CoCalc Shared Fileswww / papers / motive_visibility / motive_visibility_v7.texOpen in CoCalc with one click!
Author: William A. Stein
1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2
%
3
% motive_visibility_v2.tex
4
%
5
% August, 2001
6
%
7
% Project of William Stein, Neil Dummigan, Mark Watkins
8
%
9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
10
11
\documentclass{amsart}
12
\usepackage{amssymb}
13
\usepackage{amsmath}
14
\usepackage{amscd}
15
16
\newcommand{\edit}[1]{\footnote{#1}\marginpar{\hfill {\sf\thefootnote}}}
17
18
\newtheorem{prop}{Proposition}[section]
19
\newtheorem{defi}[prop]{Definition}
20
\newtheorem{conj}[prop]{Conjecture}
21
\newtheorem{lem}[prop]{Lemma}
22
\newtheorem{thm}[prop]{Theorem}
23
\newtheorem{cor}[prop]{Corollary}
24
\newtheorem{examp}[prop]{Example}
25
\newtheorem{remar}[prop]{Remark}
26
\newcommand{\Ker}{\mathrm {Ker}}
27
\newcommand{\Aut}{{\mathrm {Aut}}}
28
\def\id{\mathop{\mathrm{ id}}\nolimits}
29
\renewcommand{\Im}{{\mathrm {Im}}}
30
\newcommand{\ord}{{\mathrm {ord}}}
31
\newcommand{\End}{{\mathrm {End}}}
32
\newcommand{\Hom}{{\mathrm {Hom}}}
33
\newcommand{\Mor}{{\mathrm {Mor}}}
34
\newcommand{\Norm}{{\mathrm {Norm}}}
35
\newcommand{\Nm}{{\mathrm {Nm}}}
36
\newcommand{\tr}{{\mathrm {tr}}}
37
\newcommand{\Tor}{{\mathrm {Tor}}}
38
\newcommand{\Sym}{{\mathrm {Sym}}}
39
\newcommand{\Hol}{{\mathrm {Hol}}}
40
\newcommand{\vol}{{\mathrm {vol}}}
41
\newcommand{\tors}{{\mathrm {tors}}}
42
\newcommand{\cris}{{\mathrm {cris}}}
43
\newcommand{\length}{{\mathrm {length}}}
44
\newcommand{\dR}{{\mathrm {dR}}}
45
\newcommand{\lcm}{{\mathrm {lcm}}}
46
\newcommand{\Frob}{{\mathrm {Frob}}}
47
\def\rank{\mathop{\mathrm{ rank}}\nolimits}
48
\newcommand{\Gal}{\mathrm {Gal}}
49
\newcommand{\Spec}{{\mathrm {Spec}}}
50
\newcommand{\Ext}{{\mathrm {Ext}}}
51
\newcommand{\res}{{\mathrm {res}}}
52
\newcommand{\Cor}{{\mathrm {Cor}}}
53
\newcommand{\AAA}{{\mathbb A}}
54
\newcommand{\CC}{{\mathbb C}}
55
\newcommand{\RR}{{\mathbb R}}
56
\newcommand{\QQ}{{\mathbb Q}}
57
\newcommand{\ZZ}{{\mathbb Z}}
58
\newcommand{\NN}{{\mathbb N}}
59
\newcommand{\EE}{{\mathbb E}}
60
\newcommand{\TT}{{\mathbb T}}
61
\newcommand{\HHH}{{\mathbb H}}
62
\newcommand{\pp}{{\mathfrak p}}
63
\newcommand{\qq}{{\mathfrak q}}
64
\newcommand{\FF}{{\mathbb F}}
65
\newcommand{\KK}{{\mathbb K}}
66
\newcommand{\GL}{\mathrm {GL}}
67
\newcommand{\SL}{\mathrm {SL}}
68
\newcommand{\Sp}{\mathrm {Sp}}
69
\newcommand{\Br}{\mathrm {Br}}
70
\newcommand{\Qbar}{\overline{\mathbb Q}}
71
\newcommand{\Xbar}{\overline{X}}
72
\newcommand{\Ebar}{\overline{E}}
73
\newcommand{\sbar}{\overline{s}}
74
%\newcommand{\Sha}{\underline{III}}
75
%\newcommand{\Sha}{\amalg\kern-0.575em\amalg}
76
% ---- SHA ----
77
\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
78
\newcommand{\textcyr}[1]{%
79
{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
80
\selectfont #1}}
81
\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
82
83
\newcommand{\HH}{{\mathfrak H}}
84
\newcommand{\aaa}{{\mathfrak a}}
85
\newcommand{\bb}{{\mathfrak b}}
86
\newcommand{\dd}{{\mathfrak d}}
87
\newcommand{\ee}{{\mathbf e}}
88
\newcommand{\Fbar}{\overline{F}}
89
\newcommand{\CH}{\mathrm {CH}}
90
91
\begin{document}
92
\title{Constructing elements in
93
Shafarevich-Tate groups of modular motives}
94
\author{Neil Dummigan}
95
\author{William Stein}
96
\author{Mark Watkins}
97
\date{March 8th, 2002}
98
\subjclass{11F33, 11F67, 11G40.}
99
100
\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
101
Shafarevich-Tate group.}
102
103
\address{University of Sheffield\\ Department of Pure
104
Mathematics\\
105
Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
106
U.K.}
107
\address{Harvard University\\Department of Mathematics\\
108
One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
109
\address{Penn State Mathematics Department\\
110
University Park\\State College, PA 16802\\ U.S.A.}
111
112
\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
113
\email{watkins@math.psu.edu}
114
115
\maketitle
116
\section{Introduction}
117
Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
118
be the associated $L$-function. The conjecture of Birch and
119
Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$
120
at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
121
also gives an interpretation of the leading term in the Taylor
122
expansion in terms of various quantities, including the order of
123
the Shafarevich-Tate group.
