Sharedwww / tables / shacomp.texOpen in CoCalc
Author: William A. Stein
1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2
% %
3
% shacomp_v5.tex %
4
% %
5
% Visibility of Shafarevich-Tate groups of modular abelian varieties %
6
% %
7
% William A. Stein and Amod Agashe. %
8
% %
9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
10
11
\documentclass{article}
12
\hoffset=-.06\textwidth
13
\voffset=-.06\textheight
14
\textwidth=1.12\textwidth
15
\textheight=1.12\textheight
16
\bibliographystyle{amsplain}
17
\include{macros}
18
\usepackage[all]{xy}
19
20
21
\DeclareMathOperator{\Res}{Res}
22
\font\cyrbig=wncyr10 scaled \magstep 1
23
\newcommand{\BigSha}{\mbox{\cyrbig X}}
24
25
\newcommand{\marginalfootnote}[1]{\footnote{#1}\marginpar{\hfill {\sf\thefootnote}}}
26
\newcommand{\edit}[1]{\marginalfootnote{#1}}
27
28
\title{\sc Visibility of Shafarevich-Tate groups of modular abelian varieties
29
and the Birch and Swinnerton-Dyer conjecture}
30
\author{Amod Agashe\footnote{Warning: Agashe has not yet
31
read this draft of our preprint.} \and William A.~Stein}
32
33
\begin{document}
34
\UseTips
35
\maketitle
36
37
\begin{abstract}
38
We give examples of abelian subvarieties of $J_0(N)$ that have
39
nontrivial visible Shafarevich-Tate groups. These examples provide
40
evidence for the Birch and Swinnerton-Dyer conjecture, and the methods
41
used to compute them may lead to new connections between this
42
conjecture and the theory of congruences between modular forms.
43
\end{abstract}
44
45
\section{Introduction}\label{sec:intro}
46
Let~$N$ be a positive integer and let $J_0(N)$ be the Jacobian of the
47
modular curve $X_0(N)$ (see, e.g., \cite{diamond-im}). Let~$A$ be an
48
abelian subvariety of $J_0(N)$. Then the inclusion $A\hookrightarrow
49
J_0(N)$ induces a map on Galois cohomology $H^1(\Q,A) \ra
50
H^1(\Q,J_0(N))$, which restricts to a map $\Sha(A) \ra \Sha(J_0(N))$.
51
Following \cite{cremona-mazur}, we call an element $c\in\Sha(A)$
52
\defn{visible} in $J_0(N)$ if it lies in the kernel of this map.
53
When~$c$ is visible the torsor attached to~$c$ is realized as a
54
subvariety of $J_0(N)$; it is a fiber of the quotient map $J_0(N)\ra
55
J_0(N)/A.$
56
57
In~\cite{cremona-mazur}, Cremona and Mazur ask the following question:
58
If $E\subset J_0(N)$ is a elliptic curve, how much of $\Sha(E)$ is
59
visible in an abelian surface~$B$ that is itself a subvariety
60
of~$J_0(N)$? Among the $52$ examples of odd nontrivial $\Sha$ that
61
they consider, $\Sha$ is invisible in at least~$8$ cases. However,
62
they warn that the elliptic curves they consider have small conductor,
63
and the general situation is probably much different. Standard
64
conjectures imply that when $\ell$ is sufficiently large, elements of
65
order~$\ell$ in $\Sha(E)$ can not be visible in an abelian surface; it
66
is thus essential to consider visibility in higher dimensional abelian
67
varieties.
68
69
We generalize the experiment of Cremona and
70
Mazur by removing the constraint that only elliptic curves be
71
considered. The numbers become large at much smaller conductor, and
72
the proportion of~$\Sha$ that is visible declines accordingly.
73
74
In Section~\ref{sec:vis} we give a general definition of visibility,
75
which is motivated by a restriction-of-scalars construction of
76
torsors. Then we prove a theorem that gives a criterion for
77
the existence of nontrivial visible elements of Shafarevich-Tate
78
groups. Section~\ref{sec:computing} describes the algorithms we use to
79
enumerate modular abelian varieties, compute conjectural
80
orders of their Shafarevich-Tate groups, and verify the visibility
81
criterion of Section~\ref{sec:vis} when possible. Section~\ref{sec:tables}
82
presents the results of extensive numerical investigations. It provides
83
the impetus for the conjectures we make and questions we ask in
84
Section~\ref{sec:conj}.
85
86
{\bf Acknowledgment.} It is a pleasure to thank B.~Mazur for his
87
lectures and conversations about visibility, K.~Ribet for explaining
88
congruences to us, and R.~Coleman for helping us to understand
89
component groups. The authors would also like to thank B.~Conrad,
90
J.~Ellenberg, R.~Greenberg, D.~Gross, L.~Merel, B.~Poonen, and R.~Taylor
91
for many helpful discussions.
92
93
\section{Visible cohomology classes}\label{sec:vis}
94
In Section~\ref{sec:torsors} we define the notion of visibility and
95
observe that every cohomology class
96
arises from a torsor that can be constructed geometrically inside of
97
an appropriate restriction of scalars, then we define visible
98
cohomology classes. Section~\ref{sec:visthm}, which should perhaps be
99
omitted from a first reading, gives a criterion for the existence of
100
visible elements of Shafarevich-Tate groups.
