CoCalc Shared Fileswww / papers / motive_visibility / dsw_7.texOpen in CoCalc with one click!
Author: William A. Stein
1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2
%
3
% motive_visibility.tex
4
%
5
% 25 August 2002
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
\def\id{\mathop{\mathrm{ id}}\nolimits}
27
\DeclareMathOperator{\Ker}{\mathrm {Ker}}
28
\DeclareMathOperator{\Aut}{{\mathrm {Aut}}}
29
\renewcommand{\Im}{{\mathrm {Im}}}
30
\DeclareMathOperator{\ord}{ord}
31
\DeclareMathOperator{\End}{End}
32
\DeclareMathOperator{\Hom}{Hom}
33
\DeclareMathOperator{\Mor}{Mor}
34
\DeclareMathOperator{\Norm}{Norm}
35
\DeclareMathOperator{\Nm}{Nm}
36
\DeclareMathOperator{\tr}{tr}
37
\DeclareMathOperator{\Tor}{Tor}
38
\DeclareMathOperator{\Sym}{Sym}
39
\DeclareMathOperator{\Hol}{Hol}
40
\DeclareMathOperator{\vol}{vol}
41
\DeclareMathOperator{\tors}{tors}
42
\DeclareMathOperator{\cris}{cris}
43
\DeclareMathOperator{\length}{length}
44
\DeclareMathOperator{\dR}{dR}
45
\DeclareMathOperator{\lcm}{lcm}
46
\DeclareMathOperator{\Frob}{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{\nf}[1]{\mbox{\bf #1}}
75
\newcommand{\fbar}{\overline{f}}
76
77
% ---- SHA ----
78
\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
79
\newcommand{\textcyr}[1]{%
80
{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
81
\selectfont #1}}
82
\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
83
84
\newcommand{\HH}{{\mathfrak H}}
85
\newcommand{\aaa}{{\mathfrak a}}
86
\newcommand{\bb}{{\mathfrak b}}
87
\newcommand{\dd}{{\mathfrak d}}
88
\newcommand{\ee}{{\mathbf e}}
89
\newcommand{\Fbar}{\overline{F}}
90
\newcommand{\CH}{\mathrm {CH}}
91
92
\begin{document}
93
\title[Shafarevich-Tate Groups of Modular Motives]{Constructing Elements in Shafarevich-Tate Groups of Modular Motives}
94
\author{Neil Dummigan}
95
\author{William Stein}
96
\author{Mark Watkins}
97
\date{August 24th, 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
116
\begin{abstract}
117
118
We study Shafarevich-Tate groups of motives attached to modular forms
119
on $\Gamma_0(N)$ of weight bigger than~$2$. We deduce a criterion for
120
the existence of nontrivial elements of these Shafarevich-Tate groups,
121
and give $16$ examples in which the Beilinson-Bloch conjecture implies
122
the existence of such elements. We also use modular symbols and
123
observations about Tamagawa numbers to compute nontrivial conjectural
124
lower bounds on the orders of the Shafarevich-Tate groups of modular
125
motives of low level and weight at most $12$. Our methods build upon
126
Mazur's idea of visibility, but in the context of motives instead of
127
abelian varieties.
128
\end{abstract}
129
130
\maketitle
131
132
\section{Introduction}
133
Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
134
be the associated $L$-function. The conjecture of Birch and
135
Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$
136
at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
137
also gives an interpretation of the leading term in the Taylor
138
expansion in terms of various quantities, including the order of
139
the Shafarevich-Tate group of $E$.
140
141
Cremona and Mazur \cite{CM} look, among all strong Weil elliptic
142
curves over $\QQ$ of conductor $N\leq 5500$, at those with
143
non-trivial Shafarevich-Tate group (according to the Birch and
144
Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
145
group has predicted elements of prime order $m$. In most cases
146
they find another elliptic curve, often of the same conductor,
147
whose $m$-torsion is Galois-isomorphic to that of the first one,
148
and which has positive rank. The rational points on the second elliptic
149
curve produce classes in the common $H^1(\QQ,E[m])$. They show
150
\cite{CM2} that these lie in the Shafarevich-Tate group of the
151
first curve, so rational points on one curve explain elements of
152
the Shafarevich-Tate group of the other curve.
153
154
The Bloch-Kato conjecture \cite{BK} is the generalisation to
155
arbitrary motives of the leading term part of the Birch and
156
Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
157
\cite{B} generalises the part about the order of vanishing at the
158
central point, identifying it with the rank of a certain Chow
159
group.
160
161
This paper is a partial generalisation of \cite{CM} and \cite{AS}
162
from abelian varieties over $\QQ$ associated to modular forms of
163
weight~$2$ to the motives attached to modular forms of higher weight.
164
It also does for congruences between modular forms of equal weight
165
what \cite{Du2} did for congruences between modular forms of different
166
weights.
167
168
We consider the situation where two newforms~$f$ and~$g$, both of
169
even weight $k>2$ and level~$N$, are congruent modulo a maximal ideal
170
$\qq$ of odd residue characteristic, and $L(g,k/2)=0$ but
171
$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish
172
to order at least $2$ at $s=k/2$. In Section~\ref{sec:examples},
173
we give sixteen
174
examples (all with $k=4$ and $k=6$), and in each $\qq$ divides the
175
numerator of the algebraic number $L(f,k/2)/\vol_{\infty}$, where
176
$\vol_{\infty}$ is a certain canonical period.
177
178
In fact, we show how this divisibility may be deduced from the
179
vanishing of $L(g,k/2)$ using recent work of Vatsal \cite{V}. The
180
point is, the congruence between$f$ and~$g$ leads to a congruence
181
between suitable ``algebraic parts'' of the special values
182
$L(f,k/2)$ and $L(g,k/2)$. In slightly more detail, a multiplicity
183
one result of Faltings and Jordan shows that the congruence of
184
Fourier expansions leads to a congruence of certain associated
185
cohomology classes. These are then identified with the modular
186
symbols which give rise to the algebraic parts of special values.
187
If $L(g,k/2)$ vanishes then the congruence implies that
188
$L(f,k/2)/\vol_{\infty}$ must be divisible by $\qq$.
189
190
The Bloch-Kato conjecture sometimes then implies that the
191
Shafarevich-Tate group attached to~$f$ has nonzero $\qq$-torsion.
192
Under certain hypotheses and assumptions, the most substantial of
193
which is the Beilinson-Bloch conjecture relating the vanishing of
194
$L(g,k/2)$ to the existence of algebraic cycles, we are able to
195
construct some of the predicted elements of~$\Sha$ using the
196
Galois-theoretic interpretation of the congruences to transfer
197
elements from a Selmer group for~$g$ to a Selmer group for~$f$.
198
One might say that algebraic cycles for one motive explain
199
elements of~$\Sha$ for the other, or that we use congruences to
200
link the Beilinson-Bloch conjecture for one motive with the
201
Bloch-Kato conjecture for the other.
202
%In proving the local
203
%conditions at primes dividing the level, and also in examining the
204
%local Tamagawa factors at these primes, we make use of a higher weight
205
%level-lowering result due to Jordan and Livn\'e \cite{JL}.
206
207
We also compute data which, assuming the Bloch-Kato conjecture,
208
provides lower bounds for the orders of numerous Shafarevich-Tate
209
groups (see Section~\ref{sec:invis}).
210
%Our data is consistent
211
%with the fact \cite{Fl2} that the part of $\#\Sha$ coprime to the
212
%congruence modulus is necessarily a perfect square (assuming that~$\Sha$
213
%is finite).