124
125
Cremona and Mazur [2000] look, among all strong Weil elliptic
126
curves over $\QQ$ of conductor $N\leq 5500$, at those with
127
non-trivial Shafarevich-Tate group (according to the Birch and
128
Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
129
group has predicted elements of order $m$. In most cases they find
130
another elliptic curve, often of the same conductor, whose
131
$m$-torsion is Galois-isomorphic to that of the first one, and
132
which has rank two. The rational points on the second elliptic
133
curve produce classes in the common $H^1(\QQ,E[m])$. They expect
134
that these lie in the Shafarevich-Tate group of the first curve,
135
so rational points on one curve explain elements of the
136
Shafarevich-Tate group of the other curve.
137
138
The Bloch-Kato conjecture \cite{BK} is the generalisation to
139
arbitrary motives of the leading term part of the Birch and
140
Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
141
\cite{B} generalises the part about the order of vanishing at the
142
central point, identifying it with the rank of a certain Chow
143
group.
144
145
The present work may be considered as a partial generalisation of
146
the work of Cremona and Mazur, from elliptic curves over $\QQ$
147
(with which are associated modular forms of weight $2$) to the
148
motives attached to modular forms of higher weight. (See \cite{AS}
149
for a different generalisation, to modular abelian varieties of
150
higher dimension.) It may also be regarded as doing, for
151
congruences between modular forms of equal weight, what \cite{Du2}
152
did for congruences between modular forms of different weights.
153
154
We consider the situation where two newforms $f$ and $g$, both of
155
weight $k>2$ and level $N$, are congruent modulo some $\qq$,
156
$L(g,k/2)=0$ but $L(f,k/2)\neq 0$. It turns out that this forces
157
$L(g,s)$ to vanish to order at least $2$ at $s=k/2$. We are able
158
to find eleven examples (all with $k=4$ and $k=6$), and in each
159
case $\qq$ appears in the numerator of the algebraic number
160
$L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$ is a certain
161
canonical period. In fact, we show how this divisibility may be
162
deduced from the vanishing of $L(g,k/2)$ using recent work of
163
Vatsal \cite{V}. The point is, the congruence between $f$ and $g$
164
leads to a congruence between suitable ``algebraic parts'' of the
165
special values $L(f,k/2)$ and $L(g,k/2)$. If one vanishes then the
166
other is divisible by $\qq$. Under certain hypotheses, the
167
Bloch-Kato conjecture then implies that the Shafarevich-Tate group
168
attached to $f$ has non-zero $\qq$-torsion. Under certain
169
hypotheses and assumptions, the most substantial of which is the
170
Beilinson-Bloch conjecture relating the vanishing of $L(g,k/2)$ to
171
the existence of algebraic cycles, we are able to construct the
172
predicted elements of $\Sha$, using the Galois-theoretic
173
interpretation of the congruences to transfer elements from a
174
Selmer group for $g$ to a Selmer group for $f$. In proving the
175
local conditions at primes dividing the level, and also in
176
examining the local Tamagawa factors at these primes, we make use
177
of a higher weight level-lowering result due to Jordan and Livn\'e
178
\cite{JL}.
179
180
One might say that algebraic cycles for one motive explain
181
elements of $\Sha$ for the other. A main point of \cite{CM} was to
182
observe the frequency with which those elements of $\Sha$
183
predicted to exist for one elliptic curve may be explained by
184
finding a congruence with another elliptic curve containing points
185
of infinite order. One shortcoming of our work, compared to the
186
elliptic curve case, is that, due to difficulties with local
187
factors in the Bloch-Kato conjecture, we are unable to predict the
188
exact order of $\Sha$. We have to start with modular forms between
189
which there exists a congruence. However, Vatsal's work allows us
190
to explain how the vanishing of one $L$-function leads, via the
191
congruence, to the divisibility by $\qq$ of (an algebraic part of)
192
another, independent of observations of computational data. The
193
computational data does however show that there exist examples to
194
which our results apply. Moreover, it displays factors of $\qq^2$,
195
whose existence we cannot prove theoretically, but which are
196
predicted by Bloch-Kato.
197
198
\section{Motives and Galois representations}
199
Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
200
$\Gamma_0(N)$, with coefficients in an algebraic number field $E$,
201
which is necessarily totally real. A theorem of Deligne \cite{De1}
202
implies the existence, for each (finite) prime $\lambda$ of $E$,
203
of a two-dimensional vector space $V_{\lambda}$ over
204
$E_{\lambda}$, and a continuous representation
205
$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
206
such that
207
\begin{enumerate}
208
\item $\rho_{\lambda}$ is unramified at $p$ for all primes $p$ not dividing
209
$lN$ (where $\lambda \mid l$);
210
\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
211
characteristic polynomial of $\Frob_p^{-1}$ acting on
212
$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
213
\end{enumerate}
214
215
Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
216
the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
217
There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
218
both $2$-dimensional $E$-vector spaces. For details of the
219
construction see \cite{Sc}. The de Rham realisation has a Hodge
220
filtration $V_{\dR}=F^0\supset F^1=\ldots =F^{k-1}\supset
221
F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
222
cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
223
cohomology. There are natural isomorphisms $V_B\otimes
224
E_{\lambda}\simeq V_{\lambda}$. We may choose a
225
$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
226
each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
227
Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
228
There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
229
which amounts to multiplying the action of $\Frob_p$ by $p^j$.
