, of a two-dimensional vector space
124$V_{\lambda}$ over $E_{\lambda}$, and a continuous representation
125$$126\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}), 127$$
128such that
129\begin{enumerate}
130\item $\rho_{\lambda}$ is unramified at $p$ for all primes $p$ not dividing
131$lN$ (where $\lambda \mid l$);
132\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
133characteristic polynomial of $\Frob_p^{-1}$ acting on
134$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
135\end{enumerate}
136
137Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
138the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
139There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
140both $2$-dimensional $E$-vector spaces. For details of the
141construction see \cite{Sc}. The de Rham realisation has a Hodge
142filtration
143$$144 V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset F^k=\{0\}. 145$$
146The Betti realisation $V_B$ comes from singular
147cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
148cohomology. There are natural isomorphisms $V_B\otimes 149E_{\lambda}\simeq V_{\lambda}$. Using a basis for singular
150cohomology with $\ZZ$-coefficients, we get
151$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-modules $T_{\lambda}$
152inside each $V_{\lambda}$. Define
153$A_{\lambda}=V_{\lambda}/T_{\lambda}$. There are two kinds of
154twist we shall have to consider. There is the Tate twist
155$V_{\lambda}(j)$ (for an integer $j$), which amounts to
156multiplying the action of $\Frob_p$ by $p^j$. For $D$ the
158$V_{\lambda}(\chi_D)$, which is the tensor product of
159$V_{\lambda}$ with a one-dimensional space on which
160$\Gal(\Qbar/\QQ)$ acts via the quadratic character $\chi_D$.
161
162Following \cite{BK} (Section 3), for $p\neq l$ (including
163$p=\infty$) let
164$$H^1_f(\QQ_p,V_{\lambda}(\chi_D,j))=\ker (H^1(D_p,V_{\lambda}(\chi_D,j))\rightarrow 165H^1(I_p,V_{\lambda}(\chi_D,j))).$$ The subscript $f$ stands for
166finite part''. $D_p$ is a decomposition subgroup at a prime
167above $p$, $I_p$ is the inertia subgroup, and the cohomology is
168for continuous cocycles and coboundaries. For $p=l$ let
169$$H^1_f(\QQ_l,V_{\lambda}(\chi_D,j))=\ker (H^1(D_l,V_{\lambda}(\chi_D,j))\rightarrow 170H^1(D_l,V_{\lambda}(\chi_D,j)\otimes B_{\cris}))$$ (see Section 1
171of \cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
172$B_{dR}$). Let $H^1_f(\QQ,V_{\lambda}(\chi_D,j))$ be the subspace
173of elements of $H^1(\QQ,V_{\lambda}(\chi_D,j))$ whose local
174restrictions lie in $H^1_f(\QQ_p,V_{\lambda}(\chi_D,j))$ for all
175primes $p$.
176
177There is a natural exact sequence
178$$\begin{CD}0@>>>T_{\lambda}(\chi_D,j)@>>>V_{\lambda}(\chi_D,j)@>\pi>>A_{\lambda}(\chi_D,j)@>>>0\end{CD}.$$
179
180Let
181$H^1_f(\QQ_p,A_{\lambda}(\chi_D,j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(\chi_D,j))$.
182Define the $\lambda$-Selmer group \newline
183$H^1_f(\QQ,A_{\lambda}(\chi_D,j))$ to be the subgroup of elements
184of $H^1(\QQ,A_{\lambda}(\chi_D,j))$ whose local restrictions lie
185in $H^1_f(\QQ_p,A_{\lambda}(\chi_D,j))$ for all primes $p$. Note
186that the condition at $p=\infty$ is superfluous unless $l=2$.
187Define the Shafarevich-Tate group
188$$\Sha(\chi_D,j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(\chi_D,j))/\pi_*H^1_f(\QQ,V_{\lambda}(\chi_D,j)).$$
189The length of its $\lambda$-component may be taken for the
190exponent of $\lambda$ in an ideal of $O_E$, which we call
191$\#\Sha(\chi_D,j)$. We shall only concern ourselves with the case
192$j=k/2$, and write $\Sha(\chi_D)$ for $\Sha(\chi_D,j)$.
