 CoCalc Shared Fileswww / papers / motive_visibility / motive_visibility_v5.tex
Author: William A. Stein
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5% August, 2001
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7% Project of William Stein, Neil Dummigan, Mark Watkins
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21\newtheorem{lem}[prop]{Lemma}
22\newtheorem{thm}[prop]{Theorem}
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24\newtheorem{examp}[prop]{Example}
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74%\newcommand{\Sha}{\underline{III}}
75%\newcommand{\Sha}{\amalg\kern-0.575em\amalg}
76% ---- SHA ----
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83\newcommand{\HH}{{\mathfrak H}}
84\newcommand{\aaa}{{\mathfrak a}}
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89\newcommand{\CH}{\mathrm {CH}}
90
91\begin{document}
92\title{Constructing elements in
93Shafarevich-Tate groups of modular motives\\
94{\sc (NOT FOR DISTRIBUTION!)}}
95\author{Neil Dummigan}
96\author{William Stein}
97\author{Mark Watkins}
98\date{March 8th, 2002}
99\subjclass{11F33, 11F67, 11G40.}
100
101\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
102Shafarevich-Tate group.}
103
104\address{University of Sheffield\\ Department of Pure
105Mathematics\\
106Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
107U.K.}
109One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
111University Park\\State College, PA 16802\\ U.S.A.}
112
113\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
114\email{watkins@math.psu.edu}
115
116\maketitle
117\section{Introduction}
118Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
119be the associated $L$-function. The conjecture of Birch and
120Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$
121at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
122also gives an interpretation of the leading term in the Taylor
123expansion in terms of various quantities, including the order of
124the Shafarevich-Tate group.
125
126Cremona and Mazur  look, among all strong Weil elliptic
127curves over $\QQ$ of conductor $N\leq 5500$, at those with
128non-trivial Shafarevich-Tate group (according to the Birch and
129Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
130group has predicted elements of order $m$. In most cases they find
131another elliptic curve, often of the same conductor, whose
132$m$-torsion is Galois-isomorphic to that of the first one, and
133which has rank two. The rational points on the second elliptic
134curve produce classes in the common $H^1(\QQ,E[m])$. They expect
135that these lie in the Shafarevich-Tate group of the first curve,
136so rational points on one curve explain elements of the
137Shafarevich-Tate group of the other curve.
138
139The Bloch-Kato conjecture \cite{BK} is the generalisation to
140arbitrary motives of the leading term part of the Birch and
141Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
142\cite{B} generalises the part about the order of vanishing at the
143central point, identifying it with the rank of a certain Chow
144group.
145
146The present work may be considered as a partial generalisation of
147the work of Cremona and Mazur, from elliptic curves over $\QQ$
148(with which are associated modular forms of weight $2$) to the
149motives attached to modular forms of higher weight. (See \cite{AS}
150for a different generalisation, to modular abelian varieties of
151higher dimension.) It may also be regarded as doing, for
152congruences between modular forms of equal weight, what \cite{Du2}
153did for congruences between modular forms of different weights.
154
155We consider the situation where two newforms $f$ and $g$, both of
156weight $k>2$ and level $N$, are congruent modulo some $\qq$, and
157$L(g,s)$ vanishes to order at least $2$ at $s=k/2$. We are able to
158find eleven examples (all with $k=4$ and $k=6$), and in each case
159$\qq$ appears in the numerator of the algebraic number
160$L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$ is a certain
161canonical period. In fact, we show how this divisibility may be
162deduced from the vanishing of $L(g,k/2)$ using recent work of
163Vatsal \cite{V}. The point is, the congruence between $f$ and $g$
164leads to a congruence between suitable algebraic parts'' of the
165special values $L(f,k/2)$ and $L(g,k/2)$. If one vanishes then the
166other is divisible by $\qq$. Under certain hypotheses, the
167Bloch-Kato conjecture then implies that the Shafarevich-Tate group
168attached to $f$ has non-zero $\qq$-torsion. Under certain
169hypotheses and assumptions, the most substantial of which is the
170Beilinson-Bloch conjecture relating the vanishing of $L(g,k/2)$ to
171the existence of algebraic cycles, we are able to construct the
172predicted elements of $\Sha$, using the Galois-theoretic
173interpretation of the congruences to transfer elements from a
174Selmer group for $g$ to a Selmer group for $f$. In proving the
175local conditions at primes dividing the level, and also in
176examining the local Tamagawa factors at these primes, we make use
177of a higher weight level-lowering result due to Jordan and Livn\'e
178\cite{JL}.