230
231
Following \cite{BK} (Section 3), for $p\neq l$ (including
232
$p=\infty$) let
233
$$H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow
234
H^1(I_p,V_{\lambda}(j))).$$ The subscript $f$ stands for ``finite
235
part''. $D_p$ is a decomposition subgroup at a prime above $p$,
236
$I_p$ is the inertia subgroup, and the cohomology is for
237
continuous cocycles and coboundaries. For $p=l$ let
238
$$H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow
239
H^1(D_l,V_{\lambda}(j)\otimes B_{\cris}))$$ (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
242
elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
243
in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes $p$.
244
245
There is a natural exact sequence
246
$$\begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.$$
247
248
Let
249
$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
250
Define 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
254
condition at $p=\infty$ is superfluous unless $l=2$. Define the
255
Shafarevich-Tate group
256
$$\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/\pi_*H^1_f(\QQ,V_{\lambda}(j)).$$
257
The length of its $\lambda$-component may be taken for the
258
exponent 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
262
Define the group of global torsion points
263
$$\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).$$
264
This is analogous to the group of rational torsion points on an
265
elliptic curve. The length of its $\lambda$-component may be taken
266
for the exponent of $\lambda$ in an ideal of $O_E$, which we call
267
$\#\Gamma_{\QQ}$.
268
269
\section{Canonical periods}
270
From now on we assume for convenience that $N\geq 3$. We need to
271
choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
272
and deRham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
273
in 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
277
Sections 2.2 and 5.4 of \cite{DFG}.)
278
279
For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
280
module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
281
$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes
282
E_{\lambda}\simeq V_{\lambda}$. Then for $\lambda\nmid Nk!$, the
283
$O_{\lambda}$ module $T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable,
284
since it comes from $\ell$-adic cohomology with $O_{\lambda}$
285
coefficients. 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
289
Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
290
generalised elliptic curves with full level-$N$ structure. Let
291
$\mathfrak{E}$ be the universal generalised elliptic curve over
292
$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product
293
of $\mathfrak{E}$ over $M(N)$. Realising $M(N)$ as the disjoint
294
union of $\phi(N)$ copies of the quotient
295
$\Gamma(N)\backslash\mathfrak{H}^*$, and letting $\tau$ be a
296
variable on $\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is
297
isomorphic to the elliptic curve with period lattice generated by
298
$1$ and $\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a
299
variable on the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the
300
fibre product. Then $2\pi i f(\tau)\,d\tau\wedge
301
dz_1\wedge\ldots\wedge dz_{k-2}$ is a well-defined differential
302
form on (a desingularisation of) $\mathfrak{E}^{k-2}$ and
303
naturally represents a generating element of $F^{k-1}T_{\dR}$. (At
304
least, we can make our choices locally at primes dividing $Nk!$ so
305
that this is the case.) We shall call this element $e(f)$.
306
307
Under the deRham isomorphism between $V_{\dR}\otimes\CC$ and
308
$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
309
a natural action of complex conjugation on $V_B$, breaking it up
310
into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
311
Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
312
to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
313
$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
314
are rank one $O_E$-modules, but not necessarily free, since the
315
class number of $O_E$ may be greater than one. Choose non-zero
316
elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
317
the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
318
$\Omega_f^{\pm}$ by $\omega_f^{\pm}=2\pi i
319
\Omega_f^{\pm}\delta_f^{\pm}$.
320
\section{The Bloch-Kato conjecture}
321
Let $L(f,s)$ be the $L$-function attached to $f$. For
322
$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series
323
$\sum_{n=1}^{\infty}a_nn^{-s}$, but there is an analytic
324
continuation given by an integral, as described in the next
325
section. Suppose that $L(f,k/2)\neq 0$. The Bloch-Kato conjecture
326
for the motive $M_f(k/2)$ predicts that
327
$${L(f,k/2)\over \vol_{\infty}}={\left(\prod_pc_p(k/2)\right)\aaa^{\pm}\#\Sha\over (\#\Gamma_{\QQ})^2}.$$
328
Here, $\pm$ represents the parity of $(k/2)-1$, and
329
$\vol_{\infty}$ is equal to $(2\pi i)^{k/2}$ multiplied by the
330
determinant of the isomorphism $V_B^{\pm}\otimes\CC\simeq
331
(V_{\dR}/F^{k/2})\otimes\CC$, calculated with respect to the
332
lattices $O_E\delta_f^{\pm}$ and the image of $T_{\dR}$. For
333
$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be
334
$$\length H^1_f(\QQ_p,T_{\lambda}(j))_{\tors.}-\ord_{\lambda}((1-a_pp^{-j}+p^{k-1-2j})).$$
335
We omit the definition of $\ord_{\lambda}(c_p(j))$ for
336
$\lambda\mid p$, which requires one to assume Fontaine's de Rham
337
conjecture (\cite{Fo}, Appendix A6), and depends on the choices of
338
$T_{\dR}$ and $T_B$, locally at $\lambda$. (We shall mainly be
339
concerned with the $q$-part of the Bloch-Kato conjecture, where
340
$q$ is a prime of good reduction. For such primes, the de Rham
341
conjecture follows from Theorem 5.6 of \cite{Fa1}.) The above
342
formula is to be interpreted as an equality of fractional ideals
343
of $E$. (Strictly speaking, the conjecture in \cite{BK} is only
344
given for $E=\QQ$.)