193
194Define the set of global points
195$$\Gamma_{\QQ}(\chi_D)=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(\chi_D,k/2)).$$
196This is analogous to the group of rational torsion points on an
197elliptic curve. The length of its $\lambda$-component may be taken
198for the exponent of $\lambda$ in an ideal of $O_E$, which we call
199$\#\Gamma_{\QQ}(\chi_D)$.
200\section{The Bloch-Kato conjecture}
201Suppose that $L(f,k/2)\neq 0$. The Bloch-Kato conjecture for the
202motive $M_f(k/2)$ predicts that
203$${L(f,k/2)\over \vol_{\infty}}={\left(\prod_pc_p(k/2)\right)\#\Sha\over (\#\Gamma_{\QQ})^2}.$$
204The local factors $\vol_{\infty}$ and $c_p(k/2)$ depend on the
205choice of a basis for $V_{\dR}$, but their product does not. For
206$l\neq p$, $\ord_{\lambda}(c_p(\chi_D,j))$ is defined to be
207$$\length H^1_f(\QQ_p,T_{\lambda}(\chi_D,j))_{\tors.}-\ord_{\lambda}((1-\chi_D(p)a_pp^{-j}+p^{k-1-2j})).$$
208We omit the definition of $\ord_{\lambda}(c_p(\chi_D,j))$ for
209$\lambda\mid p$, which requires one to assume Fontaine's de Rham
210conjecture (\cite{Fo}, Appendix A6). The above formula is to be
211interpreted as an equality of fractional ideals of $E$. (Strictly
212speaking, the conjecture in \cite{BK} is only given for $E=\QQ$.)
213
214Similarly, if $L(f,\chi_D,k/2)\neq 0$ then the Bloch-Kato
215conjecture for $M_f(\chi_D,k/2)$ predicts that
216$${L(f,\chi_D,k/2)\over \vol_{\infty}(\chi_D)}={\left(\prod_pc_p(k/2,\chi_D)\right)\#\Sha(\chi_D)\over (\#\Gamma_{\QQ}(\chi_D))^2}.$$
217Note that if $L(f,k/2)\neq 0$ then it is necessary that
218$\chi_D(-N)=1$ if the sign in the functional equation is to allow
219$L(f,\chi_D,k/2)\neq 0$. A careful treatment of real periods such
220as $\vol_{\infty}(\chi_D)$ is given in \cite{De2}. The calculation
221of the periods of Artin motives in terms of generalized Gauss
222sums, in Section 6 of \cite{De2}, yields the following lemma.
223\begin{lem}
224If $\chi_D(-1)=1$ (i.e. if $D>0$) then
225$\vol_{\infty}(\chi_D)=\vol_{\infty}/\sqrt{D}.$
226\end{lem}
227
228\begin{lem} Let $p\nmid N$ be a prime, $j$ an integer
229and $D$ a quadratic discriminant. Then the fractional ideal
230$c_p(\chi_D,j)$ is supported at most on divisors of $2$ and
231divisors of $p$, while $c_p(j)$ is supported at most on divisors
232of $p$.
233\end{lem}
234\begin{proof}\edit{I switched to using proof environments
235instead of doing it by hand. -- was}
236As on p. 30 of \cite{Fl1}, for odd $l\neq p$,
237$\ord_{\lambda}(c_p(\chi_D,j))$ is the length of the finite
238$O_{\lambda}$-module
239$H^0(\QQ_p,H^1(I_p,T_{\lambda}(\chi_D,j))_{\tors}),$ where $I_p$
240is an inertia group at $p$. If $p$ does not divide $D$ then
241$T_{\lambda}(\chi_D,j)$ is a trivial $I_p$-module, so
242$H^1(I_p,T_{\lambda}(\chi_D,j))$ is torsion-free. In general,
243since $l\neq p$,$$H^1(I_p,T_{\lambda}(\chi_{D},j))\simeq 244T_{\lambda}(\chi_{D},j)/(1-\tau)T_{\lambda}(\chi_{D},j),$$ where
245$\tau$ is a topological generator of the tame quotient of $I_p$.