214
215
\section{Motives and Galois representations}
216
This section and the next provide definitions of some of the
217
quantities appearing later in the Bloch-Kato conjecture. Let
218
$f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
219
$\Gamma_0(N)$, with coefficients in an algebraic number field~$E$,
220
which is necessarily totally real. Let~$\lambda$ be any finite
221
prime of~$E$, and let~$\ell$ denote its residue characteristic. A
222
theorem of Deligne \cite{De1} implies the existence of a
223
two-dimensional vector space $V_{\lambda}$ over $E_{\lambda}$, and
224
a continuous representation
225
$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
226
such that
227
\begin{enumerate}
228
\item $\rho_{\lambda}$ is unramified at~$p$ for all primes~$p$ not dividing
229
$lN$, and
230
\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
231
characteristic polynomial of $\Frob_p^{-1}$ acting on
232
$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
233
\end{enumerate}
234
235
Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
236
the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
237
There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
238
both $2$-dimensional $E$-vector spaces. For details of the
239
construction see \cite{Sc}. The de Rham realisation has a Hodge
240
filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset
241
F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
242
cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
243
cohomology.
244
For each prime $\lambda$, there is a natural isomorphism
245
$V_B\otimes E_{\lambda}\simeq V_{\lambda}$. We may choose a
246
$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
247
each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
248
Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
249
There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
250
which amounts to multiplying the action of $\Frob_p$ by $p^j$.
251
252
Following \cite{BK} (Section 3), for $p\neq l$ (including
253
$p=\infty$) let
254
$$
255
H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow
256
H^1(I_p,V_{\lambda}(j))).
257
$$
258
The subscript~$f$ stands for ``finite
259
part'', $D_p$ is a decomposition subgroup at a prime above~$p$,
260
$I_p$ is the inertia subgroup, and the cohomology is for
261
continuous cocycles and coboundaries. For $p=l$ let
262
$$
263
H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow
264
H^1(D_l,V_{\lambda}(j)\otimes B_{\cris}))
265
$$
266
(see Section 1 of
267
\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
268
$B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
269
elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
270
in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes~$p$.
271
272
There is a natural exact sequence
273
$$
274
\begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.
275
$$
276
Let
277
$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
278
Define the $\lambda$-Selmer group
279
$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
280
$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
281
$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes~$p$. Note that the
282
condition at $p=\infty$ is superfluous unless $l=2$. Define the
283
Shafarevich-Tate group
284
$$
285
\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/
286
\pi_*H^1_f(\QQ,V_{\lambda}(j)).
287
$$
288
Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any
289
prime ideal~$\lambda$ is the length of the $\lambda$-component of
290
$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,
291
and write~$\Sha$ for~$\Sha(j)$.
292
293
Define the group of global torsion points
294
$$
295
\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).
296
$$
297
This is analogous to the group of rational torsion points on an
298
elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in
299
which the exponent of any prime ideal~$\lambda$ is the length of
300
the $\lambda$-component of $\Gamma_{\QQ}$.
301
302
\section{Canonical periods}
303
We assume from now on for convenience that $N\geq 3$. We need to
304
choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
305
and de Rham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
306
in a way such that $T_B$ and $T_{\dR}\otimes_{O_E}O_E[1/Nk!]$
307
agree with (respectively) the $O_E$-lattice $\mathfrak{M}_{f,B}$
308
and the $O_E[1/Nk!]$-lattice $\mathfrak{M}_{f,\dR}$ defined in
309
\cite{DFG} using cohomology, with non-constant coefficients, of
310
modular curves. (In \cite{DFG}, see especially Sections 2.2 and
311
5.4, and the paragraph preceding Lemma 2.3.)
312
313
For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
314
module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
315
$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes
316
E_{\lambda}\simeq V_{\lambda}$. Then the $O_{\lambda}$ module
317
$T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable.
318
319
Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
320
generalised elliptic curves with full level-$N$ structure. Let
321
$\mathfrak{E}$ be the universal generalised elliptic curve over
322
$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product
323
of $\mathfrak{E}$ over $M(N)$. (The motive $M_f$ is constructed
324
using a projector on the cohomology of a desingularisation of
325
$\mathfrak{E}^{k-2}$. Realising $M(N)(\CC)$ as the disjoint union
326
of $\phi(N)$ copies of the quotient
327
$\Gamma(N)\backslash\mathfrak{H}^*$ (where $\mathfrak{H}^*$ is the
328
completed upper half plane), and letting $\tau$ be a variable on
329
$\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is isomorphic to
330
the elliptic curve with period lattice generated by $1$ and
331
$\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a variable on
332
the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the fibre product.
333
Then $2\pi i f(\tau)\,d\tau\wedge dz_1\wedge\ldots\wedge dz_{k-2}$
334
is a well-defined differential form on (a desingularisation of)
335
$\mathfrak{E}^{k-2}$ and naturally represents a generating element
336
of $F^{k-1}T_{\dR}$. (At least we can make our choices locally at
337
primes dividing $Nk!$ so that this is the case.) We shall call
338
this element $e(f)$.
339
340
Under the de Rham isomorphism between $V_{\dR}\otimes\CC$ and
341
$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
342
a natural action of complex conjugation on $V_B$, breaking it up
343
into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
344
Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
345
to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
346
$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
347
are rank one $O_E$-modules, but not necessarily free, since the
348
class number of $O_E$ may be greater than one. Choose nonzero
349
elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
350
the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
351
$\Omega_f^{\pm}$ by $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$.
352
353
\section{The Bloch-Kato conjecture}\label{sec:bkconj}
354
In this section we extract from the Bloch-Kato conjecture for
355
$L(f,k/2)$ a prediction about the order of the Shafarevich-Tate
356
group, by analysing the other terms in the formula.
357
358
Let $L(f,s)$ be the $L$-function attached to~$f$. For
359
$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with
360
Euler product
361
$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but
362
there is an analytic continuation given by an integral, as
363
described in the next section. Suppose that $L(f,k/2)\neq 0$. The
364
Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the
365
following equality of fractional ideals of~$E$:
366
$$
367
\frac{L(f,k/2)}{\vol_{\infty}}=
368
\left(\prod_pc_p(k/2)\right)
369
\frac{\#\Sha}{\aaa^{\pm}(\#\Gamma_{\QQ})^2}.
370
$$
371
(Strictly speaking, the conjecture in \cite{BK}
372
is only given for $E=\QQ$.) Here, $\pm$
373
represents the parity of $(k/2)-1$, and $\vol_{\infty}$ is equal
374
to $(2\pi i)^{k/2}$ multiplied by the determinant of the
375
isomorphism $V_B^{\pm}\otimes\CC\simeq
376
(V_{\dR}/F^{k/2})\otimes\CC$, calculated with respect to the
377
lattices $O_E\delta_f^{\pm}$ and the image of $T_{\dR}$. For
378
$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be
379
\begin{align*}
380
\length&\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-
381
\ord_{\lambda}(P_p(p^{-j}))\\
382
=&\length\>\> \left(H^0(\QQ_p,A_{\lambda}(j))/H^0\left(\QQ_p, V_{\lambda}(j)^{I_p}/T_{\lambda}(j)^{I_p}\right)\right).