345
346
\begin{lem}\label{vol}
347
$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and
348
$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.
349
\end{lem}
350
\begin{proof} $\vol_{\infty}$ may be calculated as the determinant
351
of the period map from $F^{k/2}V_{\dR}\otimes\CC$ to
352
$V_B^{\pm}\otimes\CC$, with respect to lattices dual to those we
353
used above in the definition of $\vol_{\infty}$. c.f. the last
354
paragraph of 1.7 of \cite{De2}. We are using here the natural
355
pairings. Recall that the index of $O_E\delta_f^{\pm}$ in
356
$T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then the proof is completed
357
by noting that, locally away from primes dividing $Nk!$, the index
358
of $T_{\dR}$ in its dual is equal to the index of $T_B$ in its
359
dual, both being equal to the ideal denoted $\eta$ in \cite{DFG2}.
360
\end{proof}
361
\begin{lem} Let $p\nmid N$ be a prime, $j$ an integer.
362
Then the fractional ideal $c_p(j)$ is supported at most on
363
divisors of $p$.
364
\end{lem}
365
\begin{proof}
366
As on p. 30 of \cite{Fl1}, for odd $l\neq p$,
367
$\ord_{\lambda}(c_p(j))$ is the length of the finite
368
$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
369
where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
370
trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
371
torsion-free.
372
\end{proof}
373
\begin{lem}\label{local1}
374
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
375
is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
376
$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
377
that $\,p\neq -1\pmod{q}$. Suppose also that $f$ is not congruent
378
modulo $\qq$ to any newform of weight~$k$, trivial character and
379
level dividing $N/p$. Then for $j$ any integer,
380
$\ord_{\qq}(c_p(j))=0$.
381
\end{lem}
382
\begin{proof} Bearing in mind that
383
$$\ord_{\qq}((1-a_pp^{-j}+p^{k-1-2j}))=\#H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}),$$
384
and also the long exact sequence in $I_p$-cohomology arising from
385
$$\begin{CD}0@>>>T_{\qq}(j)@>>>V_{\qq}(j)@>\pi>>A_{\qq}(j)@>>>0\end{CD},$$
386
it suffices to show that
387
$$\dim H^0(I_p,A[\qq](j))=\dim H^0(I_p,V_{\qq}(j)).$$
388
If the dimensions differ then, given that $f$ is not congruent
389
modulo $\qq$ to a newform of level dividing $N/p$, Proposition 2.2
390
of \cite{L} shows that we are in the situation covered by one of
391
the three cases in Proposition 2.3 of \cite{L}. Since $p\neq
392
-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is
393
unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's
394
result that $N$ is the prime-to-$q$ part of the conductor of
395
$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses
396
the condition $q>k$) implies the existence of a newform of weight
397
$k$, trivial character and level dividing $N/p$, congruent to $g$
398
modulo $\qq$. This contradicts our hypotheses.
399
\end{proof}
400
\begin{lem}\label{at q} Let $\qq\mid q$ be a prime of $E$ such that $q\nmid
401
Nk!$ Then $\ord_{\qq}(c_q)=0$.
402
\end{lem}
403
\begin{proof} It follows from the isomorphism at the end of
404
Section 2.2 of \cite{DFG} (an application of the results of
405
\cite{Fa1}) that $T_{\qq}$ is the
406
$O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
407
filtered module $T_{\dR}\otimes O_{\lambda}$ by the functor they
408
call $\mathbb{V}$. Given this, the lemma follows from Theorem
409
4.1(iii) of \cite{BK}. (That $\mathbb{V}$ is the same as the
410
functor used in Theorem 4.1 of \cite{BK} follows from the first
411
paragraph of 2(h) of \cite{Fa1}.)
412
\end{proof}
413
414
\begin{lem} $\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0$ if $A[\lambda]$ is an
415
irreducible representation of $\Gal(\Qbar/\QQ)$.
416
\end{lem}
417
This follows trivially from the definition.
418
419
Putting together the above lemmas we arrive at the following:
420
\begin{prop}\label{sha} Assume the same hypotheses as in Lemma \ref{local1}, for
421
all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$
422
are as in the previous section. If
423
$$\ord_{\qq}(L(f,k/2)/\aaa^{\pm}\vol_{\infty})>0$$ (with numerator
424
non-zero) then the Bloch-Kato conjecture predicts that
425
$$\ord_{\qq}(\#\Sha)>0.$$
426
\end{prop}
427
428
\section{Congruences of special values}
429
Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
430
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
431
large enough to contain all the coefficients $a_n$ and $b_n$.
432
Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv
433
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Suppose
434
that $q\nmid N\phi(N)k!$. It is easy to see that we may choose the
435
$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
436
$\ord_{\qq}(\aaa^{\pm})=0$, i.e. $\delta_f^{\pm}$ generates
437
$T_B^{\pm}$ locally at $\qq$. Let us suppose that such a choice
438
has been made.
439
440
We shall now make two further assumptions:
441
\begin{enumerate}
442
\item $L(f,k/2)\neq 0$;
443
\item $L(g,k/2)=0$.