246If $p\mid D$ then thanks to the quadratic twist ramified at $p$,
247$\,\tau$ acts as multiplication by $-1$ on
248$T_{\lambda}(\chi_{D},j)$, hence
249$H^1(I_p,T_{\lambda}(\chi_{D},j))$ is trivial (remember $l\neq 2502$).
251\end{proof}
252
253The assertion about $c_p(j)$ may be proved similarly to the case
254$p\nmid D$ above.
255
256
257\begin{lem} Let $D$ be a quadratic discriminant, and $p$ a prime such that
258$\chi_D(p)=1$. Then $c_p(\chi_D,j)=c_p(j)$.
259\end{lem}
260\begin{proof}
261This is simply a consequence of the fact that the
262definition of $\ord_{\lambda}(c_p(j))$ depends only on
263$T_{\lambda}$ as a $\Gal(\Qbar_p/\QQ_p)$-module.
264\end{proof}
265
266\begin{lem} $\ord_{\lambda}(\#\Gamma_{\QQ}(\chi_D,j))=0$ unless
267the coefficients of $f=\sum a_nq^n$ satisfy the congruence
268$a_p\equiv \chi_D(p)(p^j+p^{k-1-j})\pmod{\lambda}$ for all primes
269$p\nmid lN$.
270\end{lem}
271This follows from the interpretation of $a_p$ as a trace of
272Frobenius. See \cite{SwD}.
273
274Putting together the above lemmas we arrive at the following:
275\begin{prop}\label{sha} Let $q\nmid N$ be an odd prime. Let $D>0$ be a
276quadratic discriminant such that $\chi_D(q)=1$ and $\chi_D(p)=1$
277for all primes $p\mid N$. Let $\qq\mid q$ be a prime of $E$ such
278that neither
279$$a_p\equiv \chi_D(p)(p^{k/2}+p^{(k/2)-1})\pmod{\qq}$$
280nor
281$$a_p\equiv p^{k/2}+p^{(k/2)-1}\pmod{\qq}$$
282holds for all $p\nmid qN$. If
283$$\ord_{\qq}(L(f,k/2)/\sqrt{D}L(f,\chi_D,k/2))>0$$
284(with numerator and denominator both non-zero) then the Bloch-Kato
285conjecture predicts that $\ord_{\qq}(\#\Sha)>0$.
286\end{prop}
287\begin{prop}\label{triv}
288Suppose that $p\nmid DN\phi(N)$, $p>k$ and that $f$ is not
289congruent to another cusp form for $\Gamma_0(N)$ modulo any
290divisor of $p$. Then, for $(k/2)\leq j\leq k-1$, $c_p(j)$ is
291trivial and $c_p(\chi_D,j)$ is at worst supported on divisors of
292$2$.
293\end{prop}
294This may be proved in a similar manner to Theorem 7.6 of
295\cite{Du1}. The conditions in the statement of the proposition
296ensure, among other things, that the denominator of the projector
297used to cut out the motive $M_f$ is coprime to $p$. This is
298essential if the proofs of the lemmas in Section 7 of \cite{Du1}
299are to carry across.
300
301Bearing in mind Flach's generalisation of the Cassels-Tate pairing
302\cite{Fl2}, we see the following corollary.
303\begin{cor} The Bloch-Kato conjecture predicts that, for $D$ as in
304Proposition \ref{sha}, the fractional ideal
305$(L(f,k/2)/\sqrt{D}L(f,\chi_D,k/2))$ is a square, up to divisors
306of $2$ and of primes not satisfying the conditions of Proposition
307\ref{triv}.