383
\end{align*}
384
385
We omit the definition of $\ord_{\lambda}(c_p(j))$ for
386
$\lambda\mid p$, which requires one to assume Fontaine's de Rham
387
conjecture (\cite{Fo}, Appendix A6), and depends on the choices of
388
$T_{\dR}$ and $T_B$, locally at~$\lambda$. (We shall mainly be
389
concerned with the $q$-part of the Bloch-Kato conjecture, where~$q$
390
is a prime of good reduction. For such primes, the de Rham
391
conjecture follows from Theorem 5.6 of \cite{Fa1}.)
392
393
\begin{lem}\label{vol}
394
$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and
395
$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.
396
\end{lem}
397
\begin{proof}
398
$\vol_{\infty}$ is also equal to the determinant of the period map
399
from $F^{k/2}V_{\dR}\otimes\CC$ to $V_B^{\pm}\otimes\CC$, with
400
respect to lattices dual to those we used above in the definition
401
of $\vol_{\infty}$ (c.f. the last paragraph of 1.7 of \cite{De2}).
402
We are using here natural pairings. Recall that the index of
403
$O_E\delta_f^{\pm}$ in $T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then
404
the proof is completed by noting that, locally away from primes
405
dividing $Nk!$, the index of $T_{\dR}$ in its dual is equal to the
406
index of $T_B$ in its dual, both being equal to the ideal
407
denoted~$\eta$ in \cite{DFG2}.
408
\end{proof}
409
\begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer.
410
Then the fractional ideal $c_p(j)$ is supported at most on
411
divisors of~$p$.
412
\end{lem}
413
\begin{proof}
414
As on p.~30 of \cite{Fl1}, for odd $l\neq p$,
415
$\ord_{\lambda}(c_p(j))$ is the length of the finite
416
$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
417
where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
418
trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
419
torsion free.
420
\end{proof}
421
422
\begin{lem}\label{local1}
423
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
424
is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
425
$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
426
that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not congruent
427
modulo $\qq$ to any newform of weight~$k$, trivial character and
428
level dividing $N/p$. Then any integer~$j$,
429
$\ord_{\qq}(c_p(j))=0$.
430
\end{lem}
431
\begin{proof}
432
It suffices to show that
433
$$
434
\dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)),
435
$$
436
since this ensures that
437
$H^0(I_p,A_{\qq}(j))=V_{\qq}^{I_p}/T_{\qq}^{I_p}$, hence that
438
$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}^{I_p}/T_{\qq}^{I_p})$. If
439
the dimensions differ then, given that $f$ is not congruent modulo
440
$\qq$ to a newform of level dividing $N/p$, Proposition 2.2 of
441
\cite{L} shows that we are in the situation covered by one of the
442
three cases in Proposition 2.3 of \cite{L}. Since $p\not\equiv
443
-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is
444
unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's
445
result that $N$ is the prime-to-$q$ part of the conductor of
446
$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses
447
the condition $q>k$) implies the existence of a newform of weight
448
$k$, trivial character and level dividing $N/p$, congruent to~$g$
449
modulo $\qq$. This contradicts our hypotheses.
450
\end{proof}
451
452
\begin{remar}
453
For an example of what can be done when~$f$ is congruent to
454
a form of lower level, see the first example in Section~\ref{sec:other_ex}
455
below.
456
\end{remar}
457
458
\begin{lem}\label{at q}
459
If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then
460
$\ord_{\qq}(c_q)=0$.
461
\end{lem}
462
\begin{proof}
463
It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an
464
application, at the end of Section~2.2, of the results of
465
\cite{Fa1}) that $T_{\qq}$ is the
466
$O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered
467
module $T_{\dR}\otimes O_{\qq}$ by the functor they call
468
$\mathbb{V}$. (This property is part of the definition of an
469
$S$-integral premotivic structure given in Section~1.2 of
470
\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
471
of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
472
in Theorem~4.1 of \cite{BK} follows from the first paragraph of
473
2(h) of \cite{Fa1}.)
474
\end{proof}
475
476
\begin{lem}
477
If $A[\lambda]$ is an
478
irreducible representation of $\Gal(\Qbar/\QQ)$,
479
then
480
$$\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0.$$
481
\end{lem}
482
This follows trivially from the definition.
483
484
Putting together the above lemmas we arrive at the following:
485
\begin{prop}\label{sha}
486
Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that $A[\qq]$
487
is an irreducible representation of $\Gal(\Qbar/\QQ)$, where $\qq\mid q$.
488
Assume the same hypotheses as in Lemma \ref{local1},
489
for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$
490
are as in the previous section. If
491
$$
492
\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty})>0
493
$$
494
(with numerator nonzero) then the Bloch-Kato conjecture
495
predicts that
496
$$
497
\ord_{\qq}(\#\Sha)>0.
498
$$
499
\end{prop}
500
501
\section{Congruences of special values}
502
Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
503
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
504
large enough to contain all the coefficients $a_n$ and $b_n$.
505
Suppose that $\qq\mid q$ is a prime of~$E$ such that $f\equiv
506
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$, and assume
507
that $q\nmid N\phi(N)k!$.
508
Choose $\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
509
$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates
510
$T_B^{\pm}$ locally at $\qq$.
511
Make two further assumptions:
512
$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$
513
514
\begin{prop} \label{div}
515
With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
516
\end{prop}
517
\begin{proof} This is based on some of the ideas used in Section 1 of
518
\cite{V}. Note the apparent typo in Theorem~1.13 of \cite{V},
519
which presumably should refer to ``Condition 2''. Since
520
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
521
$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm
522
1=(-1)^{(k/2)-1}$. It is well known, and easy to prove, that
523
$$
524
\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).
525
$$
526
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
527
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
528
where the integral is taken along the positive imaginary axis,
529
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
530
Thus we are reduced
531
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
532
533
Let $\mathcal{D}_0$ be the group of divisors of degree zero
534
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
535
integer $r\geq 0$, let $P_r(R)$ be the additive group of
536
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
537
groups have a natural action of $\Gamma_1(N)$. Let
538
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
539
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
540
541
Via the isomorphism (8) in Section~1.5 of \cite{V}, combined with
542
the argument in 1.7 of \cite{V}, the cohomology class
543
$\omega_f^{\pm}$ corresponds to a modular symbol $\Phi_f^{\pm}\in
544
S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
545
element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_{E,\qq})$. We are
546
now dealing with cohomology over $X_1(N)$ rather than $M(N)$,
547
which is why we insist that $q\nmid \phi(N)$. It follows from the
548
last line of Section~4.2 of \cite{St} that, up to some small
549
factorials which do not matter locally at $\qq$,
550
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
551
(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since
552
$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
553
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
554
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
555
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
556
show is divisible by $\qq$.
557
Similarly
558
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
559
(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of
560
$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
561
Therefore it would suffice to show that, for some $\mu\in O_E$,
562
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
563
$\qq$ in $S_{\Gamma_1(N)}(k,O_{E,\qq})$. It suffices to show that,
564
for some $\mu\in O_E$, the element
565
$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
566
considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
567
non-constant coefficients. This would be the case if
568
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
569
one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
570
a consequence of Theorem 2.1(1) of \cite{FJ}.
571
\end{proof}
572
\begin{remar}\label{sign}
573
The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are
574
equal. They are determined by the eigenvalue of the involution $W_N$,
575
which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and
576
$b_N$ are each $N^{k/2-1}$ times this sign and~$\qq$ has residue
577
characteristic coprime to $2N$. The common sign in the functional
578
equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of
579
$W_N$ acting on~$f$ and~$g$.