444
\end{enumerate}
445
\begin{prop}
446
With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
447
\end{prop}
448
\begin{proof} This is based on some of the ideas used in Section 1 of
449
\cite{V}. Note the apparent typo in Theorem 1.13 of \cite{V},
450
which presumably should refer to ``Condition 2''. Since
451
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
452
$\ord_{\qq}(L(f,k/2)/(2\pi i)^{k/2}\Omega_{\pm})>0$, where $\pm
453
1=(-1)^{(k/2)-1}$. It is well-known, and easy to prove, that
454
$$\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).$$
455
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
456
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
457
where the integral is taken along the positive imaginary axis,
458
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
459
Thus we are reduced
460
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
461
462
Let $\mathcal{D}_0$ be the group of divisors of degree zero
463
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
464
integer $r\geq 0$, let $P_r(R)$ be the additive group of
465
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
466
groups have a natural action of $\Gamma_0(N)$. Let
467
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
468
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
469
470
Via the isomorphism (8) in Section 1.5 of \cite{V},
471
$\omega_f^{\pm}$ corresponds to an element $\Phi_f^{\pm}\in
472
S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
473
element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$.
474
(See also Section 4.2 of \cite{St}.) Inverting $N\phi(N)$ takes
475
into account the fact that we are now dealing with $X_1(N)$ rather
476
that $M(N)$. Up to some small factorials which do not matter
477
locally at $\qq$,
478
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
479
(k/2)-1\pmod{2}}^{k-2} 2\pi i r_f(j)X^jY^{k-2-j}.$$ Since
480
$\omega_f^{\pm}=2\pi i\Omega_f^{\pm}\delta_f^{\pm}$, we see that
481
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
482
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
483
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
484
show is divisible by $\qq$.
485
Similarly
486
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
487
(k/2)-1\pmod{2}}^{k-2}2\pi i r_g(j)X^jY^{k-2-j}.$$ The coefficient
488
of $X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
489
Therefore it would suffice to show that, for some $\mu\in O_E$,
490
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
491
$\qq$ in $S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$. It suffices to show
492
that, for some $\mu\in O_E$, the element
493
$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
494
considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
495
non-constant coefficients. This would be the case if
496
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
497
one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
498
a consequence of Theorem 2.1(1) of \cite{FJ}.
499
\end{proof}
500
\begin{remar}
501
By Proposition \ref{sha} (assuming, for all $p\mid N$ the same
502
hypotheses as in Lemma \ref{local1}, together with
503
$q\nmid\phi(N)$), the Bloch-Kato conjecture now predicts that
504
$\ord_{\qq}(\#\Sha)>0$. The next section provides a conditional
505
construction of the required elements of $\Sha$.
506
\end{remar}
507
\begin{remar}\label{sign}
508
The signs in the functional equations of $L(f,s)$ and $L(g,s)$
509
have to be equal, since they are determined by the action of the
510
involution $W_N$ on the common subspace generated by the
511
reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.
512
\end{remar}
513
This is analogous to the remark at the end of Section 3 of
514
\cite{CM}. It shows that if $L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then
515
$L(g,s)$ must vanish to order at least two, as in all the examples
516
below. It is worth pointing out that there are no examples of $g$
517
of level one, and positive sign in the functional equation, such
518
that $L(g,k/2)=0$, unless Maeda's conjecture (that all the
519
normalised cuspidal eigenforms of weight $k$ and level one are
520
Galois conjugate) is false. See \cite{CF}.
521
522
\section{Constructing elements of the Shafarevich-Tate group}
523
For $f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
524
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
525
$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
526
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
527
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
528
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
529
530
Suppose that $L(g,k/2)=0$. If the sign in the functional equation
531
is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
532
\ref{sign}), this implies that the order of vanishing of $L(g,s)$
533
at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
534
conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
535
is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of
536
$\QQ$-rational, null-homologous, codimension $k/2$ algebraic
537
cycles on the motive $M_g$, modulo rational equivalence. (This
538
generalises the part of the Birch-Swinnerton-Dyer conjecture which
539
says that for an elliptic curve $E/\QQ$, the order of vanishing of
540
$L(E,s)$ at $s=1$ is equal to the rank of the Mordell-Weil group
541
$E(\QQ)$.)
542
543
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
544
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
545
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
546
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
547
get (assuming also the Beilinson-Bloch conjecture) a subspace of
548
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
549
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
550
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
551
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
552
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
553
Sections 1 and 6.5 of \cite{Fo2}.
554
555
Similarly, if $L(f,k/2)\neq 0$ then we expect that
556
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
557
coincides with the $\qq$-part of $\Sha$.
558
\begin{thm}\label{local}
559
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
560
$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
561
$A[\qq]$ and $A'[\qq]$ are irreducible representations of
562
$\Gal(\Qbar/\QQ)$ and that, for all primes $p\mid N$, $\,p\neq
563
-w_p\pmod{q}$, with $p\neq -1\pmod{q}$ if $p^2\mid N$. Suppose
564
also that neither $f$ nor $g$ is congruent modulo $\qq$ to any
565
newform of weight~$k$, trivial character and level dividing $N/p$,
566
with~$p$ any prime that divides~$N$. Then the $\qq$-torsion
567
subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at
568
least $r$.
569
\end{thm}
570
\begin{proof}
571
Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
572
continuity and rescaling we may assume that it lies in
573
$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq
574
H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction we get a non-zero
575
class $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$. By
576
irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial, so
577
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
578
we get a non-zero, $\qq$-torsion class $\gamma\in
579
H^1(\QQ,A_{\qq}(k/2))$.