308\end{cor}
309\section{Constructing elements of the Shafarevich-Tate group}
310Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
311weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
312large enough to contain all the coefficients $a_n$ and $b_n$.
313Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv 314g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. For $f$
315we have defined $V_{\lambda}$, $T_{\lambda}$ and $A_{\lambda}$.
316Let $V'_{\lambda}$, $T'_{\lambda}$ and $A'_{\lambda}$ be the
317corresponding objects for $g$. Let $A[\lambda]$ denote the
318$\lambda$-torsion in $A_{\lambda}$. Since $a_p$ is the trace of
319$\Frob_p^{-1}$ on $V_{\lambda}$, it follows from the Cebotarev
320Density Theorem that $A[\qq]$ and $A'[\qq]$, if irreducible, are
321isomorphic as $\Gal(\Qbar/\QQ)$-modules.
322
323Suppose that $L(g,k/2)=0$. If the sign in the functional equation
324is positive, this implies that the order of vanishing at $s=k/2$
325is at least $2$. According to the Beilinson-Bloch conjecture
326\cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$ is the
327rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational,
328null-homologous, codimension $k/2$ algebraic cycles on the motive
329$M_g$, modulo rational equivalence.
330
331Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
332to $H^1(\QQ,V'_{\qq}(k/2))$, and Nekov\'ar shows in \cite{Ne2}
333that its image is contained in the subspace
334$H^1_f(\QQ,V'_{\qq}(k/2))$. See also 0.13 of \cite{Ne1}. If, as
335expected, the $\qq$-adic Abel-Jacobi map is injective, we get
336(assuming also the Beilinson-Bloch conjecture) a subspace of
337$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
338vanishing of $L(g,s)$ at $s=k/2$.
339
340If $L(f,k/2)\neq 0$ then it is expected that
341$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that
342$\Sha=H^1_f(\QQ,A_{\qq}(k/2))$.\edit{Does one really get {\em all} of
343$\Sha$ as you claim, or just the $\qq$-torsion or $\qq$-part??}
344(For $j\geq k/2$, the rank of
345$H^1_f(\QQ,V_{\qq}(j))$ is expected to be the order of vanishing
346of $L(f,s)$ at $s=k-j$.)
347\begin{thm}\label{local}
348Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
349$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
350$A[\qq]$ and $A'[\qq]$ are irreducible representations of
351$\Gal(\Qbar/\QQ)$ and that, for all primes $p\mid N$, $\,p\neq \pm 3521\pmod{q}$. Suppose also that neither $f$ nor $g$ is congruent
353modulo $\qq$ to any newform\edit{I added this weight~$k$
354assertion. -- was} of weight~$k$, trivial character and level
355dividing $N/p$, with~$p$ any\edit{Change a'' to any''. -- was}
356prime that exactly divides~$N$. Then
357the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
358$\FF_{\qq}$-rank at least $r-1$.
359\end{thm}
360\begin{proof}\edit{This proof is really long.  Could it benefit by
361being broken up into more maneagable chunks (lemmas, etc.)?
362If nothing else, a summary paragraph at the beginning would be
363useful. -- was}
364Take a non-zero class $d\in 365H^1_f(\QQ,V'_{\qq}(k/2))$. By continuity and rescaling we may
366assume that it lies in $H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq 367H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction we get a non-zero
368class $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$. By
369irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial, so
370$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
371we get a non-zero class $\gamma\in H^1(\QQ,A_{\qq}(k/2))$.
372
373First we will show that $\res_p(\gamma)\in 374H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p\nmid qN$.
375Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p, 376A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
377$H^1(I_p,A'[\qq](k/2))$ injects into $H^1(I_p,A'_{\qq}(k/2))$, and
378it follows from $d\in H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of
379$\gamma$ in $H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p. 125
380of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
381contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
382to $H^1(I_p,A_{\qq}(k/2))$,  so we have shown that
383$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
384
385Next we will show that $\res_p(\gamma)\in 386H^1_f(\QQ_p,A_{\qq}(k/2))$ for $p\mid N$. Our first task is to
387show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible. It
388suffices to show that $$\dim H^0(I_p,A'[\qq](k/2))=\dim 389H^0(I_p,V'_{\qq}(k/2)).$$ If the dimensions differ then, given
390that $g$ is not congruent modulo $\qq$ to a newform of level
391strictly dividing $N$, and since $p\neq \pm 1\pmod{q}$,
392Propositions 3.1 and 2.3 of \cite{L} tell us that $A'[\qq](k/2)$
393is unramified at $p$ and that $\ord_p(N)=1$. (Here we are using
394Carayol's result that $N$ is the prime-to-$q$ part of the
395conductor of $V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of
396\cite{JL} (which uses the condition $q>k$) implies the existence
397of a newform of weight $k$, trivial character and level dividing
398$N/p$, congruent to $g$ modulo $\qq$. This contradicts our
399hypotheses, and we have established that $H^0(I_p, A'_{\qq}(k/2))$
400is $\qq$-divisible.
401
402It follows that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in
403$H^1(I_p,A[\qq](k/2))$ is zero. By inflation-restriction,
404$\res_p(c)$ then comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$.
405The order of this group is the same as the order of the group
406$H^0(\QQ_p,A[\qq](k/2))$, which we claim is trivial. By the work
407of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is
408ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As
409above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim 410H^0(I_p,A[\qq](k/2))$, so we need only consider the case where
411this common dimension is $1$. The (motivic) Euler factor at $p$
412for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts
413as multiplication by $\alpha$ on the one-dimensional space
414$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of
415\cite{Ca1} that this is the same as the Euler factor at $p$ of
416$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then
417follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where
418$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,
419$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on
420$H^0(I_p,A[\qq](k/2))$ as $\pm p^{-1}$. Since $p\neq \pm 4211\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence
422$\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in
423$H^1_f(\QQ_p,A_{\qq}(k/2))$.
424
425It remains to deal with the local condition at $p=q$. Since
426$q\nmid N$ is a prime of good reduction for the motive $M_f$,
427$\,V_{\qq}$ is a crystalline representation of
428$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V_{\qq})$ and $V_{\qq}$
429have the same dimension, where
430$D_{\cris}(V_{\qq}):=H^0(\QQ_q,V_{\qq}\otimes_{\QQ_q} B_{\cris})$.
431(This is a consequence of Theorem 5.6 of \cite{Fa1}.) It follows
432from Theorem 4.1(ii) of \cite{BK} that
433$$D_{\dR}(V_{\qq})/(F^{k/2}D_{\dR}(V_{\qq}))\simeq H^1_f(\QQ_q,V_{\qq}(k/2)),$$
434via their exponential map.
435
436Since $p$ is a prime of good reduction, the de Rham conjecture is
437a consequence of the crystalline conjecture, which follows from
438Theorem 5.6 of \cite{Fa1}. Hence
439$$(V_{\dR}\otimes_E E_{\qq})/(F^{k/2}V_{\dR}\otimes_E E_{\qq})\simeq H^1_f(\QQ_q,V_{\qq}(k/2)).$$
440Observe that the dimension of the left-hand-side is $1$.
441Meanwhile, by 3.8 of \cite{BK}, $H^1_f(\QQ_q,V_{\qq}(k/2))$ is its
442own annihilator in $H^1(\QQ_q,V_{\qq}(k/2))$ with respect to the
443Tate duality pairing. Hence $H^1_f(\QQ_q,V_{\qq}(k/2))$ has
444codimension one in $H^1(\QQ_q,V_{\qq}(k/2))$. It follows that our
445$r$-dimensional $\FF_{\qq}$-subspace of the $\qq$-torsion in
446$H^1(\QQ,A_{\qq}(k/2))$ has at least an $(r-1)$-dimensional
447subspace landing in $H^1_f(\QQ_{\qq},A_{\qq}(k/2))$.