580
\end{remar}
581
582
This is analogous to the remark at the end of Section~3 of \cite{CM},
583
which shows that if~$\qq$ has odd residue characteristic and
584
$L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then $L(g,s)$ must vanish to order
585
at least two at $s=k/2$. Note that Maeda's conjecture
586
implies that there are no examples of~$g$ of
587
level one with positive sign in their functional equation such that
588
$L(g,k/2)=0$ (see \cite{CF}).
589
590
\section{Constructing elements of the Shafarevich-Tate group}
591
Let~$f$ and~$g$ be as in the first paragraph of the previous
592
section. In the previous section we showed how the congruence
593
between $f$ and $g$ relates the vanishing of $L(g,k/2)$ to the
594
divisibility by $\qq$ of an ``algebraic part'' of $L(f,k/2)$.
595
Conjecturally the former is associated with the existence of
596
certain algebraic cycles (for $M_g$) while the latter is
597
associated with the existence of certain elements of the
598
Shafarevich-Tate group (for $M_f$, as we saw in \S 4). In this
599
section we show how the congruence, interpreted in terms of Galois
600
representations, provides a direct link between algebraic cycles
601
and the Shafarevich-Tate group.
602
603
For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
604
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
605
$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
606
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
607
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
608
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
609
610
Suppose that $L(g,k/2)=0$. If the sign in the functional equation
611
is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
612
\ref{sign}), this implies that the order of vanishing of $L(g,s)$
613
at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
614
conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
615
is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational
616
rational equivalence classes of null-homologous,
617
algebraic cycles of codimension $k/2$
618
on the motive $M_g$. (This generalises the part
619
of the Birch--Swinnerton-Dyer conjecture which says that for an
620
elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at
621
$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)
622
623
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
624
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
625
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
626
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
627
get (assuming also the Beilinson-Bloch conjecture) a subspace of
628
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
629
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
630
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
631
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
632
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
633
Sections~1 and~6.5 of \cite{Fo2}.
634
635
Similarly, if $L(f,k/2)\neq 0$ then we expect that
636
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
637
coincides with the $\qq$-part of $\Sha$.
638
\begin{thm}\label{local}
639
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
640
$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
641
$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$ and
642
that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a
643
newform of weight~$k$, trivial character and level dividing $N/p$.
644
Suppose that, for all primes $p\mid N$, $\,p\not\equiv
645
-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here
646
$w_p$ is the common eigenvalue of the Atkin-Lehner involution
647
$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of
648
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
649
\end{thm}
650
651
\begin{proof}
652
Take a nonzero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
653
continuity and rescaling we may assume that it lies in
654
$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq
655
H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction (note that
656
$T'_{\qq}(k/2)/\qq T'_{\qq}(k/2)\simeq A'[\qq](k/2)$) we get a
657
nonzero class $c\in H^1(\QQ,A'[\qq](k/2))\simeq
658
H^1(\QQ,A[\qq](k/2))$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$
659
is trivial. It follows that $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
660
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
661
we get a nonzero, $\qq$-torsion class $\gamma\in
662
H^1(\QQ,A_{\qq}(k/2))$.
663
664
Our aim is to show that $\res_p(\gamma)\in
665
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
666
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
667
668
\begin{enumerate}
669
\item {\bf $p\nmid qN$. }
670
671
Consider the $I_p$-cohomology of the short exact sequence
672
$$
673
\begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD},
674
$$
675
where~$\pi$ is multiplication by a uniformising element of
676
$O_{\qq}$. Since in this case $A'_{\qq}(k/2)$ is unramified at
677
$p$, $H^0(I_p, A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is
678
$\qq$-divisible. Therefore $H^1(I_p,A'[\qq](k/2))$ (which,
679
remember, is the same as $H^1(I_p,A[\qq](k/2))$) injects into
680
$H^1(I_p,A'_{\qq}(k/2))$. It follows from the fact that $d\in
681
H^1_f(\QQ,V'_{\qq}(k/2))$ that the image in
682
$H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is zero, hence
683
that the restriction of~$c$ (to $H^1(I_p,A'[\qq](k/2))\simeq
684
H^1(I_p,A[\qq](k/2))$) is zero. Hence the restriction of $\gamma$
685
to $H^1(I_p,A_{\qq}(k/2))$ is also zero. By line~3 of p.~125 of
686
\cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
687
contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
688
to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that
689
$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
690
691
\item {\bf $p\mid N$. }
692
693
First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
694
It suffices to show that
695
$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
696
since then the natural map from
697
$H^0(I_p,V'_{\qq}(k/2))$ to $H^0(I_p, A'_{\qq}(k/2))$ is
698
surjective; this may be done as in the proof of Lemma
699
\ref{local1}. It follows as above that the image of $c\in
700
H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is zero. Then
701
$\res_p(c)$ comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by
702
inflation-restriction. The order of this group is the same as the
703
order of the group $H^0(\QQ_p,A[\qq](k/2))$, which we claim is
704
trivial. By the work of Carayol \cite{Ca1}, $p\mid N$ implies that
705
$V_{\qq}(k/2)$ is ramified at $p$, so $\dim
706
H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim
707
H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
708
consider the case where this common dimension is $1$. The
709
(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha
710
p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication by~$\alpha$
711
on the one-dimensional space $H^0(I_p,V_{\qq}(k/2))$. It
712
follows from Theor\'eme A of \cite{Ca1} that this is the same as
713
the Euler factor at $p$ of $L(f,s)$. By the work of Atkin and
714
Lehner \cite{AL}, it then follows that $p^2\nmid N$ and
715
$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
716
$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
717
$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$ as
718
$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
719
$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
720
$\res_p(\gamma)=0$ and certainly lies in
721
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
722
723
\item {\bf $p=q$. }
724
725
Since $q\nmid N$ is a prime of good reduction for the motive
726
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
727
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
728
$V'_{\qq}$ have the same dimension, where
729
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
730
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
731
As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
732
the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
733
filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we
734
may now prove, in the same manner as Proposition 9.2 of
735
\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.
736
\end{enumerate}
737
\end{proof}
738
739
Theorem~2.7 of \cite{AS} is concerned with verifying local
740
conditions in the case $k=2$, where~$f$ and~$g$ are associated
741
with abelian varieties~$A$ and~$B$. (Their theorem also applies to
742
abelian varieties over number fields.) Our restriction outlawing
743
congruences modulo $\qq$ with cusp forms of lower level is
744
analogous to theirs forbidding~$q$ from dividing Tamagawa factors
745
$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic
746
curve with $\ord_l(j(A))<0$, consideration of a Tate
747
parametrisation shows that if $q\mid c_{A,l}$, i.e., if
748
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
749
at~$l$.)
750
751
In this paper we have encountered two technical problems which we
752
dealt with in quite similar ways:
753
\begin{enumerate}
754
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
755
\item proving local conditions at primes $p\mid N$, for an element
756
of $\qq$-torsion.
757
\end{enumerate}
758
If our only interest was in testing the Bloch-Kato conjecture at
759
$\qq$, we could have made these problems cancel out, as in Lemma
760
8.11 of \cite{DFG}, by weakening the local conditions. However, we
761
have chosen not to do so, since we are also interested in the
762
Shafarevich-Tate group, and since the hypotheses we had to assume
763
are not particularly strong. Note that, since $A[\qq]$ is
764
irreducible, the $\qq$-part of $\Sha$ does not depend on the
765
choice of $T_{\qq}$.