580
581
Our aim is to show that $\res_p(\gamma)\in
582
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
583
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
584
585
\begin{enumerate}
586
\item {\bf $p\nmid qN$. }
587
588
Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p,
589
A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
590
$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
591
$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
592
follows from $d\in H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of
593
$\gamma$ in $H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p. 125
594
of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
595
contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
596
to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that
597
$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
598
599
\item {\bf $p\mid N$. }
600
601
First we must show that $H^0(I_p, A'_{\qq}(k/2))$ is
602
$\qq$-divisible. It suffices to show that
603
$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
604
but this may be done as in the proof of Lemma \ref{local1}. It
605
follows as above that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in
606
$H^1(I_p,A[\qq](k/2))$ is zero. $\res_p(c)$ then comes from
607
$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
608
order of this group is the same as the order of the group
609
$H^0(\QQ_p,A[\qq](k/2))$, which we claim is trivial. By the work
610
of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is
611
ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As
612
above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim
613
H^0(I_p,A[\qq](k/2))$, so we need only consider the case where
614
this common dimension is $1$. The (motivic) Euler factor at $p$
615
for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts
616
as multiplication by $\alpha$ on the one-dimensional space
617
$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of
618
\cite{Ca1} that this is the same as the Euler factor at $p$ of
619
$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then
620
follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where
621
$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,
622
$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on
623
$H^0(I_p,A[\qq](k/2))$ as $\pm p^{-1}$. Since $p\neq
624
-w_p\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial.
625
Hence $\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in
626
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
627
628
\item {\bf $p=q$. }
629
630
Since $q\nmid N$ is a prime of good reduction for the motive
631
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
632
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
633
$V'_{\qq}$ have the same dimension, where
634
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
635
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
636
As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
637
the $O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
638
filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we
639
may now prove, in the same manner as Proposition 9.2 of
640
\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.
641
\end{enumerate}
642
\end{proof}
643
644
Theorem 2.7 of \cite{AS} is concerned with verifying local
645
conditions in the case $k=2$, where $f$ and $g$ are associated
646
with abelian varieties $A$ and $B$. (Their theorem also applies to
647
abelian varieties over number fields.) Our restriction outlawing
648
congruences modulo $\qq$ with cusp forms of lower level is
649
analogous to theirs forbidding $q$ from dividing Tamagawa factors
650
$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
651
curve with $\ord_l(j(A))<0$, consideration of a Tate
652
parametrisation shows that if $q\mid c_{A,l}$, i.e. if
653
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
654
at $l$.)
655
656
In this paper we have encountered two technical problems which we
657
dealt with in quite similar ways:
658
\begin{enumerate}
659
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
660
\item proving local conditions at primes $p\mid N$, for an element
661
of $\qq$-torsion.
662
\end{enumerate}
663
If our only interest was in testing the Bloch-Kato conjecture at
664
$\qq$, we could have made these problems cancel out, as in Lemma
665
8.11 of \cite{DFG}, by weakening the local conditions. However, we
666
have chosen not to do so, since we are also interested in the
667
Shafarevich-Tate group, and since the hypotheses we had to assume
668
are not particularly strong.
669
670
\section{Eleven examples}
671
\newcommand{\nf}[1]{\mbox{\bf #1}}
672
\begin{figure}
673
\caption{\label{fig:newforms}Newforms Relevant to
674
Theorem~\ref{local}}
675
$$
676
\begin{array}{|ccccc|}\hline
677
g & \deg(g) & f & \deg(f) & q's \\\hline
678
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
679
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
680
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
681
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
682
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
683
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
684
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
685
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
686
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19 \\
687
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
688
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
689
\hline
690
\end{array}
691
$$
692
\end{figure}
693
694
Table~\ref{fig:newforms} on page~\pageref{fig:newforms} lists
695
eleven pairs of newforms~$f$ and~$g$ (of equal weights and levels)
696
along with at least one prime~$q$ such that there is a prime
697
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
698
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
699
\subsection{Notation}
700
Table~\ref{fig:newforms} is laid out as follows.
701
The first column contains a label whose structure is
702
\begin{center}
703
{\bf [Level]k[Weight][GaloisOrbit]}
704
\end{center}
705
This label determines a newform $g=\sum a_n q^n$, up to Galois
706
conjugacy. For example, \nf{127k4C} denotes a newform in the third
707
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
708
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
709
\ldots)$, then by $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not dividing
710
the level, with positive trace being first in the event that the
711
two absolute values are equal, and the first Galois orbit is
712
denoted {\bf A}, the second {\bf B}, and so on. The second column
713
contains the degree of the field $\QQ(\ldots, a_n, \ldots)$. The
714
third and fourth columns contain~$f$ and its degree, respectively.
715
The fifth column contains at least one prime~$q$ such that there
716
is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that
717
the hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
718
satisfied for~$f$,~$g$, and~$\qq$.
719
720
\subsection{The first example in detail}
721
\newcommand{\fbar}{\overline{f}}
722
We describe the first line of Table~\ref{fig:newforms}
723
in more detail. See the next section for further details
724
on how the computations were performed.
725
726
Using modular symbols, we find that there is a newform
727
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
728
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
729
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
730
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
731
coefficients generate a number field~$K$ of degree~$17$, and by
732
computing the image of the modular symbol $XY\{0,\infty\}$ under
733
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
734
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
735
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
736
both equal to
737
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots\in \FF_{43}[[q]].$$
738
739
There is no form in the Eisenstein subspaces of
740
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
741
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
742
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
743
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
744
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
745
of Theorem~\ref{local}, so if $r$ is the dimension of
746
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
747
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r-1$.