101
102
\subsection{Geometric realization of torsors}\label{sec:torsors}
103
Let~$A$ be an abelian variety over a field~$K$. The following
104
definition is due to Mazur.
105
106
\begin{definition}[Visible]
107
Let $\iota:A\hookrightarrow J$
108
be an embedding of~$A$ into an abelian variety~$J$. Then the \defn{visible part
109
of $H^1(K,A)$ with respect to the embedding~$\iota$} is
110
$$\Vis_J(H^1(K,A)) = \Ker(H^1(K,A)\ra{}H^1(K,J)).$$
111
\end{definition}
112
113
The Galois cohomology group $H^1(K,A)$ has a geometric interpretation
114
as the group of classes of torsors~$X$ for~$A$ (see~\cite{lang-tate}).
115
To a cohomology class $c\in H^1(K,A)$, there is a corresponding
116
variety~$X$ over~$K$ and a map $A\cross X \ra X$ that satisfies axioms
117
similar to those for a simply transitive group action.
118
Suppose $\iota: A\ra J$ is an embedding and $c\in \Vis_J(H^1(K,A))$.
119
We have an exact sequence of abelian varieties
120
$0\ra A\ra J\ra B\ra 0$. A piece of the
121
associated long exact sequence of Galois cohomology is
122
$$\cdots \ra J(K)\ra B(K) \ra H^1(K,A) \ra H^1(K,J) \ra \cdots,$$
123
so, because $\iota_*(c)=0$, there is a point $x\in B(K)$ that maps
124
to $c\in H^1(K,A)$. Then the fiber~$X$ over~$x$ is a subvariety of~$J$,
125
which, when equipped with its natural action of~$A$, lies in the class
126
of torsors corresponding to~$c$.
127
128
\begin{proposition}
129
Every element of $H^1(K,A)$ is visible in some abelian variety~$J$.
130
\end{proposition}
131
\begin{proof}
132
Fix $c\in H^1(K,A)$. There is a finite extension~$L$ of~$K$ such that
133
$\res_L(c) = 0\in H^1(L,A)$. Let $J=\Res_{L/K}(A_L)$ be the
134
restriction of scalars down to~$K$ of the abelian variety~$A$, which
135
we view as an abelian variety over~$L$ (see \cite[\S7.6]{neronmodels}).
136
Thus~$J$ is an abelian variety over~$K$ of dimension $[L:K]\cdot \dim(A)$, and for any
137
scheme~$S$ over~$K$, we have a natural bijection $J(S) \ncisom A_L(S_L)$.
138
In particular, $J(A) = A_L(A_L)$, and there is an
139
injection $\iota:A\hookrightarrow J$ attached to $\id_{A_L}\in A_L(A_L)$.
140
Using the Shapiro lemma, one finds that there is a canonical isomorphism
141
$H^1(L,A) \isom H^1(K,J)$ and that
142
$\iota_*(c) = 0\in H^1(K,J)$.
143
\end{proof}
144
\begin{remark}
145
In \cite{cremona-mazur}, J.~de Jong gives a sophisticated proof of
146
the above proposition in the special case when~$A$ is an elliptic curve.
147
His proof involves Azumaya algebras.
148
\end{remark}
149
150
\subsection{Visible elements of Shafarevich-Tate groups}\label{sec:visthm}
151
The Shafarevich-Tate group of an abelian variety over a number
152
field measures the failure of the local-to-global principle
153
for its torsors.
154
\begin{definition}[Shafarevich-Tate group]
155
Let~$A$ be an abelian variety over~$\Q$.
156
The \defn{Shafarevich-Tate group} of $A$ is
157
$$\Sha(A) = \Ker\left(H^1(\Q,A) \ra \prod_{v} H^1(\Q_v,A)\right),$$
158
where the product is over all places of~$\Q$.
159
\end{definition}
160
161
If $\iota:A\hookrightarrow J$ is an embedding, then the
162
visible part of $\Sha(A)$ is
163
$$\Vis_J(\Sha(A)) = \Sha(A) \intersect \Vis_J(H^1(\Q,A)) = \Ker(\Sha(A)\ra \Sha(J)).$$
164
We use the following theorem to produce examples of visible
165
elements of Shafarevich-Tate groups.
166
\begin{theorem}\label{thm:shaexists}
167
Let~$A$ and~$B$ be abelian subvarieties of an abelian variety~$J$ such
168
that $A\intersect B$ is finite and $A(\Q)$ is finite.
169
Let~$N$ be an integer divisible by the primes of bad reduction for~$J$.
170
Assume that~$B$ has purely toric reduction at each prime dividing~$N$.
171
Suppose~$p$ is a prime such that
172
$$p\nmid 2N\cdot \#(J/B)(\Q)_{\tor}\cdot\#B(\Q)_{\tor}\cdot
173
\prod_{\ell\mid N} \#\Phi_{A,\ell}(\Fbar_\ell)\cdot
174
\#\Phi_{B,\ell}(\F_\ell).$$
175
Suppose furthermore that $B[p] \subset A\intersect B$.
