CoCalc Shared Fileswww / papers / motive_visibility / motive_visibility1.tex
Author: William A. Stein
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74
75\begin{document}
76\title{Shafarevich-Tate Groups of Modular Motives}
77\author{Neil Dummigan}
78\author{William A. Stein}
79%\author{??Mark Watkins (hopefully)??}
80\date{August 2001}
81\subjclass{11F33, 11F67, 11G40.}
82
83\keywords{modular form,$L$-function, Bloch-Kato conjecture,
84Shafarevich-Tate group.}
85
86\address{University of Sheffield\\ Department of Pure
87Mathematics\\
88Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
89U.K.}
90\address{Harvard University\\ Mathematics Department\\
911 Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
92%\address{Penn State Mathematics Department\\
93%University Park, State College, PA 16802\\ U.S.A.}
94
95
96\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
97%\email{[email protected]}
98
99\begin{abstract}
100\end{abstract}
101
102\maketitle
103\section{Introduction}
104
105
106\section{Motives and Galois representations}
107Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
108$\Gamma_0(N)$, with coefficients in an algebraic number field $E$.
109A theorem of Deligne \cite{De1} implies the existence, for each
110(finite) prime $\lambda$ of $E$, of a continuous representation
111$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda})$$
112($V_{\lambda}$ is a two-dimensional vector space over
113$E_{\lambda}$), such that
114\begin{enumerate}
115\item $\rho_{\lambda}$ is unramified at $p$ for all primes $p$ not dividing
116$lN$ (where $\lambda \mid l$);
117\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
118characteristic polynomial of $\Frob_p^{-1}$ acting on
119$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
120\end{enumerate}
121
122Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
123the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
124There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
125both $2$-dimensional $E$-vector spaces. For details of the
126construction see \cite{Sc}. The de Rham realisation has a Hodge
127filtration $V_{\dR}=F^0\supset F^1=\ldots =F^{k-1}\supset 128F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
129cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
130cohomology. There are natural isomorphisms $V_B\otimes 131E_{\lambda}\simeq V_{\lambda}$. Using a basis for singular
132cohomology with $\ZZ$-coefficients, we get
133$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-modules $T_{\lambda}$
134inside each $V_{\lambda}$. Define
135$A_{\lambda}=V_{\lambda}/T_{\lambda}$. There are two kinds of
136twist we shall have to consider. There is the Tate twist
137$V_{\lambda}(j)$ (for an integer $j$), which amounts to
138multiplying the action of $\Frob_p$ by $p^j$. For $D$ the
139discriminant of a quadratic field, there is the quadratic twist
140$V_{\lambda}(\chi_D)$, which is the tensor product of
141$V_{\lambda}$ with a one-dimensional space on which
142$\Gal(\Qbar/\QQ)$ acts via the quadratic character $\chi_D$.
143
144Following \cite{BK} (Section 3), for $p\neq l$ (including
145$p=\infty$) let
146$$H^1_f(\QQ_p,V_{\lambda}(\chi_D,j))=\ker (H^1(D_p,V_{\lambda}(\chi_D,j))\rightarrow 147H^1(I_p,V_{\lambda}(\chi_D,j))).$$ The subscript $f$ stands for
148finite part''. $D_p$ is a decomposition subgroup at a prime
149above $p$, $I_p$ is the inertia subgroup, and the cohomology is
150for continuous cocycles and coboundaries. For $p=l$ let
151$$H^1_f(\QQ_l,V_{\lambda}(\chi_D,j))=\ker (H^1(D_l,V_{\lambda}(\chi_D,j))\rightarrow 152H^1(D_l,V_{\lambda}(\chi_D,j)\otimes B_{\cris}))$$ (see Section 1
153of \cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
154$B_{dR}$). Let $H^1_f(\QQ,V_{\lambda}(\chi_D,j))$ be the subspace
155of elements of $H^1(\QQ,V_{\lambda}(\chi_D,j))$ whose local
156restrictions lie in $H^1_f(\QQ_p,V_{\lambda}(\chi_D,j))$ for all
157primes $p$.