766
767
\section{Examples and Experiments}
768
\label{sec:examples} This section contains tables and numerical
769
examples that illustrate the main themes of this paper. In
770
Section~\ref{sec:vistable}, we explain Table~\ref{tab:newforms},
771
which contains~$16$ examples of pairs $f,g$ such that the
772
Beilinson-Bloch conjecture and Theorem~\ref{local} together imply
773
the existence of nontrivial elements of the Shafarevich-Tate group
774
of the motive attached to~$f$. Section~\ref{sec:howdone} outlines
775
the higher-weight modular symbol computations that were used in
776
making Table~\ref{tab:newforms}. Section~\ref{sec:invis} discusses
777
Table~\ref{tab:invisforms}, which summarizes the results of an
778
extensive computation of conjectural orders of Shafarevich-Tate
779
groups for modular motives of low level and weight.
780
Section~\ref{sec:other_ex} gives specific examples in which
781
various hypotheses fail. Note that in this section ``modular
782
symbol'' has a different meaning from in \S 5, being related to
783
homology rather than cohomology. For precise definitions see
784
\cite{SV}.
785
786
\subsection{Visible $\Sha$ Table~\ref{tab:newforms}}\label{sec:vistable}
787
\begin{table}
788
\caption{\label{tab:newforms}Visible $\Sha$}\vspace{-3ex}
789
790
$$
791
\begin{array}{|c|c|c|c|c|}\hline
792
g & \deg(g) & f & \deg(f) & q's \\\hline
793
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
794
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
795
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
796
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
797
\vspace{-2ex} & & & & \\
798
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
799
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
800
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
801
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
802
\vspace{-2ex} & & & & \\
803
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\
804
\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\
805
\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\
806
\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\
807
\vspace{-2ex} & & & & \\
808
\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\
809
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
810
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
811
\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\
812
\hline
813
\end{array}
814
$$
815
\end{table}
816
817
818
Table~\ref{tab:newforms} on page~\pageref{tab:newforms} lists
819
sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
820
along with at least one prime~$q$ such that there is a prime
821
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
822
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
823
The notation is as follows.
824
The first column contains a label whose structure is
825
\begin{center}
826
{\bf [Level]k[Weight][GaloisOrbit]}
827
\end{center}
828
This label determines a newform $g=\sum a_n q^n$, up to Galois
829
conjugacy. For example, \nf{127k4C} denotes a newform in the third
830
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
831
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
832
\ldots)$, then by the sequence of absolute values $|\mbox{\rm
833
Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
834
being first in the event that the two absolute values are equal,
835
and the first Galois orbit is denoted {\bf A}, the second {\bf B},
836
and so on. The second column contains the degree of the field
837
$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
838
contain~$f$ and its degree, respectively. The fifth column
839
contains at least one prime~$q$ such that there is a prime
840
$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
841
hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
842
satisfied for~$f$,~$g$, and~$\qq$.
843
844
For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
845
prime $q$ appears in the $q$-column, meaning $q^2$ divides the
846
order of the group $S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp})$, defined
847
at the end of 7.3 below.
848
849
850
We describe the first line of Table~\ref{tab:newforms}
851
in more detail. See the next section for further details
852
on how the computations were performed.
853
854
Using modular symbols, we find that there is a newform
855
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
856
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
857
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
858
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
859
coefficients generate a number field~$K$ of degree~$17$, and by
860
computing the image of the modular symbol $XY\{0,\infty\}$ under
861
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
862
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
863
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
864
both equal to
865
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7
866
+ \cdots\in \FF_{43}[[q]].$$
867
868
There is no form in the Eisenstein subspaces of
869
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
870
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
871
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
872
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
873
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
874
of Theorem~\ref{local}, so if $r$ is the dimension of
875
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
876
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
877
878
Recall that since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that
879
$r\geq 2$. Then, since $L(f,k/2)\neq 0$, we expect that the
880
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to
881
the $\qq$-torsion subgroup of $\Sha$. Admitting these assumptions,
882
we have constructed the $\qq$-torsion in $\Sha$ predicted by the
883
Bloch-Kato conjecture.
884
885
For particular examples of elliptic curves one can often find and
886
write down rational points predicted by the Birch and
887
Swinnerton-Dyer conjecture. It would be nice if likewise one could
888
explicitly produce algebraic cycles predicted by the
889
Beilinson-Bloch conjecture in the above examples. Since
890
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
891
0.3.2 of \cite{Z}), so ought to be trivial in
892
$\CH_0^{k/2}(M_g)\otimes\QQ$.
893
894
\subsection{How the computation was performed}\label{sec:howdone}
895
We give a brief summary of how the computation was performed. The
896
algorithms that we used were implemented by the second author, and
897
most are a standard part of MAGMA (see \cite{magma}).
898
899
Let~$g$,~$f$, and~$q$ be some data from a line of
900
Table~\ref{tab:newforms} and let~$N$ denote the level of~$g$. We
901
verified the existence of a congruence modulo~$q$, that
902
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
903
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
904
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
905
906
To prove there is a congruence, we showed that the corresponding
907
{\em integral} spaces of modular symbols satisfy an appropriate
908
congruence, which forces the existence of a congruence on the
909
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
910
irreducible by computing a set that contains all possible residue
911
characteristics of congruences between~$g$ and any Eisenstein
912
series of level dividing~$N$, where by congruence, we mean a
913
congruence for all Fourier coefficients of index~$n$ with
914
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
915
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
916
listing a basis of such~$h$ and finding the possible congruences,
917
where again we disregard the Fourier coefficients of index not
918
coprime to~$N$.
919
920
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
921
modular symbol ${\mathbf
922
e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$ under a map
923
with the same kernel as the period mapping, and found that the
924
image was~$0$. The period mapping sends the modular
925
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
926
so that ${\mathbf e}$ maps to~$0$ implies that
927
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
928
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g)
929
=(-1)^{k/2} g$ which, because of the functional equation, implies
930
that $L'(g,\frac{k}{2})=0$. Table~\ref{tab:newforms} is of
931
independent interest because it includes examples of modular forms
932
of even weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
933
the functional equation. We found no such examples of weights
934
$\geq 8$.
935
936
\subsection{Conjecturally nontrivial $\Sha$}\label{sec:invis}
937
In this section we apply some of the results of
938
Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders
939
of Shafarevich-Tate groups of many modular motives. The results of
940
this section suggest that~$\Sha$ of a modular motive is usually not
941
``visible at level~$N$'', i.e., explained by congruences at level~$N$,
942
which agrees with the observations of \cite{CM} and \cite{AS}. For
943
example, when $k>6$ we find many examples of conjecturally
944
nontrivial~$\Sha$ but no examples of nontrivial visible~$\Sha$.
945
946
For any newform~$f$, let $L(M_f/\QQ,s) = \prod_{i=1}^{d}
947
L(f^{(i)},s)$ where $f^{(i)}$ runs over the
948
$\Gal(\Qbar/\QQ)$-conjugates of~$f$. Let~$T$ be the complex torus
949
$\CC^d/\mathcal{L}$, where the lattice $\mathcal{L}$ is defined by
950
integrating integral cuspidal modular symbols against the
951
conjugates of~$f$. Let $\Omega_{M_f/\QQ}$ denote the volume of
952
the $-1$ eigenspace $T^{-}=\{z \in T : \overline{z}=-z\}$ for
953
complex conjugation on~$T$.