176
Then
177
$$B(\Q)/pB(\Q)\hookrightarrow \Vis_J(\Sha(A)).$$
178
\end{theorem}
179
The proof proceeds in four steps. First the torsion and
180
$p$-congruence hypothesis is used to produce an injection $B(\Q)/p
181
B(\Q)\hookrightarrow \Vis_J(H^1(\Q,A))$. Next we perform a local
182
analysis at each place~$v$ of~$\Q$, which proceeds in three steps. At
183
places~$v$ of bad reduction, we use the Mumford-Tate uniformization;
184
at odd primes of good reduction we apply an exactness theorem about
185
N\'eron models; when~$2$ is a place of bad reduction, we modify the
186
situation by a $2$-isogeny and apply another exactness theorem.
187
188
189
\begin{proof}
190
The quotient $J/A$ is an abelian variety~$C$. The long exact sequence
191
of Galois cohomology associated to the short exact sequence
192
$$0 \ra A \ra J \ra C \ra 0$$
193
begins
194
$$0\ra A(\Q) \ra J(\Q) \ra C(\Q) \xrightarrow{\,\delta\,}
195
H^1(\Q,A) \ra \cdots.$$
196
Let~$\phi$ be map $B\ra C$, which is obtained by composing
197
the inclusion $B\hookrightarrow J$ with the quotient map $J\ra C$.
198
Since $B[p]\subset A$, we see that~$\phi$ factors through multiplication by~$p$.
199
We thus obtain the following commutative diagram:
200
$$\xymatrix{
201
& B\ar[d] \ar[r]^{p}& B\ar[d]\\
202
A\ar[r]&J\ar[r]&C.}$$
203
Using that $B(\Q)[p]=0$, we
204
obtain the following diagram, all of whose rows and columns are exact:
205
$$\xymatrix{
206
& K_0\ar[d] & K_1\ar[d]& K_2\ar[d]\\
207
0 \ar[r] & B(\Q) \ar[r]^{p}\ar[d] & B(\Q)\ar[dr]^{\pi} \ar[r]\ar[d]
208
& B(\Q)/pB(\Q)\ar[r]\ar[d] & 0\\
209
0 \ar[r] & J(\Q)/A(\Q)\ar[r]\ar[d] & C(\Q) \ar[r] & \delta(C(\Q)) \ar[r] & 0\\
210
& K_3,
211
}$$
212
where $K_0$, $K_1$ and $K_2$ are the indicated kernels and $K_3$ is the
213
indicated cokernel.
214
The snake lemma gives an exact sequence
215
$$K_0\ra K_1 \ra K_2 \ra K_3.$$
216
Because $B\ra C$ is an isogeny, $K_1\subset B(\Q)_{\tor}$.
217
Since $B(\Q)[p]=0$ and $K_2$ is a $p$-torsion group, the map
218
$K_1\ra K_2$ is the~$0$ map.
219
The quotient
220
$J(\Q)/B(\Q)$ has no $p$-torsion because
221
it is a subgroup of $(J/B)(\Q)$; also, $A(\Q)$ is a finite group,
222
so $K_3 = J(\Q)/(A(\Q)+B(\Q))$ has no $p$-torsion, and the map
223
$K_2\ra K_3$ must be the~$0$ map.
224
We conclude that $K_2=0$.
225
226
The above argument shows that $B(\Q)/p B(\Q)$ is a subgroup of
227
$H^1(\Q,A)$; however, the latter group contains infinitely many elements of
228
order~$p$, whereas $\Sha(A)[p]$ is a finite group, so we must work
229
harder in order to deduce that $B(\Q)/p B(\Q)$ actually lies in
230
$\Sha(A)[p]$. Let $x\in B(\Q)$; we must show
231
that $\pi(x)\in \Sha(A)[p]$. It suffices to show that
232
$\res_v(\pi(x))=0$ for all places~$v$ of~$\Q$.
233
234
At the archimedian place $v=\infty$, the restriction
235
$\res_v(\pi(x))$ is killed by~$2$ and the odd prime~$p$,
236
hence $\res_v(\pi(x))=0$.
237
238
Suppose that~$v$ is a place at which~$J$ has bad reduction.
239
By hypothesis, $B$ has purely toric reduction,
240
so over $\Q_v^{\ur}$
241
there is an isomorphism $B\isom\Gm^d/\Gamma$
242
of $\Gal(\Qbar_v/\Q_v^{\ur})$-modules,
243
for some ``lattice'' $\Gamma$. For example, when
244
$\dim B=1$, this is the Tate curve representation of~$B$.
245
Let~$n$ be the order of the component group of~$B$ at~$v$; thus~$n$
246
equals the order of the cokernel of the valuation
247
map $\Gamma\ra \Z^d$. Choose a representative
248
$P=(x_1,\ldots,x_d)\in\Gm^d$ for the point~$x$.
249
Let $n'=\#\Phi_{B,v}(\F_v)$.
250
Then since~$P$ is rational over $\Q_v$,
251
$n'P$ can be adjusted by elements of~$\Gamma$
252
so that each of its components $x_i\in\Gm$ has valuation~$0$;
253
since~$p$ does not equal the residue characteristic of~$v$,
254
it follows that there is a point $Q\in\Gm^d(\Q_v^{\ur})$
255
such that $pQ = n'P$.
256
Thus the cohomology class $\res_v(\pi(n'x))$ is unramified at~$v$.
257
By \cite[Prop.~3.8]{milne:duality},
258
$$H^1(\Q_v^{\ur}/\Q_v,A(\Q_v^{\ur}))
259
=H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}(\Fpbar)),$$
260
where $\Phi_{A,v}$ is the component group of~$A$ at~$v$.