179
180One might say that algebraic cycles for one motive explain
181elements of $\Sha$ for the other. A main point of \cite{CM} was to
182observe the frequency with which those elements of $\Sha$
183predicted to exist for one elliptic curve may be explained by
184finding a congruence with another elliptic curve containing points
185of infinite order. One shortcoming of our work, compared to the
186elliptic curve case, is that, due to difficulties with local
187factors in the Bloch-Kato conjecture, we are unable to predict the
188exact order of $\Sha$. We have to start with modular forms between
189which there exists a congruence. However, Vatsal's work allows us
190to explain how the vanishing of one $L$-function leads, via the
191congruence, to the divisibility by $\qq$ of (an algebraic part of)
192another, independent of observations of computational data. The
193computational data does however show that there exist examples to
194which our results apply. Moreover, it displays factors of $\qq^2$,
195whose existence we cannot prove theoretically, but which are
196predicted by Bloch-Kato, given the existence of a non-degenerate
197alternating pairing on $\Sha$ \cite{Fl2}.
198
199\section{Motives and Galois representations}
200Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
201$\Gamma_0(N)$, with coefficients in an algebraic number field $E$,
202which is necessarily totally real. A theorem of Deligne \cite{De1}
203implies the existence, for each (finite) prime $\lambda$ of $E$,
204of a two-dimensional vector space $V_{\lambda}$ over
205$E_{\lambda}$, and a continuous representation
206$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
207such that
208\begin{enumerate}
209\item $\rho_{\lambda}$ is unramified at $p$ for all primes $p$ not dividing
210$lN$ (where $\lambda \mid l$);
211\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
212characteristic polynomial of $\Frob_p^{-1}$ acting on
213$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
214\end{enumerate}
215
216Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
217the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
218There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
219both $2$-dimensional $E$-vector spaces. For details of the
220construction see \cite{Sc}. The de Rham realisation has a Hodge
221filtration $V_{\dR}=F^0\supset F^1=\ldots =F^{k-1}\supset 222F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
223cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
224cohomology. There are natural isomorphisms $V_B\otimes 225E_{\lambda}\simeq V_{\lambda}$. We may choose a
226$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
227each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
228There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
229which amounts to multiplying the action of $\Frob_p$ by $p^j$.
230
231Following \cite{BK} (Section 3), for $p\neq l$ (including
232$p=\infty$) let
233$$H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow 234H^1(I_p,V_{\lambda}(j))).$$ The subscript $f$ stands for finite
235part''. $D_p$ is a decomposition subgroup at a prime above $p$,
236$I_p$ is the inertia subgroup, and the cohomology is for
237continuous cocycles and coboundaries. For $p=l$ let
238$$H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow 239H^1(D_l,V_{\lambda}(j)\otimes B_{\cris}))$$ (see Section 1 of
240\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
241$B_{dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
242elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
243in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes $p$.
244
245There is a natural exact sequence
246$$\begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.$$
247
248Let
249$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
250Define the $\lambda$-Selmer group \newline
251$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
252$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
253$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes $p$. Note that the
254condition at $p=\infty$ is superfluous unless $l=2$. Define the
255Shafarevich-Tate group
256$$\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/\pi_*H^1_f(\QQ,V_{\lambda}(j)).$$
257The length of its $\lambda$-component may be taken for the
258exponent of $\lambda$ in an ideal of $O_E$, which we call
259$\#\Sha(j)$. We shall only concern ourselves with the case
260$j=k/2$, and write $\Sha$ for $\Sha(j)$.
261
262Define the set of global points
263$$\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).$$
264This is analogous to the group of rational torsion points on an
265elliptic curve. The length of its $\lambda$-component may be taken
266for the exponent of $\lambda$ in an ideal of $O_E$, which we call
267$\#\Gamma_{\QQ}$.
268
269\section{Canonical periods}
270From now on we assume for convenience that $N\geq 3$. We need to
271choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
272and deRham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
273in any way such that $T_B\otimes_{O_E}O_E[1/Nk!]$ and
274$T_{\dR}\otimes_{O_E}O_E[1/Nk!]$ agree with the
275$O_E[1/Nk!]$-lattices $\mathfrak{M}_{f,B}$ and
276$\mathfrak{M}_{f,\dR}$ defined in \cite{DFG}. (See especially
277Sections 2.2 and 5.4 of \cite{DFG}.)
278
279For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
280module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
281$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes 282E_{\lambda}\simeq V_{\lambda}$. Then for $\lambda\nmid Nk!$, the
283$O_{\lambda}$ module $T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable,
284since it comes from $\ell$-adic cohomology with $O_{\lambda}$
285coefficients. We may assume that $T_{\lambda}$ is
286$\Gal(\Qbar/\QQ)$-stable for all finite $\lambda$, by adjusting
287$T_B$ locally at primes $\lambda\mid Nk!$ if necessary.