448
449To justify this argument carefully, we should check that the
450natural map from $H^1(\QQ_q,V_{\qq}(k/2))$ to
451$H^1(\QQ_{\qq},A_{\qq}(k/2))$ is surjective. Its cokernel injects
452into \newline $H^2(\QQ_{\qq},T_{\qq}(k/2))$, which is dual to
453$H^0(\QQ_{\qq},A_{\qq}(k/2))$ (via local Tate duality). It
454suffices to check that $H^0(\QQ_{\qq},A[\qq](k/2))$ is trivial.
455This follows from the description of $A[\qq](k/2)$ as a
456$\Gal(\Qbar_q/\QQ_q)$-module provided by theorems of Deligne and
457Fontaine (Theorems 2.5 and 2.6 of \cite{Ed}).
458\end{proof}
459
460Theorem 2.7 of \cite{AS} is concerned with verifying local
461conditions in the case $k=2$, where $f$ and $g$ are associated
462with abelian varieties $A$ and $B$. (Their theorem also applies to
463abelian varieties over number fields.) Our restriction outlawing
464congruences modulo $\qq$ with cusp forms of lower level is
465analogous to theirs forbidding $q$ from dividing Tamagawa factors
466$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
467curve with $\ord_l(j(A))<0$, consideration of a Tate
468parametrisation shows that if $q\mid c_{A,l}$, i.e. if
469$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
470at $l$.)
471
472\section{Eleven examples of Theorem~\ref{local}}
473\newcommand{\nf}[1]{\mbox{\bf #1}}
474\begin{figure}
475\caption{\label{fig:newforms}Newforms Satisfying Theorem~\ref{local}}
476$$477\begin{array}{|ccccc|}\hline 478 f & \deg(f) & g & \deg(g) & q's \\\hline 479\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43\\ 480\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23\\ 481\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29\\ 482\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13\\ 483\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17\\ 484\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11\\ 485\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73\\ 486\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23\\ 487\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19\\ 488\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59\\ 489\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73\\ 490\hline 491\end{array} 492$$
493\end{figure}
494
495
496Table~\ref{fig:newforms} on page~\pageref{fig:newforms}
497lists eleven pairs of newforms~$f$ and~$g$
498along with at least one prime~$q$ such that there is a
499prime $\qq\mid q$ such that by Theorem~\ref{local}
500the Beilinson-Bloch conjecture
501implies that
502$$\dim_{\FF_\qq} \#H^1_f(\QQ,A_{\qq}(k/2))[\qq] > 0.$$
503Recall that $H^1_f(\QQ,A_{\qq}(k/2))[\qq]$ is a piece of the
504finite cohomology associated to the motive attached to~$g$.
505As discussed just before Theorem~\ref{local}, since $L(g,\frac{k}{2})\neq 0$,
506we expect that $H^1_f(\QQ,A_{\qq}(k/2))[\qq]$ equals the $\qq$-torsion
507of the Shafarevich-Tate group of the motive attached to~$g$.
508(Note also that the subscript of $f$ means finite'' and has nothing
509to do with the newform that we denote by $f$''.)
510
511\subsection{Notation}
512Table~\ref{fig:newforms} is laid out as follows.
513The first column contains a label whose structure is
514\begin{center}
515{\bf [Level]k[Weight][GaloisOrbit]}
516\end{center}
517This label determines a newform $f=\sum a_n q^n$, up to Galois conjugacy.
518For example, \nf{127k4C} denotes a
519newform in the third Galois orbit of
520newforms in $S_4(\Gamma_0(127))$.
521The Galois orbits are ordered
522first by the degree of $\QQ(\ldots, a_n, \ldots)$,
523then by $|\mbox{\rm Tr}(a_p(f))|$ for~$p$ not dividing the
524level, with positive trace being first in the event that the two absolute
525values are equal, and the first Galois orbit is denoted {\bf A}, the
526second {\bf B}, and so on.