748
749
Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,
750
since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup
751
of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion
752
subgroup of $\Sha$. Admitting these assumptions, we have
753
constructed the $\qq$-torsion in $\Sha$ predicted by the
754
Bloch-Kato conjecture.
755
756
For particular examples of elliptic curves one can often find and
757
write down rational points predicted by the Birch and
758
Swinnerton-Dyer conjecture. It would be nice if likewise one could
759
explicitly produce algebraic cycles predicted by the
760
Beilinson-Bloch conjecture in the above examples. Since
761
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
762
0.3.2 of \cite{Z}), so ought to be trivial in
763
$\CH_0^{k/2}(M_g)\otimes\QQ$.
764
765
\subsection{Some remarks on how the computation was performed}
766
We give a brief summary of how the computation was performed. The
767
algorithms that we used were implemented by the second author, and
768
most are a standard part of the MAGMA V2.8 (see \cite{magma}).
769
770
Let~$g$,~$f$, and~$q$ be some data from a line of
771
Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We
772
verified the existence of a congruence modulo~$q$, that
773
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
774
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
775
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
776
777
To prove there is a congruence, we showed that the corresponding
778
{\em integral} spaces of modular symbols satisfy an appropriate
779
congruence, which forces the existence of a congruence on the
780
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
781
irreducible by computing a set that contains all possible residue
782
characteristics of congruences between~$g$ and any Eisenstein
783
series of level dividing~$N$, where by congruence, we mean a
784
congruence for all Fourier coefficients of index~$n$ with
785
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
786
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
787
listing a basis of such~$h$ and finding the possible congruences,
788
where again we disregard the Fourier coefficients of index not
789
coprime to~$N$.
790
791
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
792
modular symbol ${\mathbf
793
e}=X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}\{0,\infty\}$ under a
794
map with the same kernel as the period mapping, and found that the
795
image was~$0$. The period mapping sends the modular
796
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
797
so that ${\mathbf e}$ maps to~$0$ implies that
798
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
799
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g) = g$
800
which, because of the functional equation, implies that
801
$L'(g,\frac{k}{2})=0$.
802
803
In the course of our search, we found~$17$ plausible examples and
804
had to discard~$6$ of them because either the representation
805
$\rho_{f,\qq}$ arose from lower level, was reducible, or $p\equiv
806
\pm 1\pmod{q}$ for some $p\mid N$. Table~\ref{fig:newforms} is
807
of independent interest because it includes examples of modular
808
forms of prime level and weight $>2$ with a zero at $\frac{k}{2}$
809
that is not forced by the functional equation. We found no such
810
examples of weights $\geq 8$.
811
812
813
814
\begin{thebibliography}{AL}
815
\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
816
$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
817
\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
818
Shafarevich-Tate groups of abelian varieties: evidence for the
819
Birch and Swinnerton-Dyer conjecture, in preparation.
820
\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
821
{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
822
\bibitem[BCP]{magma}
823
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
824
{T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
825
235--265, Computational algebra and number theory (London, 1993).
826
\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
827
of motives, The Grothendieck Festschrift Volume I, 333--400,
828
Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
829
\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques
830
associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
831
\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
832
\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
833
Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em
834
Duke Math. J. }{\bf 59 }(1989), 785--801.
835
\bibitem[CM]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
836
Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
837
13--28.
838
\bibitem[CF]{CF} B. Conrey, D. Farmer, On the non-vanishing of
839
$L_f(s)$ at the center of the critical strip, preprint.
840
\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
841
$\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math.
842
{\bf 179, } 139--172, Springer, 1969.
843
\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
844
d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
845
part 2, 313--346.
846
\bibitem[DFG1]{DFG} F. Diamond, M. Flach, L. Guo, Adjoint motives
847
of modular forms and the Tamagawa number conjecture, preprint.
848
{{\sf
849
http://www.andromeda.rutgers.edu/\~{\mbox{}}liguo/lgpapers.html}}
850
\bibitem[DFG2]{DFG2} F. Diamond, M. Flach, L. Guo, The Bloch-Kato
851
conjecture for adjoint motives of modular forms, {\em Math. Res.
852
Lett. }{\bf 8 }(2001), 437--442.
853
\bibitem[Du1]{Du1} N. Dummigan, Period ratios of modular forms, {\em
854
Math. Ann. }{\bf 318 }(2000), 621--636.
855
\bibitem[Du2]{Du3} N. Dummigan, Symmetric square $L$-functions and
856
Shafarevich-Tate groups, {\em Experiment. Math. }{\bf 10 }(2001),
857
383--400.
858
\bibitem[Du3]{Du2} N. Dummigan, Congruences of modular forms and
859
Selmer groups, {\em Math. Res. Lett. }{\bf 8 }(2001), 479--494.
860
\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
861
Galois representations, {\em in }Algebraic analysis, geometry and
862
number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
863
Press, Baltimore, 1989.
864
\bibitem[FJ]{FJ} G. Faltings, B. Jordan, Crystalline cohomology
865
and $\GL(2,\QQ)$, {\em Israel J. Math. }{\bf 90 }(1995), 1--66.
866
\bibitem[Fl1]{Fl2} M. Flach, A generalisation of the Cassels-Tate
867
pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.