954
955
956
{\begin{table}
957
\vspace{-2ex}
958
\caption{\label{tab:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}
959
\vspace{-4ex}
960
961
$$
962
\begin{array}{|c|c|c|c|}\hline
963
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
964
\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\
965
\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\
966
\nf{263k4B} & 39 & 41^{2} & 263 \\
967
\nf{269k4C} & 39 & 23^{2} & 269 \\
968
\nf{271k4B} & 39 & 29^{2} & 271 \\
969
\nf{281k4B} & 40 & 29^{2} & 281 \\
970
\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\
971
\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\
972
\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\
973
\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\
974
\hline
975
\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\
976
\nf{101k6B} & 24 & 17^{2} & 101 \\
977
\nf{103k6B} & 24 & 23^{2} & 103 \\
978
\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\
979
\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\
980
\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\
981
\nf{157k6B} & 34 & 251^{2} & 157 \\
982
\nf{167k6B} & 40 & 41^{2} & 167 \\
983
\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\
984
\nf{173k6B} & 39 & 71^{2} & 173 \\
985
\nf{181k6B} & 40 & 107^{2} & 181 \\
986
\nf{191k6B} & 46 & 85091^{2} & 191 \\
987
\nf{193k6B} & 41 & 31^{2} & 193 \\
988
\nf{199k6B} & 46 & 200329^2 & 199 \\
989
\hline
990
\nf{47k8B} & 16 & 19^{2} & 47 \\
991
\nf{59k8B} & 20 & 29^{2} & 59 \\
992
\nf{67k8B} & 20 & 29^{2} & 67 \\
993
\nf{71k8B} & 24 & 379^{2} & 71 \\
994
\nf{73k8B} & 22 & 197^{2} & 73 \\
995
\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\
996
\nf{79k8B} & 25 & 307^{2} & 79 \\
997
\nf{83k8B} & 27 & 1019^{2} & 83 \\
998
\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\
999
\nf{89k8B} & 29 & 44491^{2} & 89 \\
1000
\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\
1001
\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\
1002
\nf{103k8B} & 32 & 75367^{2} & 103 \\
1003
\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\
1004
\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\
1005
\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\
1006
\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\
1007
\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\
1008
\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\
1009
\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\
1010
\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\
1011
\hline
1012
\end{array}
1013
$$
1014
\end{table}
1015
\begin{table}
1016
$$
1017
\begin{array}{|c|c|c|c|}\hline
1018
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
1019
\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\
1020
\nf{121k8G} & 12 & 13^{2} & 3, 11 \\
1021
\nf{121k8H} & 12 & 19^{2} & 5, 11 \\
1022
\nf{125k8D} & 16 & 179^{2} & 5 \\
1023
\nf{127k8B} & 39 & 59^{2} & 127 \\
1024
\nf{128k8F} & 4 & 11^{2} & 1 \\
1025
\nf{131k8B} & 43 & 241^{2} \!\cdot\! 817838201^{2}&131\\
1026
\nf{134k8C} & 11 & 61^{2} & 11, 17, 41, 67, 71, 421, 2356138931854759 \\
1027
\nf{137k8B} & 42 & 71^{2} \!\cdot\! 749093^{2} & 137 \\
1028
\nf{139k8B} & 43 & 47^{2} \!\cdot\! 89^{2} \!\cdot\! 1021^{2} & 139 \\
1029
\nf{141k8C} & 14 & 13^{2} & 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\
1030
\nf{142k8B} & 10 & 11^{2} & 3, 53, 71, 56377, 1965431024315921873 \\
1031
\nf{143k8C} & 19 & 307^{2} & 3, 11, 13, 89, 199, 409, 178397,
1032
639259, 17440535
1033
97287 \\
1034
\nf{143k8D} & 21 & 109^{2} & 3, 7, 11, 13, 61, 79, 103, 173, 241,
1035
769, 36583
1036
\\
1037
\nf{145k8C} & 17 & 29587^{2} & 5, 11, 29, 107, 251623, 393577,
1038
518737, 9837145
1039
699 \\
1040
\nf{146k8C} & 12 & 3691^{2} & 11, 73, 269, 503, 1673540153, 11374452082219 \\
1041
\nf{148k8B} & 11 & 19^{2} & 3, 37 \\
1042
\nf{149k8B} & 47 & 11^{4} \!\cdot\! 40996789^{2} & 149\\
1043
1044
\hline
1045
1046
\nf{43k10B} & 17 & 449^{2} & 43 \\
1047
\nf{47k10B} & 20 & 2213^{2} & 47 \\
1048
\nf{53k10B} & 21 & 673^{2} & 53 \\
1049
\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\
1050
\nf{59k10B} & 25 & 37^{2} & 59 \\
1051
\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\
1052
\nf{64k10K} & 2 & 19^{2} & 3 \\
1053
\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\
1054
\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\
1055
\nf{71k10B} & 30 & 1103^{2} & 71 \\
1056
1057
\hline
1058
\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\
1059
\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\
1060
\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\
1061
\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\
1062
\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\
1063
\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\
1064
\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\
1065
\nf{49k12H} & 12 & 271^{2} & 7 \\
1066
\hline
1067
\end{array}
1068
$$
1069
\end{table}
1070
1071
The following lemma may be proved by tensoring the space of
1072
integral modular symbols with $O_E$, then decomposing it in such a
1073
way that $\Omega_{M_f/\QQ}$ becomes the determinant of a diagonal
1074
matrix with $d$ non-zero entries. We omit the somewhat awkward
1075
details.
1076
\begin{lem}\label{lem:lrat}
1077
If $p\nmid Nk!$ is a non-congruence prime for $f$ then the
1078
$p$-parts of
1079
$$
1080
\frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}\qquad\text{and}\qquad
1081
\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\right)
1082
$$
1083
are equal, where $\vol_\infty$ is as in Section~\ref{sec:bkconj}.
1084
\end{lem}
1085
For the rest of this section, {\em we officially assume the
1086
Bloch-Kato conjecture.}
1087
1088
Let~$\mathcal{S}$ be the set of newforms with~level $N$ and weight~$k$
1089
satisfying either $k=4$ and $N\leq 321$, or $k=6$ and $N\leq 199$, or
1090
$k=8$ and $N\leq 133$, or $k=10$ and $N\leq 72$, or $k=12$ and $N\leq
1091
49$. Given $f\in \mathcal{S}$, let~$B$ be the lower bound on $\#\Sha$ defined
1092
as follows:
1093
\begin{enumerate}
1094
\item Let $L_1$ be the rational number $L(M_f,k/2)/\Omega_{M_f}$.
1095
If $L_1=0$ let $B=1$ and terminate.
1096
\item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$.
1097
\item Let $L_3$ be the part of $L_2$ that is coprime to
1098
$p+1$ for every prime~$p$ such that $p^2\mid N$.
1099
\item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic
1100
of any prime of
1101
congruence between~$f$ and a form of weight~$k$ and
1102
lower level. (By congruence here, we mean a congruence for coefficients
1103
$a_n$ with $n$ coprime to the level of~$f$.)
1104
\item Let $B$ be the part of $L_4$ coprime to the residue characteristic
1105
of any prime of congruence
1106
between~$f$ and an Eisenstein series. (This eliminates
1107
residue characteristics of reducible representations.)
1108
\end{enumerate}
1109
Proposition~\ref{sha} and Lemma~\ref{lem:lrat} imply that if
1110
$\ord_p(B)
1111
> 0$, and $p$ is not a congruence prime for $f$, then
1112
$\ord_p(\#\Sha) > 0$. We have left the congruence primes in $B$ in
1113
the starred examples since the squares are still suggestive.