527For the purposes of the present paper, the ordering is not important;
528if you repeat the experiment, you will quickly find which form
529must be~$f$ and which must be~$g$.
530The second column contains the degree
531of the field $\QQ(\ldots, a_n, \ldots)$.  The third and forth
532columns contain~$g$ and its degree, respectively.
533The fifth columns contains at least one prime~$q$
534such that there is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$,
535and so that the hypothesis of Theorem~\ref{local} are satisfied
536for~$f$,~$g$, and~$\qq$.
537
538\subsection{The first example in detail}
539\newcommand{\fbar}{\overline{f}}
540We describe the first line of Table~\ref{fig:newforms}
541in more detail.  See the next section for further details
542on how the computations were performed.
543
544Using modular symbols, we find that there is a newform
545$$f = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots 546\in S_4(\Gamma_0(127))$$
547with $L(f,2)=0$.
548Because $W_{127}(f)=f$,  the functional equation has sign~$+1$,
549so $L'(f,2)=0$ as well.
550We also find a newform $g \in S_4(\Gamma_0(127))$ whose
551Fourier coefficients generate a number field~$K$ of degree~$17$,
552and by computing the image of the modular symbol
553$XY\{0,\infty\}$ under the period mapping, we find
554that $L(g,2)\neq 0$.  The newforms~$f$ and~$g$ are congruent
555modulo a prime $\qq$ of~$K$ of residue characteristic~$43$.
556The mod~$\qq$ reductions of~$f$ and~$g$ are both equal to
557$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots\in \FF_{43}[[q]].$$
558There is no form in the Eisenstein subspaces of $M_4(\Gamma_0(127))$
559whose Fourier coefficients of index~$n$, with $(n,127)=1$, are
560congruent modulo $43$ to those of $\fbar$, so
561$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible.
562Since $127$ is prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$
563does not arise from a
564level~$1$ form of weight~$4$.  Thus, if we assume the Beilinson-Bloch
565conjecture, Theorem~\ref{local} implies that the Shafarevich-Tate
566group attached to~$g$ has order divisible by~$43$.
567
568\subsection{Some remarks on how the computation was performed}
569We give a brief summary of how the computation was performed.  The
570algorithms that we used were implemented by the second author, and
571most are a standard part of the MAGMA V2.8 (see \cite{magma}).
572
573Let~$f$,~$g$, and~$q$ be some data from a line of
574Table~\ref{fig:newforms} and let~$N$ denote the level of~$f$.
575We verified the existence of a congruence modulo~$q$, that
576$L(f,\frac{k}{2})=L'(f,\frac{k}{2})=0$ and $L(g,\frac{k}{2})\neq 0$,
577and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does not arise from
578any $S_k(\Gamma_0(N/p))$, as follows:
579
580To prove there is a congruence, we showed that the corresponding {\em
581integral} spaces of modular symbols satisfy an appropriate congruence,
582which forces the existence of a congruence on the level of Fourier
583expansions.  We showed that $\rho_{f,\qq}$ is irreducible by computing
584a set that contains all possible residue characteristics of congruences
585between~$f$ and any Eisenstein series of level dividing~$N$, where
586by congruence, we mean a congruence for all Fourier coefficients of index~$n$
587with $(n,N)=1$.
588Similarly, we checked that~$f$ is not congruent to any form~$h$ of
589level $N/p$ for any~$p$ that exactly divides~$N$ by listing a basis of
590such~$h$ and finding the possibly congruences, where again we
591disregard the Fourier coefficients of index not coprime to~$N$.
592
593To verify that $L(f,\frac{k}{2})=0$, we computed the image of the
594modular symbol ${\mathbf 595e}=X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}\{0,\infty\}$ under a map
596with the same kernel as the period mapping, and found that the image
597was~$0$.  The period mapping sends the modular symbol~${\mathbf e}$
598to a nonzero multiple of $L(f,\frac{k}{2})$, so that ${\mathbf e}$
599maps to~$0$ implies that $L(f,\frac{k}{2})=0$. In a similar way,
600we verified that $L(g,\frac{k}{2})\neq 0$.  Next, we checked
601that $W_N(f) = f$ which, because of the functional equation, implies
602that $L'(f,\frac{k}{2})=0$.