158
159There is a natural exact sequence
160$$\begin{CD}0@>>>T_{\lambda}(\chi_D,j)@>>>V_{\lambda}(\chi_D,j)@>\pi>>A_{\lambda}(\chi_D,j)@>>>0\end{CD}.$$
161
162Let
163$H^1_f(\QQ_p,A_{\lambda}(\chi_D,j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(\chi_D,j))$.
164Define the $\lambda$-Selmer group \newline
165$H^1_f(\QQ,A_{\lambda}(\chi_D,j))$ to be the subgroup of elements
166of $H^1(\QQ,A_{\lambda}(\chi_D,j))$ whose local restrictions lie
167in $H^1_f(\QQ_p,A_{\lambda}(\chi_D,j))$ for all primes $p$. Note
168that the condition at $p=\infty$ is superfluous unless $l=2$.
169Define the Shafarevich-Tate group
170$$\Sha(\chi_D,j)=\oplus_{\lambda}H^1_f(\QQ,A_l(\chi_D,j))/\pi_*H^1_f(\QQ,V_l(\chi_D,j)).$$
171The length of its $\lambda$-component may be taken for the
172exponent of $\lambda$ in an ideal of $O_E$, which we call
173$\#\Sha(\chi_D,j)$. We shall only concern ourselves with the case
174$j=k/2$, and write $\Sha(\chi_D)$ for $\Sha(\chi_D,j)$.
175
176Define the set of global points
177$$\Gamma_{\QQ}(\chi_D)=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(\chi_D,k/2)).$$
178This is analogous to the group of rational torsion points on an
179elliptic curve. The length of its $\lambda$-component may be taken
180for the exponent of $\lambda$ in an ideal of $O_E$, which we call
181$\#\Gamma_{\QQ}(\chi_D)$.
182\section{The Bloch-Kato conjecture}
183The Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts that
184$$\frac{L(f,k/2)}{\vol_{\infty}}=\frac{\left(\prod_pc_p(k/2)\right)\cdot\#\Sha}{(\#\Gamma_{\QQ})^2}.$$
185The local factors $\vol_{\infty}$ and $c_p(k/2)$ depend on the
186choice of a basis for $V_{\dR}$, but their product does not. For
187$l\neq p$, $\ord_{\lambda}(c_p(\chi_D,j))$ is defined to be
188$$\length H^1_f(\QQ_p,T_{\lambda}(\chi_D,j))_{\tors.}-\ord_{\lambda}((1-\chi_D(p)a_pp^{-j}+p^{k-1-2j})).$$
189We omit the definition of $\ord_{\lambda}(c_p(\chi_D,j))$ for
190$\lambda\mid p$, which requires one to assume Fontaine's de Rham
191conjecture (\cite{Fo}, Appendix A6). The above formula is to be
192interpreted as an equality of fractional ideals of $E$. (Strictly
193speaking, the conjecture in \cite{BK} is only given for $E=\QQ$.)
194
195Similarly, the Bloch-Kato conjecture for $M_f(\chi_D,k/2)$
196predicts that
197$$\frac{L(f,\chi_D,k/2)}{\vol_{\infty}(\chi_D)}=\frac{\left(\prod_pc_p(k/2,\chi_D)\right)\#\Sha(\chi_D)}{(\#\Gamma_{\QQ}(\chi_D))^2}.$$
198A careful treatment of real periods
199such as $\vol_{\infty}(\chi_D)$ is given in \cite{De2}. The
200calculation of the periods of Artin motives in terms of
201generalized Gauss sums, in Section 6 of \cite{De2}, yields the
202following lemma.
203\begin{lem}
204If $\chi_D(-1)=1$ (i.e. if $D>0$) then
205$\vol_{\infty}(\chi_D)=\vol_{\infty}/\sqrt{D}.$
206\end{lem}
207
208\begin{lem} Let $p\nmid N$ be a prime, $j$ an integer
209and $D$ a quadratic discriminant. Then the fractional ideal
210$c_p(\chi_D,j)$ is supported at most on divisors of $2$ and
211divisors of $p$, while $c_p(j)$ is supported at most on divisors
212of $p$.