288
289Let $X_0(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
290elliptic curves with cyclic $N$-isogenies. Let $\mathfrak{E}$ be
291the universal elliptic curve over $X_0(N)$. Let
292$\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product of
293$\mathfrak{E}$ over $X_0(N)$. Realising $X_0(N)$ as the quotient
294$\Gamma_0(N)\backslash\mathfrak{H}^*$, and letting $\tau$ be a
295variable on $\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is
296isomorphic to the elliptic curve with period lattice generated by
297$1$ and $\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a
298variable on the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the
299fibre product. Then $2\pi i f(\tau)\,d\tau\wedge 300dz_1\wedge\ldots\wedge dz_{k-2}$ is a well-defined differential
301form on (a desingularisation of) $\mathfrak{E}^{k-2}$ and
302naturally represents a generating element of $F^{k-1}T_{\dR}$. (At
303least, we can make our choices locally at primes dividing $Nk!$ so
304that this is the case.) We shall call this element $e(f)$.
305
306Under the deRham isomorphism between $V_{\dR}\otimes\CC$ and
307$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
308a natural action of complex conjugation on $V_B$, breaking it up
309into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
310Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
311to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
312$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
313are rank one $O_E$-modules, but not necessarily free, since the
314class number of $O_E$ may be greater than one. Choose non-zero
315elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
316the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
317$\Omega_f^{\pm}$ by $\omega_f^{\pm}=2\pi i 318\Omega_f^{\pm}\delta_f^{\pm}$.
319\section{The Bloch-Kato conjecture}
320Let $L(f,s)$ be the $L$-function attached to $f$. For
321$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series
322$\sum_{n=1}^{\infty}a_nn^{-s}$, but there is an analytic
323continuation given by an integral, as described in the next
324section. Suppose that $L(f,k/2)\neq 0$. The Bloch-Kato conjecture
325for the motive $M_f(k/2)$ predicts that
326$${L(f,k/2)\over \vol_{\infty}}={\left(\prod_pc_p(k/2)\right)\aaa^{\pm}\#\Sha\over (\#\Gamma_{\QQ})^2}.$$
327Here, $\pm$ represents the parity of $(k/2)-1$, and
328$\vol_{\infty}$ is equal to $(2\pi i)^{k/2}\Omega_{\pm}$. For
329$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be
330$$\length H^1_f(\QQ_p,T_{\lambda}(j))_{\tors.}-\ord_{\lambda}((1-a_pp^{-j}+p^{k-1-2j})).$$
331We omit the definition of $\ord_{\lambda}(c_p(j))$ for
332$\lambda\mid p$, which requires one to assume Fontaine's de Rham
333conjecture (\cite{Fo}, Appendix A6), and depends on the choices of
334$T_{\dR}$ and $T_B$, locally at $\lambda$. (We shall mainly be
335concerned with the $q$-part of the Bloch-Kato conjecture, where
336$q$ is a prime of good reduction. For such primes, the de Rham
337conjecture follows from Theorem 5.6 of \cite{Fa1}.) The above
338formula is to be interpreted as an equality of fractional ideals
339of $E$. (Strictly speaking, the conjecture in \cite{BK} is only
340given for $E=\QQ$.)
341
342\begin{lem} Let $p\nmid N$ be a prime, $j$ an integer.
343Then the fractional ideal $c_p(j)$ is supported at most on
344divisors of $p$.
345\end{lem}
346\begin{proof}
347As on p. 30 of \cite{Fl1}, for odd $l\neq p$,
348$\ord_{\lambda}(c_p(j))$ is the length of the finite
349$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
350where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
351trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
352torsion-free.
353\end{proof}
354\begin{lem}\label{local1}
355Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
356is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
357$\qq\mid q$, and that, for all primes $p\mid N$, $\,p\neq \pm 3581\pmod{q}$. Suppose also that $f$ is not congruent modulo $\qq$ to
359any newform of weight~$k$, trivial character and level dividing
360$N/p$, with~$p$ any prime that exactly divides~$N$. Then for any
361$p\mid N$, and $j$ an integer, $\ord_{\qq}(c_p(j))=0$.
362\end{lem}
363\begin{proof} Bearing in mind that
364$$\ord_{\qq}((1-a_pp^{-j}+p^{k-1-2j}))=\#H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}),$$
365and also the long exact sequence in $I_p$-cohomology arising from
366$$\begin{CD}0@>>>T_{\qq}(j)@>>>V_{\qq}(j)@>\pi>>A_{\qq}(j)@>>>0\end{CD},$$
367it suffices to show that
368$$\dim H^0(I_p,A[\qq](j))=\dim H^0(I_p,V_{\qq}(j)).$$
369If the dimensions differ then, given that $f$ is not congruent
370modulo $\qq$ to a newform of level strictly dividing $N$, and
371since $p\neq \pm 1\pmod{q}$, Propositions 3.1 and 2.3 of \cite{L}
372tell us that $A[\qq](j)$ is unramified at $p$ and that
373$\ord_p(N)=1$. (Here we are using Carayol's result that $N$ is the
374prime-to-$q$ part of the conductor of $V_{\qq}$ \cite{Ca1}.) But
375then Theorem 1 of \cite{JL} (which uses the condition $q>k$)
376implies the existence of a newform of weight $k$, trivial
377character and level dividing $N/p$, congruent to $g$ modulo $\qq$.