261
Since~$p$ does not divide $\#\Phi_{A,v}(\Fpbar)$,
262
and $\pi(n'x)$ has order~$p$, it follows that
263
$$0=\res_v(\pi(n'x))=n'\res_v(\pi(x)).$$
264
Since the order of $\res_v(\pi(x))$ is coprime to~$n'$,
265
we conclude that $\res_v(\pi(x))=0$.
266
267
Next suppose that~$J$ has good reduction at~$\ell$
268
and that~$\ell$ is {\em odd}.
269
Let $\cA$, $\cJ$, $\cC$, be the N\'eron models
270
of~$A$,~$J$,~$C$, respectively.
271
Since~$\ell$ is odd, $1=e<\ell-1$, so we may apply
272
\cite[Thm.~7.5.4]{neronmodels} to conclude that
273
the sequence of group schemes
274
$$0\ra \cA \ra \cJ\ra \cC \ra 0$$
275
is exact, in the sense that it
276
is exact as a sequence of sheaves on the
277
\'etale site (see the proof of~\cite[Thm.~7.5.4]{neronmodels}).
278
Thus it is exact on the stalks, so by~\cite[2.9(d)]{milne:etale}
279
the sequence
280
$$0\ra \cA(\Z_v^{\ur})\ra \cB(\Z_v^{\ur}) \ra \cC(\Z_v^{\ur})\ra 0$$
281
is exact. By the N\'eron mapping property, the sequence
282
$$0\ra A(\Q_v^{\ur})\ra B(\Q_v^{\ur}) \ra C(\Q_v^{\ur})\ra 0$$
283
is also exact, so
284
$\res_v(\pi(x))$ is unramified.
285
By \cite[Prop.~I.3.8]{milne:duality},
286
$$H^1(\Q_v^{\ur}/\Q_v,A) \isom H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}(\Fbar_v))=0,$$
287
since~$A$ has good reduction at~$v$.
288
Hence $\res_v(\pi(x))=0$.
289
290
If~$J$ has bad reduction at~$v=2$, then we already dealt with~$2$ above.
291
Consider the case when~$J$ has good reduction at~$2$. The
292
absolute ramification index~$e$ of $\Z_2$ is~$1$, which is
293
{\em not} less than $2-1=1$, so we can not apply \cite[Thm.~7.5.4]{neronmodels}.
294
However, we can modify everything by an isogeny of degree a power
295
of~$2$ and apply a different theorem, as follows.
296
The $2$-primary subgroup~$\Psi$ of $A\intersect B$ is rational
297
over~$\Q$. The abelian varieties
298
$\tilde{J}=J/\Psi$, $\tilde{A}=A/\Psi$, and
299
$\tilde{B}=B/\Psi$ also satisfy the hypothesis of
300
the theorem we wish to prove.
301
By \cite[Prop.~7.5.3(a)]{neronmodels}, the corresponding sequence of
302
N\'eron models
303
$$0\ra\tilde{\cA}\ra\tilde{\cJ}\ra\tilde{\cC}\ra 0$$
304
is exact, so the sequence
305
$$0\ra \tilde{A}(\Q_v^{\ur})\ra\tilde{J}(\Q_v^{\ur})
306
\ra\tilde{C}(\Q_v^{\ur})\ra 0$$
307
is exact. Thus the image of
308
$\res_v(\pi(x))$ in $H^1(\Q_v,\tilde{A})$ is unramified.
309
It equals~$0$, again by \cite[Prop.~3.8]{milne:duality},
310
since the component group of $\tilde{A}$ at~$v$ has order a power
311
of~$2$, whereas $\pi(x)$ has odd prime order~$p$.
312
Thus $\res_v(\pi(x))=0$, since
313
the kernel of $H^1(\Q_v,A)\ra H^1(\Q_v,\tilde{A})$ is a
314
finite group of $2$-power order.
315
\end{proof}
316
317
\subsubsection{Visibility when $A$ also has positive rank}
318
In Theorem~\ref{thm:shaexists}, if the condition that $A(\Q)$ has rank~$0$ is removed,
319
then the proof can be easily modified to show that the kernel of
320
$B(\Q)/p B(\Q) \ra \Vis_J(\Sha(A))$
321
has dimension at most the rank of $A(\Q)$.
322
323
According to \cite{cremona:algs},
324
the smallest conductor elliptic curve~$E$ of rank~$3$ is found in
325
$J=J_0(5077)$. The number $5077$ is prime, and~$J$ decomposes
326
up to isogeny as
327
$A \cross B \cross E,$
328
where each of~$A$, $B$, and~$E$ are abelian subvarieties of~$J$
329
associated to newforms, which have
330
dimensions $205$, $216$, and~$1$, respectively.
331
The modular degree of~$E$ is $1984=2^6\cdot 31$, and
332
the sign of the Atkin-Lehner involution on~$E$ is the same
333
as its sign on~$A$, so $E[31]\subset A$.
334
The numerator of $(5077-1)/12$ is $3^2\cdot 47$, so
335
$31$ is coprime to the orders of any relevant component groups or torsion.
336
Thus $\Vis_J(\Sha(A))$ contains $(\Z/31\Z)^2$.
337
338
339
340
\section{Guide to computing on $J_0(N)$}\label{sec:computing}
341
The Jacobian $J_0(N)$ is equipped with an action of the Hecke algebra~$\T$.