213\end{lem}
214\begin{proof}
215As on p. 30 of \cite{Fl1}, for odd $l\neq p$,
216$\ord_{\lambda}(c_p(\chi_D,j))$ is the length of the finite
217$O_{\lambda}$-module
218$H^0(\QQ_p,H^1(I_p,T_{\lambda}(\chi_D,j))_{\tors}),$ where $I_p$
219is an inertia group at $p$. If $p$ does not divide $D$ then
220$T_{\lambda}(\chi_D,j)$ is a trivial $I_p$-module, so
221$H^1(I_p,T_{\lambda}(\chi_D,j))$ is torsion-free. In general,
222since $l\neq p$,$$H^1(I_p,T_{\lambda}(\chi_{D},j))\simeq 223T_{\lambda}(\chi_{D},j)/(1-\tau)T_{\lambda}(\chi_{D},j),$$ where
224$\tau$ is a topological generator of the tame quotient of $I_p$.
225If $p\mid D$ then thanks to the quadratic twist ramified at $p$,
226$\,\tau$ acts as multiplication by $-1$ on
227$T_{\lambda}(\chi_{D},j)$, hence
228$H^1(I_p,T_{\lambda}(\chi_{D},j))$ is trivial (remember $l\neq 2292$).
230
231The assertion about $c_p(j)$ may be proved similarly to the case
232$p\nmid D$ above.
233\end{proof}
234
235\begin{lem} Let $D$ be a quadratic discriminant, and $p$ a prime such that
236$\chi_D(p)=1$. Then $c_p(\chi_D,j)=c_p(j)$.
237\end{lem}
238\begin{proof}
239This is simply a consequence of the fact that the
240definition of $\ord_{\lambda}(c_p(j))$ depends only on
241$T_{\lambda}$ as a $\Gal(\Qbar_p/\QQ_p)$-module.
242\end{proof}
243
244\begin{lem} $\ord_{\lambda}(\#\Gamma_{\QQ}(\chi_D,j))=0$ unless
245the coefficients of $f=\sum a_nq^n$ satisfy the congruence
246$a_p\equiv \chi_D(p)(p^j+p^{k-1-j})\pmod{\lambda}$ for all primes
247$p\nmid lN$.
248\end{lem}
249This follows from the interpretation of $a_p$ as a trace of
250Frobenius. See \cite{SwD}.
251
252Putting together the above lemmas we arrive at the following:
253\begin{prop}\label{sha} Let $q\nmid N$ be an odd prime. Let $D>0$ be a
254quadratic discriminant such that $\chi_D(q)=1$ and $\chi_D(p)=1$
255for all primes $p\mid N$. Let $\qq\mid q$ be a prime of $E$ such
256that neither
257$$a_p\equiv \chi_D(p)(p^{k/2}+p^{(k/2)-1})\pmod{\qq}$$
258nor
259$$a_p\equiv p^{k/2}+p^{(k/2)-1}\pmod{\qq}$$
260holds for all $p\nmid qN$. If
261$$\ord_{\qq}(L(f,k/2)/\sqrt{D}L(f,\chi_D,k/2))>0$$
262then the Bloch-Kato conjecture predicts that
263$\ord_{\qq}(\#\Sha)>0$.
264\end{prop}
265\begin{prop}\label{triv}
266Suppose that $p\nmid DN$, $p>k$ and that $f$ is not
267congruent to another cusp form for $\Gamma_0(N)$ modulo any
268divisor of $p$. Then, for $(k/2)\leq j\leq k-1$, $c_p(j)$ is
269trivial and $c_p(\chi_D,j)$ is at worst supported on divisors of
270$2$.
271\end{prop}
272This may be proved in a similar manner to Theorem 7.6 of
273\cite{Du1}. The third condition in the statement of the
274proposition ensures that the denominator of the projector used to
275cut out the motive $M$ is coprime to $p$. Bearing in mind Flach's
276generalisation of the Cassels-Tate pairing \cite{Fl2}, we see the
277following corollary.
278\begin{cor} The Bloch-Kato conjecture predicts that, for $D$ as in
279Proposition \ref{sha}, the fractional ideal
280$(L(f,k/2)/\sqrt{D}L(f,\chi_D,k/2))$ is a square, up to divisors
281of $2$ and of primes not satisfying the conditions of Proposition
282\ref{triv}.