1114
1115
We computed~$B$ for every newform in~$\mathcal{S}$. There are
1116
many examples in which $L_3$ is large, but~$B$ is not, and this is
1117
because of Tamagawa factors. For example, {\bf 39k4C} has
1118
$L_3=19$, but $B=1$ because of a $19$-congruence with a form of
1119
level~$13$; in this case we must have $19\mid c_{13}(2)$, where
1120
$c_{13}(2)$ is as in Section~\ref{sec:bkconj}. See
1121
Section~\ref{sec:other_ex} for more details. Also note that in
1122
every example~$B$ is a perfect square, which is consistent with
1123
the fact \cite{Fl2} that the order of $\Sha$ (if finite) is
1124
necessarily a perfect square. That our computed value of~$B$
1125
should be a square is not a priori obvious.
1126
1127
For simplicity, we discard residue characteristics instead of primes
1128
of rings of integers, so our definition of~$B$ is overly conservative.
1129
For example,~$5$ occurs in row~$2$ of Table~\ref{tab:newforms} but not
1130
in Table~\ref{tab:invisforms}, because \nf{159k4E} is Eisenstein at
1131
some prime above~$5$, but the prime of congruences of
1132
characteristic~$5$ between \nf{159k4B} and \nf{159k4E} is not
1133
Eisenstein.
1134
1135
1136
The newforms for which $B>1$ are given in
1137
Table~\ref{tab:invisforms}. The second column of the table records
1138
the degree of the field generated by the Fourier coefficients
1139
of~$f$. The third contains~$B$. Let~$W$ be the intersection of
1140
the span of all conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and
1141
$W^{\perp}$ the Petersson orthogonal complement of~$W$ in
1142
$S_k(\Gamma_0(N),\ZZ)$. Then the fourth column contains the odd
1143
prime divisors of $\#(S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp}))$, which
1144
are exactly the possible primes of congruence for~$f$. We place a
1145
$*$ next to the four entries of Table~\ref{tab:invisforms} that
1146
also occur in Table~\ref{tab:newforms}.
1147
1148
\subsection{Examples in which hypotheses fail}\label{sec:other_ex}
1149
We have some other examples where forms of
1150
different levels are congruent.
1151
However, Remark~\ref{sign} does not
1152
apply, so that one of the forms could have an odd functional
1153
equation, and the other could have an even functional equation.
1154
For instance, we have a $19$-congruence between the
1155
newforms $g=\nf{13k4A}$ and $f=\nf{39k4C}$ of Fourier
1156
coefficients coprime to $39$.
1157
Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$
1158
has {\it odd} functional equation.
1159
Here~$f$ fails the condition about not being congruent
1160
to a form of lower level, so in Lemma~\ref{local1} it is possible that
1161
$\ord_{\qq}(c_{19}(2))>0$. In fact this does happen. Because
1162
$V'_{\qq}$ (attached to~$g$ of level $81$) is unramified at $p=19$,
1163
$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
1164
two-dimensional. As in (2) of the proof of Theorem~\ref{local},
1165
one of the eigenvalues of $\Frob_p^{-1}$ acting on this
1166
two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
1167
$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
1168
$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
1169
$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
1170
$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
1171
Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
1172
our example here with $p=7$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
1173
non-trivial when $w_p=-1$, so (2) of the proof of Theorem~\ref{local}
1174
does not work. This is just as well, since had it
1175
worked we would have expected
1176
$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
1177
shows not to be the case.
1178
1179
In the following example, the divisibility between the levels is the
1180
other way round. There is a $7$-congruence between $g=\nf{122k6A}$
1181
and $f=\nf{61k6B}$, both $L$-functions have even functional equation,
1182
and $L(g,3)=0$. In the proof of Theorem~\ref{local},
1183
there is a problem with the local condition at $p=2$. The map from
1184
$H^1(I_2,A'[\qq](3))$ to $H^1(I_2,A'_{\qq}(3))$ is not necessarily
1185
injective, but its kernel is at most one dimensional, so we still get
1186
the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
1187
$\FF_{\qq}$-rank at least~$1$ (assuming $r\geq 2$), and thus get
1188
(probable) elements of $\Sha$ for \nf{61k6B}. In particular, these
1189
elements of $\Sha$ are {\it invisible} at level 61. When the levels
1190
are different we are no longer able to apply Theorem 2.1 of
1191
\cite{FJ}. However, we still have the congruences of integral modular
1192
symbols required to make the proof of Proposition \ref{div} go
1193
through. Indeed, as noted above, the congruences of modular forms were
1194
found by producing congruences of modular symbols. Despite these
1195
congruences of modular symbols, Remark 5.3 does not apply, since there
1196
is no reason to suppose that $w_N=w_{N'}$, where $N$ and $N'$ are the
1197
distinct levels.
1198
1199
Finally, there are two examples where we have a form $g$ with even
1200
functional equation such that $L(g,k/2)=0$, and a congruent form
1201
$f$ which has odd functional equation; these are a 23-congruence
1202
between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
1203
between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
1204
$\ord_{s=2}L(f,s)=1$, it ought to be the case that
1205
$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
1206
similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
1207
the appropriate modification of Theorem \ref{local} does not
1208
necessarily provide us with non-trivial $\qq$-torsion in $\Sha$.
1209
It only tells us that the $\qq$-torsion subgroup of
1210
$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It
1211
could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$
1212
appears in the conjectural formula for the first derivative of the
1213
complex $L$ function, evaluated at $s=k/2$, but in combination
1214
with a regulator that we have no way of calculating.
1215
1216
Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
1217
associated with $f$ and $g$ by the construction of Mazur, Tate and
1218
Teitelbaum \cite{MTT}, each divided by a suitable canonical
1219
period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
1220
quite clear what to make of this. This divisibility may be proved
1221
as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
1222
times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
1223
normalised) are congruent $\bmod{\,\qq}$, as a result of the
1224
congruence between the modular symbols out of which they are
1225
constructed. Integrating an appropriate function against these
1226
measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
1227
to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
1228
since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
1229
where the signs in the functional equations of $L(g,s)$ and
1230
$L_q(g,s)$ are the same, positive in this instance. (According to
1231
the proposition in Section 18 of \cite{MTT}, the signs differ
1232
precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)
1233
1234
We also found some examples for which the conditions of Theorem~\ref{local}
1235
were not met. For example, we have a $7$-congruence between
1236
\nf{639k4B} and \nf{639k4H}, but $w_{71}=-1$, so that $71\equiv
1237
-w_{71}\pmod{7}$. There is a similar problem with a $7$-congruence
1238
between \nf{260k6A} and \nf{260k6E} --- here $w_{13}=1$ so that
1239
$13\equiv -w_{13}\pmod{7}$. Finally, there is a $5$-congruence between
1240
\nf{116k6A} and \nf{116k6D}, but here the prime~$5$ is less than the
1241
weight~$6$.
1242
1243
\begin{thebibliography}{AL}
1244
\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
1245
$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
1246
\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
1247
Shafarevich-Tate groups of abelian varieties, preprint.
1248
\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
1249
{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
1250
\bibitem[BCP]{magma}
1251
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
1252
{T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
1253
235--265, Computational algebra and number theory (London, 1993).