379\end{proof}
380\begin{lem} Let $\qq\mid q$ be a prime of $E$ such that $q\nmid 381Nk!$ Then $\ord_{\qq}(c_q)=0$.
382\end{lem}
383\begin{proof} It follows from the isomorphism at the end of
384Section 2.2 of \cite{DFG} (an application of the results of
385\cite{Fa1}) that $T_{\qq}$ is the
386$O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
387filtered module $T_{\dR}\otimes O_{\lambda}$ by the functor they
388call $\mathbb{V}$. Given this, the lemma follows from Theorem
3894.1(iii) of \cite{BK}.
390\end{proof}
391
392\begin{lem} $\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0$ if $A[\lambda]$ is an
393irreducible representation of $\Gal(\Qbar/\QQ)$.
394\end{lem}
395This follows trivially from the definition.
396
397Putting together the above lemmas we arrive at the following:
398\begin{prop}\label{sha} Let $q\nmid N$ be an odd prime such that $q>k$.
399Let $\qq\mid q$ be a prime of $E$ such that $A[\qq]$ is an
400irreducible representation of $\Gal(\Qbar/\QQ)$. Choose $T_{\dR}$
401and $T_B$ which locally at $\qq$ are as in the previous section.
402If $$\ord_{\qq}(L(f,k/2)/\aaa^{\pm}\vol_{\infty})>0$$ (with
403numerator non-zero) then the Bloch-Kato conjecture predicts that
404$\ord_{\qq}(\#\Sha)>0$.
405\end{prop}
406
407\section{Congruences of special values}
408Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
409weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
410large enough to contain all the coefficients $a_n$ and $b_n$.
411Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv 412g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. It is
413easy to see that we may choose the $\delta_f^{\pm}\in T_B^{\pm}$
414in such a way that $\ord_{\qq}(\aaa^{\pm})=0$, i.e.
415$\delta_f^{\pm}$ generates $T_B^{\pm}$ locally at $\qq$. Let us
416suppose that such a choice has been made.
417
418We shall now make two further assumptions:
419\begin{enumerate}
420\item $L(f,k/2)\neq 0$;
421\item $L(g,k/2)=0$.
422\end{enumerate}
423\begin{prop}
424With assumptions as above,
425$\ord_{\qq}(L(f,k/2)/\aaa^{\pm}\vol_{\infty})>0$.
426\end{prop}
427\begin{proof} This is based on some of the ideas used in Section 1 of
428\cite{V}.  Note the apparent typo in Theorem 1.13 of \cite{V},
429which presumably should refer to Condition 2''. Since
430$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
431$\ord_{\qq}(L(f,k/2)/(2\pi i)^{k/2}\Omega_{\pm})>0$, where $\pm 4321=(-1)^{(k/2)-1}$. It is well-known, and easy to prove, that
433$$\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).$$
434Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
435$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
436where the integral is taken along the positive imaginary axis,
437then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
438Thus we are reduced
439to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
440
441Let $\mathcal{D}_0$ be the group of divisors of degree zero
442supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
443integer $r\geq 0$, let $P_r(R)$ be the additive group of
444homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
445groups have a natural action of $\Gamma_0(N)$. Let
446$S_{\Gamma_0(N)}(k,R):=\Hom_{\Gamma_0(N)}(\mathcal{D}_0,P_{k-2}(R))$
447be the $R$-module of weight $k$ modular symbols for $\Gamma_0(N)$.
448
449Via the isomorphism (8) in Section 1.5 of \cite{V},
450$\omega_f^{\pm}$ corresponds to an element $\Phi_f^{\pm}\in 451S_{\Gamma_0(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
452element $\Delta_f^{\pm}\in S_{\Gamma_0(N)}(k,O_E)$.
453$$\Phi_f^{\pm}([\infty]-)=\sum_{j=0,j\equiv 454(k/2)-1\pmod{2}}^{k-2}r_f(j)X^jY^{k-2-j}.$$ Since
455$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
456$$\Delta_f^{\pm}([\infty]-)=\sum_{j=0,j\equiv 457(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
458coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
459show is divisible by $\qq$.
460Similarly
461$$\Phi_g^{\pm}([\infty]-)=\sum_{j=0,j\equiv 462(k/2)-1\pmod{2}}^{k-2}r_g(j)X^jY^{k-2-j}.$$ The coefficient of
463$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
464Therefore it would suffice to show that, for some $\mu\in O_E$,
465the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
466$\qq$ in $S_{\Gamma_0(N)}(k,O_E)$. It suffices to show that, for
467some $\mu\in O_E$, the element $\delta_f^{\pm}-\mu\delta_g^{\pm}$
468is divisible by $\qq$, considered as an element of $\qq$-adic
469cohomology with coefficients. This would be the case if
470$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
471one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
472a consequence of Theorem 2.1(1) of \cite{FJ}.