603
604%In the course of our search, we found~$17$ plausible examples and had
605%to discard~$6$ of them because either the representation $\rho_{f,\qq}$ arose
606%from lower level, was irreducible, or $p\equiv \pm 1\pmod{q}$ for
607%some  $p\mid N$.  Table~\ref{fig:newforms}
608%is of independent interest because it
609%includes examples of modular forms of prime level and weight
610%$>2$ with a zero at $\frac{k}{2}$ that is not forced
611%by the functional equation.  We found no such examples
612%of weights $\geq 8$.
613
614\section{Some Remarks on Visibility}
615It would be nice to add a section on visibility.  I.e.,
616systematically list examples where the rational part of the central
617value has a big square factor, which {\em probably} should be
618nontrivial $\Sha$, and say whether or not the theory of this paper
619predicts that it is visible.  Answer the question:
620Does it look like all Sha be visible for motives?'' probably
621no''.
622
623
624
625
626
627
628
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699\end{thebibliography}
700
701
702\end{document}
703\subsection{Plan for finishing the computation}
704
705What I (William) have to do with the below stuff:\\
706\begin{enumerate}
707\item Fill in the missing blanks at these levels:
708$$453, 639, 657, 681, 684, 95, 116, 122, 260$$
709\item Make a MAGMA program that has a table of forms to try.
710\item First, unfortunately, for some reason the
711labels are sometimes wrong, e.g., \nf{159k4A} should
712have been \nf{159k4B}, so I have to check all the
713rational forms to see which has $L(1)=0$.
714\item Try each level using the MAGMA program (this will take a long time to run).
715\item Finish the Table of examples'' above.
716\end{enumerate}
717
718
719\begin{verbatim}
720
721
722(This stuff below is being integrated into the above, as I do
723the required (rather time consuming) computations.)
724**************************************************
725FROM Mark Watkins:
726
727Here's a list of stuff; the left-hand (sinister) column
728contains forms with a double zero at the central point,
729whilst the right-hand (dexter) column contains forms which
730have a large square factor in LRatio at 2. The middle column
731is the large prime factor in ModularDegree of the LHS,
732and/or the lpf in the LRatio at 2 of the RHS.
733
734It seems that 567k4L has invisible Sha possibly.
735The ModularDegree for 639k4B has no large factors.
736
737127k4A  43  127k4C
738159k4A  23  159k4E
739365k4A  29  365k4E
740369k4A  13  369k4I
741453k4A  17  453k4E
742453k4A  23
743465k4A  11  465k4H
744477k4A  73  477k4L
745567k4A  23  567k4G
746        13  567k4L
747581k4A  19  581k4E
748639k4B  --
749
750
751Forms with spurious zeros to do:
752
753657k4A
754681k4A
755684k4B
75695k6A   31,59
757116k6A  --
758122k6A  73
759260k6A
760
761If we allow 5 and 7 to be small primes, then we get more
762visibility info.
763
764159k4A   5  159k4E
765369k4A   5  369k4I
766453k4A   5  453k4E
767639k4B   7  639k4H
768
769Maybe the general theorem does not apply to primes which are so small,
770but we might be able to show that they are OK in these specific cases.
771
772I checked the first three weight 6 examples with AbelianIntersection,
773but actually computing the LRatio bogs down too much. Will need
774new version.
775
77695k6A   31     95k6D
77795k6A   59     95k6D
778116k6A   5    116k6D
779122k6A  73    122k6D
780
781Incidentally, the LRatio for 581k4E has a power of 19^4.
782Perhaps not surprising, as the ModularDegree of 581k4A
783has a factor of 19^2. But maybe it means a bigger Sha.
784\end{verbatim}
785