1254
\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
1255
of motives, The Grothendieck Festschrift Volume I, 333--400,
1256
Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
1257
\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques
1258
associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
1259
\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
1260
\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
1261
Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em
1262
Duke Math. J. }{\bf 59 }(1989), 785--801.
1263
\bibitem[CM1]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
1264
Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
1265
13--28.
1266
\bibitem[CM2]{CM2} J. E. Cremona, B. Mazur, Appendix to A. Agashe,
1267
W. Stein, Visible evidence for the Birch and Swinnerton-Dyer
1268
conjecture for modular abelian varieties of rank zero, preprint.
1269
\bibitem[CF]{CF} B. Conrey, D. Farmer, On the non-vanishing of
1270
$L_f(s)$ at the center of the critical strip, preprint.
1271
\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
1272
$\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math.
1273
{\bf 179, } 139--172, Springer, 1969.
1274
\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
1275
d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
1276
part 2, 313--346.
1277
\bibitem[DFG1]{DFG} F. Diamond, M. Flach, L. Guo, Adjoint motives
1278
of modular forms and the Tamagawa number conjecture, preprint.
1279
{{\sf
1280
http://www.andromeda.rutgers.edu/\~{\mbox{}}liguo/lgpapers.html}}
1281
\bibitem[DFG2]{DFG2} F. Diamond, M. Flach, L. Guo, The Bloch-Kato
1282
conjecture for adjoint motives of modular forms, {\em Math. Res.
1283
Lett. }{\bf 8 }(2001), 437--442.
1284
\bibitem[Du1]{Du3} N. Dummigan, Symmetric square $L$-functions and
1285
Shafarevich-Tate groups, {\em Experiment. Math. }{\bf 10 }(2001),
1286
383--400.
1287
\bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and
1288
Selmer groups, {\em Math. Res. Lett. }{\bf 8 }(2001), 479--494.
1289
\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
1290
Galois representations, {\em in }Algebraic analysis, geometry and
1291
number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
1292
Press, Baltimore, 1989.
1293
\bibitem[FJ]{FJ} G. Faltings, B. Jordan, Crystalline cohomology
1294
and $\GL(2,\QQ)$, {\em Israel J. Math. }{\bf 90 }(1995), 1--66.
1295
\bibitem[Fl1]{Fl2} M. Flach, A generalisation of the Cassels-Tate
1296
pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.
1297
\bibitem[Fl2]{Fl1} M. Flach, On the degree of modular parametrisations,
1298
S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
1299
ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
1300
Boston Berlin, 1993.
1301
\bibitem[Fo1]{Fo} J.-M. Fontaine, Sur certains types de
1302
repr\'esentations $p$-adiques du groupe de Galois d'un corps
1303
local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
1304
}{\bf 115 }(1982), 529--577.
1305
\bibitem[Fo2]{Fo2} J.-M. Fontaine, Valeurs sp\'eciales des
1306
fonctions $L$ des motifs, S\'eminaire Bourbaki, Vol. 1991/92. {\em
1307
Ast\'erisque }{\bf 206 }(1992), Exp. No. 751, 4, 205--249.
1308
\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture ``epsilon''
1309
for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
1310
51--56.
1311
\bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois
1312
representations coming from modular forms, {\em J. Number Theory
1313
}{\bf 31 }(1989), 133--141.
1314
\bibitem[MTT]{MTT} B. Mazur, J. Tate, J. Teitelbaum, On $p$-adic
1315
analogues of the conjectures of Birch and Swinnerton-Dyer, {\em
1316
Invent. Math. }{\bf 84 }(1986), 1--48.
1317
\bibitem[Ne]{Ne2} J. Nekov\'ar, $p$-adic Abel-Jacobi maps and $p$-adic
1318
heights. The arithmetic and geometry of algebraic cycles (Banff,
1319
AB, 1998), 367--379, CRM Proc. Lecture Notes, 24, Amer. Math.
1320
Soc., Providence, RI, 2000.
1321
\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
1322
{\em Invent. Math. }{\bf 100 }(1990), 419--430.
1323
\bibitem[SV]{SV} W. A. Stein, H. A. Verrill, Cuspidal modular
1324
symbols are transportable, {\em L.M.S. Journal of Computational
1325
Mathematics }{\bf 4 }(2001), 170--181.
1326
\bibitem[St]{St} G. Stevens, $\Lambda$-adic modular forms of
1327
half-integral weight and a $\Lambda$-adic Shintani lifting.
1328
Arithmetic geometry (Tempe, AZ, 1993), 129--151, Contemp. Math.,
1329
174, Amer. Math. Soc., Providence, RI, 1994.
1330
\bibitem[V]{V} V. Vatsal, Canonical periods and congruence
1331
formulae, {\em Duke Math. J. }{\bf 98 }(1999), 397--419.
1332
\bibitem[Z]{Z} S. Zhang, Heights of Heegner cycles and derivatives of $L$-series,
1333
{\em Invent. Math. }{\bf 130 }(1997), 99--152.
1334
\end{thebibliography}
1335
1336
1337
\end{document}
1338
127k4A 43 127k4C 17 [43]
1339
159k4A 5,23 159k4E 8 [5]x[23]
1340
365k4B 29 365k4E 18 [29]x[5] (extra factor of 5 divides the level)
1341
369k4A 5,13 369k4J 9 [5]x[13]x[2]
1342
453k4A 5,17 453k4E 23 [5]x[17]
1343
453k4A 23 151k4A 30 Odd func eq for g
1344
465k4A 11 465k4H 7 [11]x[5]x[2]
1345
477k4A 73 477k4M 12 [73]x[2]
1346
567k4A 23 567k4I 8 [23]x[3]
1347
81k4A 13 567k4L 12 Odd func eq for f, Theorem 4.1 gives nothing.
1348
581k4A 19,19 581k4E 34 [19^2]x[4] (possible BIG Sha, as 19^2 divides MD)
1349
639k4A 7 639k4H 12 [7]
1350
657k4A 5 657k4C 7 [5]x[3]x[2] (see next note)
1351
657k4A 5 657k4G 12 [5]x[4] (does 657k4A make both these visible?)
1352
681k4A 43 227k4A 23 Odd func eq for g
1353
681k4A 59 681k4D 30 [59]x[3]x[2]
1354
684k4C 7,7 684k4K 4 [7^2]x[2] (see note to 581k4A)
1355
95k6A 31,59 95k6D 9 [31]x[59]
1356
116k6A 5 116k6D 6 [5]x[29]x[2]
1357
122k6A 7 61k6B 14 7^2 appears in L(61k6B,3)
1358
122k6A 73 122k6C 6 [73]x[3] (guess that 3 is a bad prime now)
1359
260k6A 7,17 260k6E 4 [7]x[17]x[4] <-- Did not compute MD or LROP
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
\nf{263k4B} & & 41^2 & \\
1373
\nf{269k4C} & & 23^2 & \\
1374
\nf{271k4B} & & 29^2 &\\
1375
\nf{281k4B} & & 29^2\\
1376
\hline
1377
\nf{101k6B} & & 17^2 & 101\\
1378
\nf{103k6B} & & 23^2\\
1379
\nf{111k6C} & & 11^2\\
1380
\nf{153k6G} & & 7^2\\
1381
\nf{157k6B} & & 252^2\\
1382
\nf{167k6B} & & 41^2\\
1383
\nf{172k6B} & & 7^2\\
1384
\nf{173k6B} & & 71^2\\
1385
\nf{181k6B} & & 107^2\\
1386
\hline
1387
\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571\\
1388