342
Let $f\in S_2(\Gamma_0(N))$ be a newform, and let $I_f\subset\T$
343
be the annihilator of~$f$. The abelian variety~$A_f$ attached
344
to~$f$ is the quotient $J_0(N)/I_f J_0(N)$. Thus $A_f$ is an
345
abelian variety of dimension equal to the number of Galois
346
conjugates of~$f$ and equipped with a faithful action of $\T/I_f$.
347
For the remainder of this section, $A=A_f$ denotes the optimal quotient
348
of $J_0(N)$ attached to the annihilator~$I=I_f$ of a newform~$f$.
349
350
\subsection{The Birch and Swinnerton-Dyer conjecture}
351
The Birch and Swinnerton-Dyer conjecture, as generalized by
352
Tate in~\cite{tate:bsd}, furnishes a conjectural formula
353
for the order of the Shafarevich-Tate group of any new
354
optimal quotient~$A$.
355
In general, it is difficult given~$A$ to compute
356
the conjectural order of~$\Sha(A)$.
357
However, the situation is more optimistic
358
when~$A$ is a new modular abelian variety
359
such that $L(A,1)\neq 0$.
360
For these~$A$ we have devised an algorithm
361
that we use to compute the odd part of the
362
conjectural order of $\Sha(A)$ in many cases.
363
The following is a special case of a much more general conjecture.
364
\begin{conjecture}[Birch and Swinnerton-Dyer]\label{conj:bsd}
365
Suppose $L(A,1)\neq 0$. Then
366
$$\frac{L(A,1)}{\Omega_A} =
367
\frac{\#\Sha(A)\cdot\prod_{p\mid N} c_p}
368
{\# A(\Q)\cdot\#\Adual(\Q)}.$$
369
\end{conjecture}
370
When $L(A,1)\neq 0$, work of Kolyvagin and Logachev
371
\cite{kolyvagin-logachev:finiteness,kolyvagin-logachev:totallyreal}
372
implies that $A(\Q)$, $\Adual(\Q)$, and $\Sha(A)$ are all finite,
373
so the quantities appearing in the above formula make sense.
374
Here $c_p=\#\Phi_{A,p}(\F_p)$, the positive real number~$\Omega_A$ is
375
the measure of~$A(\R)$ with respect to a basis of differentials on
376
the N\'eron model of~$A$, and~$\Adual$ is the abelian variety
377
dual of~$A$.
378
The algorithms described below enable us in my cases to compute the
379
conjectural order of $\Sha(A)$. However, for question of visibility,
380
we instead need to compute the order of $\Sha(\Adual)$. This is no
381
different because the Cassels-Tate pairing implies that
382
$\#\Sha(A) = \#\Sha(\Adual)$.
383
384
\subsection{Modular symbols}\label{modsym}
385
It is not possible to compute very much about $J_0(N)$ without
386
modular symbols, which provide a finite presentation for the homology
387
group $H_1(X_0(N),\Z)$ in terms of paths between elements of
388
$\P^1(\Q) = \Q\union \{\infty\}$.
389
390
The \defn{modular symbol} defined by a pair $\alpha,\beta\in\P^1(\Q)$
391
is denoted $\{\alpha,\beta\}$. This modular symbol should be viewed as
392
the homology class, relative to the cusps,
393
of a geodesic path from~$\alpha$ to~$\beta$ in $\h^*$.
394
The homology group relative to the cusps is a slight enlargement
395
of the usual homology group, in that
396
we allow paths with endpoints in $\P^1(\Q)$ instead of restricting
397
to closed loops.
398
We declare that modular symbols satisfy
399
the following homology relations:
400
if $\alpha,\beta,\gamma \in \Q\union\{\infty\}$, then
401
$$\{\alpha,\beta\} + \{\beta,\gamma\} + \{\gamma,\alpha\} = 0.$$
402
Furthermore, the space of modular symbols is torsion free, so, e.g.,
403
$\{\alpha,\alpha\} = 0$ and
404
$\{\alpha,\beta\} = -\{\beta,\alpha\}$.
405
406
Denote by~$\sM_2$ the free abelian group with basis the set of
407
symbols $\{\alpha,\beta\}$ modulo the three-term homology relations
408
above and modulo any torsion.
409
There is a left action of $\GL_2(\Q)$ on $\sM_2$, whereby
410
a matrix~$g$ acts by
411
$$g\{\alpha, \beta\} = \{g(\alpha), g(\beta)\},$$
412
and~$g$ acts on~$\alpha$ and~$\beta$ by a linear fractional
413
transformation.
414
The space $\sM_2(N)$ of \defn{modular symbols for $\Gamma_0(N)$}
415
is the quotient of $\sM_2$ by the submodule
416
generated by the infinitely many elements
417
of the form $x - g(x)$, for~$x$ in ~$\sM_2$
418
and~$g$ in $\Gamma_0(N)$, and modulo any torsion.
419
A \defn{modular symbol for $\Gamma_0(N)$} is an element of
420
this space. We frequently denote the equivalence
421
class that defines a modular symbol by giving a
422
representative element.
423
424
In \cite{manin:parabolic}, Manin proved that there is
425
a natural injection $H_1(X_0(N),\Z)\hookrightarrow \sM_2(N)$.
426
The image of $H_1(X_0(N),\Z)$ in $\sM_2(N)$ can be identified as follows.