283\end{cor}
284\section{Constructing elements of the Shafarevich-Tate group}
285Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
286weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
287large enough to contain all the coefficients $a_n$ and $b_n$.
288Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv 289g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. For $f$
290we have defined $V_{\lambda}$, $T_{\lambda}$ and $A_{\lambda}$.
291Let $V'_{\lambda}$, $T'_{\lambda}$ and $A'_{\lambda}$ be the
292corresponding objects for $g$. Let $A[\lambda]$ denote the
293$\lambda$-torsion in $A_{\lambda}$. Since $a_p$ is the trace of
294$\Frob_p^{-1}$ on $V_{\lambda}$, it follows from the Cebotarev
295Density Theorem that $A[\qq]$ and $A'[\qq]$ are isomorphic as
296$\Gal(\Qbar/\QQ)$-modules.
297
298Suppose that $L(g,k/2)=0$. If the sign in the functional equation
299is positive, this implies that the order of vanishing at $s=k/2$
300is at least $2$. According to the Beilinson-Bloch conjecture
301\cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$ is the
302rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational,
303null-homologous algebraic cycles on the motive $M_g$, modulo
304rational equivalence.
305
306Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
307to $H^1(\QQ,V'_{\qq}(k/2))$, and Nekov\'ar shows in \cite{Ne2}
308that its image is contained in the subspace
309$H^1_f(\QQ,V'_{\qq}(k/2))$. See also 0.13 of \cite{Ne1}. If, as
310expected, the $\qq$-adic Abel-Jacobi map is injective, we get
311(assuming also the Beilinson-Bloch conjecture) a subspace of
312$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
313vanishing of $L(g,s)$ at $s=k/2$.
314
315If $L(f,k/2)\neq 0$ then it is expected that
316$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that
317$\Sha=H^1_f(\QQ,A_{\qq}(k/2))$.
318\begin{thm}\label{local}
319Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
320$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
321$A[\qq]$ and $A'[\qq]$ are irreducible representations of
322$\Gal(\Qbar/\QQ)$ and that, for all primes $p\mid N$, $\,p\neq \pm 3231\pmod{q}$. Suppose also that neither $f$ nor $g$ is congruent
324modulo $\qq$ to any newform of trivial character and level
325dividing $N/p$, with $p$ a prime exactly dividing $N$. Then the
326$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
327$\FF_{\qq}$-rank at least $r-1$.
328\end{thm}\edit{There's something wrong with the statement
329of this theorem, as none of the objects depend on~$g$.
330I think $H^1_f(\QQ,A_{\qq}(k/2))$
331should be replaced by $H^1_g(\QQ,A_{\qq}(k/2))$.}
332\begin{proof}Take a non-zero class $d\in 333H^1_f(\QQ,V'_{\qq}(k/2))$, by continuity and rescaling we may
334assume that it lies in $H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq 335H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction we get a non-zero
336class $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$.
337Since $a_p\equiv p^{k/2}+p^{(k/2)-1}\pmod{\qq}$ (for all $p\nmid 338qN$) does not hold (by irreducibility), $H^0(\QQ,A[\qq](k/2))$ is
339trivial, so $H^1(\QQ,A[\qq](k/2))$ injects into
340$H^1(\QQ,A_{\qq}(k/2))$, and we get a non-zero class $\gamma\in 341H^1(\QQ,A_{\qq}(k/2))$.
342
343First we will show that $\res_p(\gamma)\in 344H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p\nmid qN$.
345Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p, 346A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
347$H^1(I_p,A'[\qq](k/2))$ injects into $H^1(I_p,A'_{\qq}(k/2))$, and
348it follows from $d\in H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of
349$\gamma$ in $H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p. 125
350of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
351contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
352to $H^1(I_p,A_{\qq}(k/2))$,  so we have shown that
353$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
354
355Next we will show that $\res_p(\gamma)\in 356H^1_f(\QQ_p,A_{\qq}(k/2))$ for $p\mid N$. Our first task is to
357show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible. It
358suffices to show that $$\dim H^0(I_p,A'[\qq](k/2))=\dim 359H^0(I_p,V'_{\qq}(k/2)).$$ If the dimensions differ then, given
360that $g$ is not congruent modulo $\qq$ to a newform of level
361strictly dividing $N$, and since $p\neq \pm 1\pmod{q}$,
362Propositions 3.1 and 2.3 of \cite{L} tell us that $A'[\qq](k/2)$
363is unramified at $p$ and that $\ord_p(N)=1$. (Here we are using
364Carayol's result that $N$ is the prime-to-$q$ part of the
365conductor of $V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of
366\cite{JL} implies the existence of a newform of trivial character
367and level dividing $N/p$, congruent to $g$ modulo $\qq$. This
368contradicts our hypotheses, and we have established that $H^0(I_p, 369A'_{\qq}(k/2))$ is $\qq$-divisible.