868
\bibitem[Fl2]{Fl1} M. Flach, On the degree of modular parametrisations,
869
S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
870
ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
871
Boston Berlin, 1993.
872
\bibitem[Fo1]{Fo} J.-M. Fontaine, Sur certains types de
873
repr\'esentations $p$-adiques du groupe de Galois d'un corps
874
local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
875
}{\bf 115 }(1982), 529--577.
876
\bibitem[Fo2]{Fo2} J.-M. Fontaine, Valeurs sp\'eciales des
877
fonctions $L$ des motifs, S\'eminaire Bourbaki, Vol. 1991/92. {\em
878
Ast\'erisque }{\bf 206 }(1992), Exp. No. 751, 4, 205--249.
879
\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture ``epsilon''
880
for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
881
51--56.
882
\bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois
883
representations coming from modular forms, {\em J. Number Theory
884
}{\bf 31 }(1989), 133--141.
885
886
\bibitem[Ne1]{Ne1} J. Nekov\'ar, Kolyvagin's method for Chow groups of
887
Kuga-Sato varieties, {\em Invent. Math. }{\bf 107 }(1992),
888
99--125.
889
\bibitem[Ne2]{Ne2} J. Nekov\'ar, $p$-adic Abel-Jacobi maps and $p$-adic
890
heights. The arithmetic and geometry of algebraic cycles (Banff,
891
AB, 1998), 367--379, CRM Proc. Lecture Notes, 24, Amer. Math.
892
Soc., Providence, RI, 2000.
893
\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
894
{\em Invent. Math. }{\bf 100 }(1990), 419--430.
895
\bibitem[St]{St} G. Stevens, $\Lambda$-adic modular forms of
896
half-integral weight and a $\Lambda$-adic Shintani lifting.
897
Arithmetic geometry (Tempe, AZ, 1993), 129--151, Contemp. Math.,
898
174, Amer. Math. Soc., Providence, RI, 1994.
899
\bibitem[SwD]{SwD} H. P. F. Swinnerton-Dyer, On $\ell$-adic representations and
900
congruences for coefficients of modular forms,{\em Modular
901
functions of one variable} III, Lect. Notes Math. {\bf 350, }
902
Springer, 1973.
903
\bibitem[V]{V} V. Vatsal, Canonical periods and congruence
904
formulae, {\em Duke Math. J. }{\bf 98 }(1999), 397--419.
905
\bibitem[Z]{Z} S. Zhang, Heights of Heegner cycles and derivatives of $L$-series,
906
{\em Invent. Math. }{\bf 130 }(1997), 99--152.
907
\end{thebibliography}
908
909
910
\end{document}
911
\subsection{Plan for finishing the computation}
912
913
What I (William) have to do with the below stuff:\\
914
\begin{enumerate}
915
\item Fill in the missing blanks at these levels:
916
$$453, 639, 657, 681, 684, 95, 116, 122, 260$$
917
\item Make a MAGMA program that has a table of forms to try.
918
\item First, unfortunately, for some reason the
919
labels are sometimes wrong, e.g., \nf{159k4A} should
920
have been \nf{159k4B}, so I have to check all the
921
rational forms to see which has $L(1)=0$.
922
\item Try each level using the MAGMA program (this will take a long time to run).
923
\item Finish the ``Table of examples'' above.
924
\end{enumerate}
925
926
927
\begin{verbatim}
928
929
930
(This stuff below is being integrated into the above, as I do
931
the required (rather time consuming) computations.)
932
**************************************************
933
FROM Mark Watkins:
934
935
Here's a list of stuff; the left-hand (sinister) column
936
contains forms with a double zero at the central point,
937
whilst the right-hand (dexter) column contains forms which
938
have a large square factor in LRatio at 2. The middle column
939
is the large prime factor in ModularDegree of the LHS,
940
and/or the lpf in the LRatio at 2 of the RHS.
941
942
It seems that 567k4L has invisible Sha possibly.
943
The ModularDegree for 639k4B has no large factors.
944
945
127k4A 43 127k4C
946
159k4A 23 159k4E
947
365k4A 29 365k4E
948
369k4A 13 369k4I
949
453k4A 17 453k4E
950
453k4A 23
951
465k4A 11 465k4H
952
477k4A 73 477k4L
953
567k4A 23 567k4G
954
13 567k4L
955
581k4A 19 581k4E
956
639k4B --
957
958
959
Forms with spurious zeros to do:
960
961
657k4A
962
681k4A
963
684k4B
964
95k6A 31,59
965
116k6A --
966
122k6A 73
967
260k6A
968
969
If we allow 5 and 7 to be small primes, then we get more
970
visibility info.
971
972
159k4A 5 159k4E
973
369k4A 5 369k4I
974
453k4A 5 453k4E
975
639k4B 7 639k4H
976
977
Maybe the general theorem does not apply to primes which are so small,
978
but we might be able to show that they are OK in these specific cases.
979
980
I checked the first three weight 6 examples with AbelianIntersection,
981
but actually computing the LRatio bogs down too much. Will need
982
new version.
983
984
95k6A 31 95k6D
985
95k6A 59 95k6D
986
116k6A 5 116k6D
987
122k6A 73 122k6D
988
989
Incidentally, the LRatio for 581k4E has a power of 19^4.
990
Perhaps not surprising, as the ModularDegree of 581k4A
991
has a factor of 19^2. But maybe it means a bigger Sha.
992
\end{verbatim}
993