473\end{proof}
474\begin{remar}
475By Proposition \ref{sha}, the Bloch-Kato conjecture now predicts
476that $\ord_{\qq}(\#\Sha)>0$. The next section provides a
477conditional construction of the required elements of $\Sha$.
478\end{remar}
479\begin{remar}\label{sign}
480The signs in the functional equations of $L(f,s)$ and $L(g,s)$
481have to be equal, since they are determined by the action of the
482involution $W_N$ on the common subspace generated by the
483reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.
484\end{remar}
485This is analogous to the remark at the end of Section 3 of
486\cite{CM}. It shows that if $L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then
487$L(g,s)$ must vanish to order at least two, as in all the examples
488below. It is worth pointing out that there are no examples of $g$
489of level one such that $L(g,k/2)=0$, unless Maeda's conjecture
490(that all the normalised cuspidal eigenforms are Galois conjugate)
491is false. See \cite{CF}.
492
493\section{Constructing elements of the Shafarevich-Tate group}
494For $f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
495$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
496$A'_{\lambda}$ be the corresponding objects for $g$. Let
497$A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$. Since
498$a_p$ is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows
499from the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
500irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
501
502Suppose that $L(g,k/2)=0$. If the sign in the functional equation
503is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
504\ref{sign}), this implies that the order of vanishing of $L(g,s)$
505at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
506conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
507is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of
508$\QQ$-rational, null-homologous, codimension $k/2$ algebraic
509cycles on the motive $M_g$, modulo rational equivalence. (This
510generalises the part of the Birch-Swinnerton-Dyer conjecture which
511says that for an elliptic curve $E/\QQ$, the order of vanishing of
512$L(E,s)$ at $s=1$ is equal to the rank of the Mordell-Weil group
513$E(\QQ)$.)
514
515Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
516to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
517subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
518If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
519get (assuming also the Beilinson-Bloch conjecture) a subspace of
520$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
521vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
522conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
523equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
524follow from the conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
525Sections 1 and 6.5 of \cite{Fo2}.
526
527Similarly, if $L(f,k/2)\neq 0$ then we expect that
528$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
529coincides with the $\qq$-part of $\Sha$.
530\begin{thm}\label{local}
531Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
532$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
533$A[\qq]$ and $A'[\qq]$ are irreducible representations of
534$\Gal(\Qbar/\QQ)$ and that, for all primes $p\mid N$, $\,p\neq \pm 5351\pmod{q}$. Suppose also that neither $f$ nor $g$ is congruent
536modulo $\qq$ to any newform of weight~$k$, trivial character and
537level dividing $N/p$, with~$p$ any prime that exactly divides~$N$.
538Then the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
539$\FF_{\qq}$-rank at least $r-1$.
540\end{thm}
541\begin{proof}
542Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
543continuity and rescaling we may assume that it lies in
544$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq 545H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction we get a non-zero
546class $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$. By
547irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial, so
548$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
549we get a non-zero, $\qq$-torsion class $\gamma\in 550H^1(\QQ,A_{\qq}(k/2))$.
551
552Our aim is to show that $\res_p(\gamma)\in 553H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
554consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
555The case $p=q$ does not quite work, which is the reason for the
556$r-1$'' in the statement of the theorem.
557
558\begin{enumerate}
559\item {\bf $p\nmid qN$. }
560
561Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p, 562A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
563$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
564$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
565follows from $d\in H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of
566$\gamma$ in $H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p. 125
567of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
568contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
569to $H^1(I_p,A_{\qq}(k/2))$,  so we have shown that
570$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
571
572\item {\bf $p\mid N$. }
573
574First we must show that $H^0(I_p, A'_{\qq}(k/2))$ is
575$\qq$-divisible. It suffices to show that
576$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
577but this may be done as in the proof of Lemma\ref{local1}. It
578follows as above that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in
579$H^1(I_p,A[\qq](k/2))$ is zero. $\res_p(c)$ then comes from
580$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
581order of this group is the same as the order of the group
582$H^0(\QQ_p,A[\qq](k/2))$, which we claim is trivial. By the work
583of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is
584ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As
585above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim 586H^0(I_p,A[\qq](k/2))$, so we need only consider the case where
587this common dimension is $1$. The (motivic) Euler factor at $p$
588for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts
589as multiplication by $\alpha$ on the one-dimensional space
590$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of
591\cite{Ca1} that this is the same as the Euler factor at $p$ of
592$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then
593follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where
594$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,
595$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on
596$H^0(I_p,A[\qq](k/2))$ as $\pm p^{-1}$. Since $p\neq \pm 5971\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence
598$\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in
599$H^1_f(\QQ_p,A_{\qq}(k/2))$.