427
Let $\sB_2(N)$ denote the free abelian group whose basis is the finite set
428
$\Gamma_0(N)\backslash \P^1(\Q)$.
429
The \defn{boundary map} $\delta: \sM_2(N)\ra \sB_2(N)$
430
sends $\{\alpha,\beta\}$ to $[\beta]-[\alpha]$, where $[\beta]$
431
denotes the basis element of $\sB_2(N)$ corresponding to $\beta\in\P^1(\Q)$.
432
The kernel $\sS_2(N)$ of~$\delta$ is the subspace of
433
\defn{cuspidal} modular symbols.
434
An element of $\sS_2(N)$ can be thought of as a linear
435
combination of paths
436
in $\h^*$ whose endpoints are cusps, and whose images in $X_0(N)$
437
are a linear combination of loops.
438
We thus obtain a canonical isomorphism $\varphi:\sS_2(N)\ra H_1(X_0(N),\Z)$.
439
440
Part of the utility of modular symbols comes from the classical Abel-Jacobi
441
theorem, which allows us to view $J_0(N)(\C)$ as the quotient
442
$\C^g/H_1(X_0(N),\Z)$,
443
where $H_1(X_0(N),\Z)$ is embedded in
444
$\C^d\ncisom\Hom(S_2(\Gamma_0(N)),\C)$ using the integration pairing.
445
Thus modular symbols give an explicit description of $J_0(N)(\C)$
446
and of its constituent parts as modules over the Hecke algebra.
447
We can also compute Hecke operators using modular symbols.
448
449
For further introductory remarks on modular symbols, see~\cite{stein:modsyms},
450
and for detailed instructions as to how to compute the space of modular symbols
451
and the action of Hecke operators on it, see~\cite{cremona:algs}.
452
453
\subsection{Computing with quotients and subvarieties of $J_0(N)$}
454
First, we describe how to enumerate the newforms of level~$N$. Then
455
we define the modular degree, whose square annihilates the visible part
456
of~$\Sha$. Finally, we describe how to intersect abelian subvarieties
457
of $J_0(N)$.
458
459
\subsubsection{Enumerating quotients}
460
Let $H_1(X_0(N),\Z)^+$ denote the $+1$-eigenspace for the action
461
of the involution induced by complex conjugation.
462
We list all newforms of a given level~$N$ by decomposing the new
463
subspace of $H_1(X_0(N),\Q)^+$ under the action of the the
464
Hecke operators. First we compute the characteristic polynomial of~$T_2$,
465
and use it to break up the full space. We apply this process
466
recursively with $T_3, T_5, \ldots$ until either we have exceeded the
467
bound coming from~\cite{sturm:cong}, or we have found a Hecke
468
operator~$T_n$ whose characteristic polynomial is irreducible. After
469
computing the decomposition, we order the newforms in a way that
470
extends the systematic ordering in~\cite{cremona:algs}: First sort by
471
dimension, with smallest dimension first; within each dimension, sort
472
in binary by the signs of the Atkin-Lehner involutions, e.g., $+++$,
473
$++-$, $+-+$, $+--$, $-++$, etc. When two forms have the same sign
474
sequence, order by $|\Tr(a_p)|$ with ties broken by taking the
475
positive trace first.
476
477
We denote a Galois conjugacy class of newforms by a bold symbol such
478
as $\mathbf{389E}$, which consists of the level and the isogeny class,
479
where $\mathbf{A}$ denotes the first class, $\mathbf{B}$ the second,
480
and so on.
481
482
As discussed in \cite[pg.~5]{cremona:algs}, for certain small levels
483
the above ordering when restricted to elliptic curves does not agree
484
with the ordering used in Cremona's tables. For example, in the
485
present paper our $\mathbf{446B}$ is Cremona's $\mathbf{446D}$.
486
487
\subsubsection{The modular degree\label{modpolar}}
488
A \defn{polarization}~$\lambda$ of an abelian variety~$A$ over~$\Q$ is an isogeny
489
$\lambda:A\ra \Adual$ such that $\lambda_{\Qbar}$
490
arises from an ample invertible sheaf on $A_{\Qbar}$ (see, e.g.,
491
\cite[\S13]{milne:abvars}).
492
Since $J_0(N)$ is a Jacobian, it possesses a canonical
493
polarization arising from the $\theta$-divisor, and this
494
polarization induces the \defn{modular polarization}
495
$\theta: \Adual\ra A$ of $\Adual$.
496
$$\[email protected]=3pc{
497
{\Adual}\[email protected]{^(->}[r]^{\pi^{\vee}\qquad}\ar[dr]^{\theta} &
498
J_0(N)^{\vee} \isom J_0(N)\,\,\,\[email protected]{->>}[d]^{\pi}\\
499
&A.}$$
500
If we view $\Adual$ as an abelian subvariety of $J_0(N)$, then the
501
kernel of~$\theta$ is the intersection of $\Adual$ with $I J_0(N)$; thus
502
the kernel of $\theta$ measures intersections between $\Adual$ and other
503
factors of $J_0(N)$.
504
\begin{definition}[Modular degree]
505
The \defn{modular degree} $m_A$ of $A$ is
506
$\sqrt{\deg(\theta)}$.
507
\end{definition}
508
\noindent{}By \cite[Thm.~13.3]{milne:abvars}, $\deg(\theta)$ is a perfect
509
square, so $m_A$ is an integer.