370
371It follows that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in
372$H^1(I_p,A[\qq](k/2))$ is zero. By inflation-restriction,
373$\res_p(c)$ then comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$.
374The order of this group is the same as the order of the group
375$H^0(\QQ_p,A[\qq](k/2))$, which we claim is trivial. By the work
376of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is
377ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As
378above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim 379H^0(I_p,A[\qq](k/2))$, so we need only consider the case where
380this common dimension is $1$. The (motivic) Euler factor at $p$
381for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts
382as multiplication by $\alpha$ on the one-dimensional space
383$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of
384\cite{Ca1} that this is the same as the Euler factor at $p$ of
385$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then
386follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where
387$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,
388$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on
389$H^0(I_p,A[\qq](k/2))$ as $\pm p^{-1}$. Since $p\neq \pm 3901\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence
391$\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in
392$H^1_f(\QQ_p,A_{\qq}(k/2))$.
393
394It remains to deal with the local condition at $p=q$. Since
395$q\nmid N$ is a prime of good reduction for the motive $M_f$,
396$\,V_{\qq}$ is a crystalline representation of
397$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V_{\qq})$ and $V_{\qq}$
398have the same dimension, where
399$D_{\cris}(V_{\qq}):=H^0(\QQ_q,V_{\qq}\otimes_{\QQ_q} B_{\cris})$.
400(This is a consequence of Theorem 5.6 of \cite{Fa1}.) It follows
401from Theorem 4.1(ii) of \cite{BK} that
402$$D_{\dR}(V_{\qq})/F^{k/2}D_{\dR}(V_{\qq})\simeq H^1_f(\QQ_q,V_{\qq}(k/2)),$$
403via their exponential map.
404
405Since $p$ is a prime of good reduction, the de Rham conjecture is
406a consequence of the crystalline conjecture, which follows from
407Theorem 5.6 of \cite{Fa1}. Hence
408$$V_{\dR}\otimes_E E_{\qq}/F^{k/2}V_{\dR}\otimes_E E_{\qq}\simeq H^1_f(\QQ_q,V_{\qq}(k/2)).$$
409Observe that the dimension of the left-hand-side is $1$.
410Meanwhile, by 3.8 of \cite{BK}, $H^1_f(\QQ_q,V_{\qq}(k/2))$ is its
411own annihilator in $H^1(\QQ_q,V_{\qq}(k/2))$ with respect to the
412Tate duality pairing. Hence $H^1_f(\QQ_q,V_{\qq}(k/2))$ has
413codimension one in $H^1(\QQ_q,V_{\qq}(k/2))$. It follows that our
414$r$-dimensional $\FF_{\qq}$-subspace of the $\qq$-torsion in
415$H^1(\QQ,A_{\qq}(k/2))$ has at least an $(r-1)$-dimensional
416subspace landing in $H^1_f((\QQ_{\qq},A_{\qq}(k/2))$.
417
418To justify this argument carefully, we should check that the
419natural map from $H^1(\QQ_q,V_{\qq}(k/2))$ to
420$H^1((\QQ_{\qq},A_{\qq}(k/2))$ is surjective. Its cokernel injects
421into \newline $H^2((\QQ_{\qq},T_{\qq}(k/2))$, which is dual to
422$H^0((\QQ_{\qq},A_{\qq}(k/2))$ (via local Tate duality). It
423suffices to check that $H^0((\QQ_{\qq},A[\qq](k/2))$ is trivial.