600
601\item {\bf $p=q$. }
602
603Since $q\nmid N$ is a prime of good reduction for the motive
604$M_g$, $\,V'_{\qq}$ is a crystalline representation of
605$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
606$V'_{\qq}$ have the same dimension, where
607$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q} 608B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
609It follows from Theorem 4.1(ii) of \cite{BK} that
610$$D_{\dR}(V'_{\qq})/(F^{k/2}D_{\dR}(V'_{\qq}))\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)),$$
611via their exponential map.
612
613Since $p$ is a prime of good reduction, the de Rham conjecture is
614a consequence of the crystalline conjecture, which follows from
615Theorem 5.6 of \cite{Fa1}. Hence
616$$(V'_{\dR}\otimes_E E_{\qq})/(F^{k/2}V'_{\dR}\otimes_E E_{\qq})\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)).$$
617Observe that the dimension of the left-hand-side is $1$. Hence
618there is a subspace of $H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension at
619least $r-1$, mapping to zero in $H^1(\QQ_q,V'_{\qq}(k/2))$. It
620follows that our $r$-dimensional $\FF_{\qq}$-subspace of the
621$\qq$-torsion in $H^1(\QQ,A_{\qq}(k/2))$ has at least an
622$(r-1)$-dimensional subspace landing in
623$H^1_f(\QQ_{\qq},A_{\qq}(k/2))$ (in fact, mapping to zero).
624
625\end{enumerate}
626\end{proof}
627
628Theorem 2.7 of \cite{AS} is concerned with verifying local
629conditions in the case $k=2$, where $f$ and $g$ are associated
630with abelian varieties $A$ and $B$. (Their theorem also applies to
631abelian varieties over number fields.) Our restriction outlawing
632congruences modulo $\qq$ with cusp forms of lower level is
633analogous to theirs forbidding $q$ from dividing Tamagawa factors
634$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
635curve with $\ord_l(j(A))<0$, consideration of a Tate
636parametrisation shows that if $q\mid c_{A,l}$, i.e. if
637$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
638at $l$.)
639
640In this paper we have encountered two technical problems which we
641dealt with in quite similar ways:
642\begin{enumerate}
643\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
644\item proving local conditions at primes $p\mid N$, for an element
645of $\qq$-torsion.
646\end{enumerate}
647If our only interest was in testing the Bloch-Kato conjecture at
648$\qq$, we could have made these problems cancel out, as in Lemma
6498.11 of \cite{DFG}, by weakening the local conditions. However, we
650have chosen not to do so, since we are also interested in the
651Shafarevich-Tate group, and since the hypotheses we had to assume
652are not particularly strong.
653
654\section{Eleven examples}
655\newcommand{\nf}{\mbox{\bf #1}}
656\begin{figure}
657\caption{\label{fig:newforms}Newforms Relevant to
658Theorem~\ref{local}}
659$$660\begin{array}{|ccccccc|}\hline 661 g & \deg(g) & f & \deg(f) & q's \\\hline 662\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\ 663\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\ 664\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\ 665\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\ 666\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\ 667\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\ 668\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\ 669\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\ 670\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19 \\ 671\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\ 672\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\ 673\hline 674\end{array} 675$$
676\end{figure}
677
678Table~\ref{fig:newforms} on page~\pageref{fig:newforms} lists
679eleven pairs of newforms~$f$ and~$g$ (of equal weights and levels)
680along with at least one prime~$q$ such that there is a prime
681$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
682$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
683\subsection{Notation}
684Table~\ref{fig:newforms} is laid out as follows.
685The first column contains a label whose structure is
686\begin{center}
687{\bf [Level]k[Weight][GaloisOrbit]}
688\end{center}
689This label determines a newform $g=\sum a_n q^n$, up to Galois
690conjugacy. For example, \nf{127k4C} denotes a newform in the third
691Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
692orbits are ordered first by the degree of $\QQ(\ldots, a_n, 693\ldots)$, then by $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not dividing
694the level, with positive trace being first in the event that the
695two absolute values are equal, and the first Galois orbit is
696denoted {\bf A}, the second {\bf B}, and so on. The second column
697contains the degree of the field $\QQ(\ldots, a_n, \ldots)$. The
698third and fourth columns contain~$f$ and its degree, respectively.
699The fifth column contains at least one prime~$q$ such that there
700is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that
701the hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
702satisfied for~$f$,~$g$, and~$\qq$.
703
704\subsection{The first example in detail}
705\newcommand{\fbar}{\overline{f}}
706We describe the first line of Table~\ref{fig:newforms}
707in more detail.  See the next section for further details
708on how the computations were performed.
709
710Using modular symbols, we find that there is a newform
711$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots 712\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
713the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
714also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
715coefficients generate a number field~$K$ of degree~$17$, and by
716computing the image of the modular symbol $XY\{0,\infty\}$ under
717the period mapping, we find that $L(f,2)\neq 0$.  The newforms~$f$
718and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
719characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
720both equal to
721$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots\in \FF_{43}[[q]].$$
722
723There is no form in the Eisenstein subspaces of
724$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
725$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
726$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
727prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
728level~$1$ form of weight~$4$. Thus we have checked the hypotheses
729of Theorem~\ref{local}, so if $r$ is the dimension of
730$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
731$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r-1$.