510
For an algorithm to compute $m_A$, see~\cite{kohel-stein:ants4}.
511
512
The modular degree is of interest because its square
513
annihilates the visible cohomology classes.
514
\begin{proposition}
515
$\displaystyle \Vis_{J_0(N)}(H^1(\Q,\Adual))\subset H^1(\Q,\Adual)[m_A^2]$
516
\end{proposition}
517
\begin{proof}
518
Let $\delta$ be the composite map
519
$\Adual \ra J_0(N)\ra A$. There is a map $\hat{\delta}:A\ra \Adual$
520
such that $\hat{\delta}\circ\delta$ is multiplication by $\deg(\delta)=m_A^2$.
521
Thus
522
$\Ker(H^1(\Q,\Adual)\ra H^1(\Q,J_0(N)))$
523
is contained in $H^1(\Q,\Adual)[m_A^2]$.
524
\end{proof}
525
\begin{remark}
526
When~$A$ has dimension one, the visible part of $H^1(\Q,\Adual)$ is contained
527
in $H^1(\Q,A)[m_A]$. Is this true for~$A$ of all dimensions?
528
\end{remark}
529
530
\subsubsection{Intersecting complex tori}\label{sec:intersect}
531
Consider a complex torus $J=V/\Lambda$, and let
532
$A=V_A/\Lambda_A$ and $B=V_B/\Lambda_B$ be subtori whose
533
intersection $A\intersect B$ is finite.
534
Here $V_A$ and $V_B$ are subspaces of~$V$ and $\Lambda_A$ and $\Lambda_B$
535
are submodules of~$\Lambda$.
536
\begin{proposition}\label{prop:intersection}
537
There is a natural isomorphism of groups
538
$$A\intersect B \isom
539
\left(\frac{\Lambda}{\Lambda_A + \Lambda_B}\right)_{\tor.}$$
540
\end{proposition}
541
\begin{proof}
542
There is an exact sequence
543
$$0\ra A\intersect B \ra A \oplus B \ra J.$$
544
Consider the diagram
545
$$\xymatrix{
546
& {\Lambda_A \oplus\Lambda_B}\ar[d] \ar[r] & {\Lambda} \ar[r]\ar[d]&
547
{\Lambda/(\Lambda_A + \Lambda_B)}\ar[d]\\
548
& {V_A \oplus V_B}\ar[d] \ar[r] & V \ar[r]\ar[d] & {V/(V_A+V_B)}\ar[d]\\
549
{A\intersect B}\ar[r] & A\oplus B\ar[r] & J \ar[r] & J/(A+ B).}$$
550
The snake lemma\index{Snake lemma} gives an exact sequence
551
$$0 \ra
552
A\intersect B \ra
553
\Lambda/(\Lambda_A + \Lambda_B) \ra
554
V/(V_A+V_B).$$
555
Since $V/(V_A+V_B)$ is a $\C$-vector space, the torsion
556
part of $\Lambda/(\Lambda_A + \Lambda_B)$ must map to~$0$.
557
No non-torsion in $\Lambda/(\Lambda_A + \Lambda_B)$ could
558
map to~$0$, because if it did then $A\intersect B$ would not
559
be finite. The lemma follows.
560
\end{proof}
561
562
The following formula for the intersection of~$n$
563
subtori is obtained in a similar way.
564
\begin{proposition}
565
For $i=1,\ldots,n$ let $A_i = V_i/\Lambda_i$ be a subtorus of
566
$J=V/\Lambda$, and assume that each pairwise intersection
567
$A_i \intersect A_j$ is finite.
568
Then
569
$$A_1\intersect \cdots \intersect A_n
570
\isom
571
\left(\frac{\Lambda\oplus \cdots \oplus \Lambda}
572
{f(\Lambda_1\oplus\cdots\oplus \Lambda_n)}\right),$$
573
where $f(x_1,\ldots,x_n)=(x_1-x_2,x_2-x_3,x_3-x_4,\ldots,x_{n-1}-x_n)$.
574
\end{proposition}
575
576
577
\subsection{Computing the conjectural order of $\BigSha(A)$}
578
In this section, we describe how in many cases we can compute the
579
conjectural order of $\Sha(A)$ when $L(A,1)\neq 0$, at least up to a
580
power of~$2$.
581
582
In Section~\ref{sec:torsion}, we bound $\#A(\Q)$ and $\#\Adual(\Q)$.
583
We compute each $c_p$ in Section~\ref{sec:tamagawa},
584
for each~$p$ with $p\mid\mid N$. When $p^2\mid{}N$, it is
585
possible to bound $c_p$; see, e.g., \cite[Cor.~15.2.1]{silverman:aec}
586
where one finds that when $\dim A=1$ and $p^2\mid N$, we have $c_p\leq 4$.
587
In Section~\ref{sec:bsdratio}, we use modular symbols to compute
588
the rational number $L(A,1)/\Omega_A$, up to a bounded Manin constant.
589
590
\subsubsection{Torsion subgroup}\label{sec:torsion}
591
We obtain an upper bound on $\#A(\Q)_{\tor}$ and $\#\Adual(\Q)_{\tor}$
592
as follows.
593
The characteristic polynomial $\chi_p(X)$ of the Hecke operator~$T_p$
594
acting on~$A$ is a monic polynomial having integer coefficients
595
and degree equal to the dimension of~$A$.