424This follows from the description of $A[\qq](k/2)$ as a
425$\Gal(\Qbar_q/\QQ_q)$-module provided by theorems of Deligne and
426Fontaine (Theorems 2.5 and 2.6 of \cite{Ed}).
427\end{proof}
428
429Theorem 2.7 of \cite{AS} is concerned with verifying local
430conditions in the case $k=2$, where $f$ and $g$ are associated
431with abelian varieties $A$ and $B$. (Their theorem also applies to
432abelian varieties over number fields.) Our restriction outlawing
433congruences modulo $\qq$ with cusp forms of lower level is
434analogous to theirs forbidding $q$ from dividing Tamagawa factors
435$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
436curve with $\ord_l(j(A))<0$, consideration of a Tate
437parametrisation shows that if $q\mid c_{A,l}$, i.e. if
438$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
439at $l$.)
440
441\section{Some Data}
442\subsection{Prime Level $127$}
443Using modular symbols, we find that there is a newform
444$$f = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots 445\in S_4(\Gamma_0(127))$$
446with $L(f,2)=0$.
447(In fact, $L'(f,2)=0$, as well.)
448%The sign of the Atkin-Lehner involution $W_{127}$ on~$f$
449%is $+1$, so $L'(f,2)=0$ as well (look up reference).
450There is also a newform $g \in S_4(\Gamma_0(127))$ whose
451Fourier coefficients generate a number field~$K$ of degree~$17$,
452and $L(g,2)\neq 0$.
453The newforms~$f$ and~$g$ are congruent
454modulo a prime $\qq$ of~$K$ of residue characteristic~$43$.
455The mod~$\qq$ reductions of~$f$ and~$g$ are both equal to
456$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots.$$
457The residue characteristics of congruence primes
458between the cuspidal and Eisenstein subspaces are~$2$,~$3$,~$7$,
459and $1613$, so (reference) $\rho_{f,\qq}\approx\rho_{g,\qq}$
460is irreducible.
461Since $S_4(\SL_2(\ZZ))=0$, we see that~$\fbar$ does not arise from a
462level~$1$ form of weight~$4$.  Thus, if we assume the Beilinson-Bloch
463conjecture, Theorem~\ref{local} implies that $\Sha(M_g)$ has
464order divisible by~$43$.
465
466\begin{verbatim}
467FROM Mark Watkins:
468
469Here's a list of stuff; the left-hand (sinister) column
470contains forms with a double zero at the central point,
471whilst the right-hand (dexter) column contains forms which
472have a large square factor in LRatio at 2. The middle column
473is the large prime factor in ModularDegree of the LHS,
474and/or the lpf in the LRatio at 2 of the RHS.
475
476It seems that 567k4L has invisible Sha possibly.
477The ModularDegree for 639k4B has no large factors.
478
479127k4A  43  127k4C
480159k4A  23  159k4E
481365k4A  29  365k4E
482369k4A  13  369k4I
483453k4A  17  453k4E
484453k4A  23
485465k4A  11  465k4H
486477k4A  73  477k4L
487567k4A  23  567k4G
488        13  567k4L
489581k4A  19  581k4E
490639k4B  --
491
492
493Forms with spurious zeros to do:
494
495657k4A
496681k4A
497684k4B
49895k6A   31,59
499116k6A  --
500122k6A  73
501260k6A
502
503If we allow 5 and 7 to be small primes, then we get more
504visibility info.
505
506159k4A   5  159k4E
507369k4A   5  369k4I
508453k4A   5  453k4E
509639k4B   7  639k4H
510
511Maybe the general theorem does not apply to primes which are so small,
512but we might be able to show that they are OK in these specific cases.
513
514I checked the first three weight 6 examples with AbelianIntersection,
515but actually computing the LRatio bogs down too much. Will need
516new version.
517
51895k6A   31     95k6D
51995k6A   59     95k6D
520116k6A   5    116k6D
521122k6A  73    122k6D
522
523Incidentally, the LRatio for 581k4E has a power of 19^4.
524Perhaps not surprising, as the ModularDegree of 581k4A
525has a factor of 19^2. But maybe it means a bigger Sha.
526\end{verbatim}
527
528
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595\end{document}
596