732
733Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,
734since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup
735of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion
736subgroup of $\Sha$. Admitting these assumptions, we have
737constructed the $\qq$-torsion in $\Sha$ predicted by the
738Bloch-Kato conjecture.
739
740For particular examples of elliptic curves one can often find and
741write down rational points predicted by the Birch and
742Swinnerton-Dyer conjecture. It would be nice if likewise one could
743explicitly produce algebraic cycles predicted by the
744Beilinson-Bloch conjecture in the above examples. Since
745$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
7460.3.2 of \cite{Z}), so ought to be trivial in
747$\CH_0^{k/2}(M_g)\otimes\QQ$.
748
749\subsection{Some remarks on how the computation was performed}
750We give a brief summary of how the computation was performed.  The
751algorithms that we used were implemented by the second author, and
752most are a standard part of the MAGMA V2.8 (see \cite{magma}).
753
754Let~$g$,~$f$, and~$q$ be some data from a line of
755Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We
756verified the existence of a congruence modulo~$q$, that
757$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq 7580$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
759not arise from any $S_k(\Gamma_0(N/p))$, as follows:
760
761To prove there is a congruence, we showed that the corresponding
762{\em integral} spaces of modular symbols satisfy an appropriate
763congruence, which forces the existence of a congruence on the
764level of Fourier expansions.  We showed that $\rho_{g,\qq}$ is
765irreducible by computing a set that contains all possible residue
766characteristics of congruences between~$g$ and any Eisenstein
767series of level dividing~$N$, where by congruence, we mean a
768congruence for all Fourier coefficients of index~$n$ with
769$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
770form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
771listing a basis of such~$h$ and finding the possible congruences,
772where again we disregard the Fourier coefficients of index not
773coprime to~$N$.
774
775To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
776modular symbol ${\mathbf 777e}=X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}\{0,\infty\}$ under a
778map with the same kernel as the period mapping, and found that the
779image was~$0$.  The period mapping sends the modular
780symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
781so that ${\mathbf e}$ maps to~$0$ implies that
782$L(g,\frac{k}{2})=0$. In a similar way, we verified that
783$L(f,\frac{k}{2})\neq 0$.  Next, we checked that $W_N(g) = g$
784which, because of the functional equation, implies that
785$L'(g,\frac{k}{2})=0$.
786
787In the course of our search, we found~$17$ plausible examples and
788had to discard~$6$ of them because either the representation
789$\rho_{f,\qq}$ arose from lower level, was reducible, or $p\equiv 790\pm 1\pmod{q}$ for some  $p\mid N$.  Table~\ref{fig:newforms} is
791of independent interest because it includes examples of modular
792forms of prime level and weight $>2$ with a zero at $\frac{k}{2}$
793that is not forced by the functional equation.  We found no such
794examples of weights $\geq 8$.
795
796
797
798\begin{thebibliography}{AL}
799\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
800$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
801\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
802Shafarevich-Tate groups of abelian varieties: evidence for the
803Birch and Swinnerton-Dyer conjecture, in preparation.
804\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
805{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
806\bibitem[BCP]{magma}
807W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
808  {T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
809  235--265, Computational algebra and number theory (London, 1993).
810\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
811of motives, The Grothendieck Festschrift Volume I, 333--400,
812Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
813\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques
814associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
815\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
816\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
817Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em
818Duke Math. J. }{\bf 59 }(1989), 785--801.
819\bibitem[CM]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
820Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
82113--28.
822\bibitem[CF]{CF} B. Conrey, D. Farmer, On the non-vanishing of
823$L_f(s)$ at the center of the critical strip, preprint.
824\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
825$\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math.
826{\bf 179, } 139--172, Springer, 1969.
827\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
828d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
829part 2, 313--346.
830\bibitem[DFG]{DFG} F. Diamond, M. Flach, L. Guo, Adjoint motives
831of modular forms and the Tamagawa number conjecture, preprint.
832{{\sf
833http://www.andromeda.rutgers.edu/\~{\mbox{}}krm/liguo/lgpapers.html}}
834\bibitem[Du1]{Du1} N. Dummigan, Period ratios of modular forms, {\em
835Math. Ann. }{\bf 318 }(2000), 621--636.
836\bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and
837Selmer groups, {\em Math. Res. Lett. }{\bf 8 }(2001), 479--494.
838\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
839Galois representations, {\em in }Algebraic analysis, geometry and
840number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
841Press, Baltimore, 1989.
842\bibitem[FJ]{FJ} G. Faltings, B. Jordan, Crystalline cohomology
843and $\GL(2,\QQ)$, {\em Israel J. Math. }{\bf 90 }(1995), 1--66.
844\bibitem[Fl1]{Fl2} M. Flach, A generalisation of the Cassels-Tate
845pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.
846\bibitem[Fl2]{Fl1} M. Flach, On the degree of modular parametrisations,
847S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
848ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
849Boston Berlin, 1993.
850\bibitem[Fo1]{Fo} J.-M. Fontaine, Sur certains types de
851repr\'esentations $p$-adiques du groupe de Galois d'un corps
852local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
853}{\bf 115 }(1982), 529--577.
854\bibitem[Fo2]{Fo2} J.-M. Fontaine, Valeurs sp\'eciales des
855fonctions $L$ des motifs, S\'eminaire Bourbaki, Vol. 1991/92. {\em
856Ast\'erisque }{\bf 206 }(1992), Exp. No. 751, 4, 205--249.
857\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture epsilon''
858for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
85951--56.
860\bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois
861representations coming from modular forms, {\em J. Number Theory
862}{\bf 31 }(1989), 133--141.
863
864\bibitem[Ne1]{Ne1} J. Nekov\'ar, Kolyvagin's method for Chow groups of
865Kuga-Sato varieties, {\em Invent. Math. }{\bf 107 }(1992),
86699--125.
867\bibitem[Ne2]{Ne2} J. Nekov\'ar, $p$-adic Abel-Jacobi maps and $p$-adic
868heights. The arithmetic and geometry of algebraic cycles (Banff,
869AB, 1998), 367--379, CRM Proc. Lecture Notes, 24, Amer. Math.
870Soc., Providence, RI, 2000.
871\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
872{\em Invent. Math. }{\bf 100 }(1990), 419--430.
873\bibitem[SwD]{SwD} H. P. F. Swinnerton-Dyer, On $\ell$-adic representations and
874congruences for coefficients of modular forms,{\em Modular
875functions of one variable} III, Lect. Notes Math. {\bf 350, }
876Springer, 1973.
877\bibitem[V]{V} V. Vatsal, Canonical periods and congruence
878formulae, {\em Duke Math. J. }{\bf 98 }(1999), 397--419.
879\bibitem[Z]{Z} S. Zhang, Heights of Heegner cycles and derivatives of $L$-series,
880{\em Invent. Math. }{\bf 130 }(1997), 99--152.
881\end{thebibliography}
882
883
884\end{document}
885\subsection{Plan for finishing the computation}
886
887What I (William) have to do with the below stuff:\\
888\begin{enumerate}
889\item Fill in the missing blanks at these levels:
890$$453, 639, 657, 681, 684, 95, 116, 122, 260$$
891\item Make a MAGMA program that has a table of forms to try.
892\item First, unfortunately, for some reason the
893labels are sometimes wrong, e.g., \nf{159k4A} should
894have been \nf{159k4B}, so I have to check all the
895rational forms to see which has $L(1)=0$.
896\item Try each level using the MAGMA program (this will take a long time to run).
897\item Finish the Table of examples'' above.
898\end{enumerate}
899
900
901\begin{verbatim}
902
903
904(This stuff below is being integrated into the above, as I do
905the required (rather time consuming) computations.)
906**************************************************
907FROM Mark Watkins:
908
909Here's a list of stuff; the left-hand (sinister) column
910contains forms with a double zero at the central point,
911whilst the right-hand (dexter) column contains forms which
912have a large square factor in LRatio at 2. The middle column
913is the large prime factor in ModularDegree of the LHS,
914and/or the lpf in the LRatio at 2 of the RHS.
915
916It seems that 567k4L has invisible Sha possibly.
917The ModularDegree for 639k4B has no large factors.
918
919127k4A  43  127k4C
920159k4A  23  159k4E
921365k4A  29  365k4E
922369k4A  13  369k4I
923453k4A  17  453k4E
924453k4A  23
925465k4A  11  465k4H
926477k4A  73  477k4L
927567k4A  23  567k4G
928        13  567k4L
929581k4A  19  581k4E
930639k4B  --
931
932
933Forms with spurious zeros to do:
934
935657k4A
936681k4A
937684k4B
93895k6A   31,59
939116k6A  --
940122k6A  73
941260k6A
942
943If we allow 5 and 7 to be small primes, then we get more
944visibility info.
945
946159k4A   5  159k4E
947369k4A   5  369k4I
948453k4A   5  453k4E
949639k4B   7  639k4H
950
951Maybe the general theorem does not apply to primes which are so small,
952but we might be able to show that they are OK in these specific cases.
953
954I checked the first three weight 6 examples with AbelianIntersection,
955but actually computing the LRatio bogs down too much. Will need
956new version.
957
95895k6A   31     95k6D
95995k6A   59     95k6D
960116k6A   5    116k6D
961122k6A  73    122k6D
962
963Incidentally, the LRatio for 581k4E has a power of 19^4.
964Perhaps not surprising, as the ModularDegree of 581k4A
965has a factor of 19^2. But maybe it means a bigger Sha.
966\end{verbatim}
967