596
\begin{proposition}
597
Both $\#A(\Q)_{\tor}$ and $\#\Adual(\Q)_{\tor}$ divide
598
$$\gcd \{ \chi_p(p+1) : (p,2N)=1,\, \text{\rm $p$ prime} \}.$$
599
\end{proposition}
600
\begin{proof}
601
Use the Eichler-Shimura relation and that for primes~$p$ for which $p\nmid 2N$ the maps
602
$A(\Q)_{\tor} \ra \tilde{A}(\F_p)$ and
603
$\Adual(\Q)_{\tor} \ra \tilde{A}^{\vee}(\F_p)$
604
are both injective, and that
605
$\#\Adual(\F_p)_{\tor}=\#\tilde{A}^{\vee}(\F_p)$.
606
\end{proof}
607
608
The difference of two cusps $\alp,\beta \in{} X_0(N)$ defines
609
a point $(\alp)-(\beta) \in J_0(N)(\C)$. Manin observed
610
in \cite{manin:parabolic} that $(0)-(\infty)$ is rational.
611
The order of the image of $(0)-(\infty)$ in $A(\Q)$ can be computed as follows.
612
Let
613
$$V=\Hom(S_2(\Gamma_0(N)),\C),$$
614
and $V_I = \Hom(S_2(\Gamma_0(N))[I],\C)$.
615
The integration pairing $\langle f, \gamma \rangle = 2\pi i \int_\gamma f(z)dz$
616
between homology and cusp forms gives rise to a map $P: H_1(X_0(N),\Q)\ra V_I$.
617
By the Abel-Jacobi theory (see, e.g., \cite[Thm~IV.2.2]{lang:modular}),
618
$A(\C) \isom V_I/P(H_1(X_0(N),\Z))$.
619
\begin{proposition}
620
The order of the image of $(\alp)-(\beta)$ in $A(\C)$
621
equals the order of the image of the modular symbol $\{\alp,\beta\}$
622
in $P(H_1(X_0(N),\Q))/P(H_1(X_0(N),\Z)).$
623
\end{proposition}
624
The quotient appearing in the proposition can be computed algebraically
625
by replacing~$P$ by a map with the same kernel as~$P$. Such a map can
626
be computed using the Hecke operators (see \cite[\S3.7]{stein:phd}).
627
628
\subsubsection{Tamagawa numbers}\label{sec:tamagawa}
629
Suppose~$p$ is a prime that exactly divides~$N$ and let $\Phi_{A,p}$
630
denote the component group of~$A$ at~$p$.
631
We have an exact sequence,
632
$$0\ra \cA_{\Fp}^0\ra \cA_{\Fp} \ra \Phi_{A,p}\ra 0,$$
633
where $\cA_{\Fp}$ is the closed fiber of the N\'eron model of~$A$ over $\Z_p$ and $\cA_{\Fp}^0$
634
is the component of $\cA_{\Fp}$ that contains the identity.
635
A formula for $\#\Phi_{A,p}(\Fbar_p)$ and, up to a power of $2$,
636
for $\#\Phi_{A,p}(\F_p)$,
637
is given in \cite{kohel-stein:ants4} and \cite{stein:compgroup}.
638
639
\subsubsection{Rational part of the special value}\label{sec:bsdratio}
640
As in Section~\ref{sec:torsion},
641
let $P : H_1(X_0(N),\Z) \ra \Hom(S_2(\Gamma_0(N))[I],\C)$
642
be the map induced by integration.
643
Let $P(H_1(X_0(N),\Z))^+$ denote the $+1$-eigenspace for the action
644
of the involution induced by complex conjugation on the image of~$P$.
645
646
\begin{theorem}\label{thm:ratpart}
647
$$\frac{L(A,1)}{\Omega_{A}}
648
= [P(H_1(X_0(N),\Z))^+ : P(\T\{0,\infty\})]/(c_\infty\cdot c_A),$$
649
where $c_\infty$ is the number of components of $A(\R)$ and $c_A$
650
is the Manin constant of~$A$, as defined below.
651
\end{theorem}
652
In order to define the Manin constant of~$A$,
653
let $\cA$ denote the N\'eron model of~$A$ over~$\Z$.
654
\begin{definition}[Manin constant]
655
The \defn{Manin constant}~$c_A$ of~$A$ is the index
656
$$c_A := [S_2(\Gamma_0(N);\Z)[I]:H^0(\cA,\Omega_{\cA/\Z})].$$
657
\end{definition}
658
In the definition, we have implicitly mapped $H^0(\cA,\Omega_{\cA/\Z})$ into
659
$S_2(\Gamma_0(N);\Q)$ using the composition of the following maps:
660
$$H^0(\cA,\Omega_{\cA/\Z}) \ra
661
H^0(\cJ,\Omega_{\cJ/\Z})[I] \ra
662
H^0(J,\Omega_{J/\Q})[I] \ra
663
S_2(\Gamma_0(N);\Q)[I].$$
664
For a discussion of why $H^0(\cA,\Omega_{\cA/\Z})$ is in fact
665
contained in $S_2(\Gamma_0(N);\Z)[I]$, see \cite{agashe-stein:manin}.
666
\begin{theorem}
667
If $\ell \mid c_A$ then $\ell^2 \mid 4N$.
668
\end{theorem}
669
\begin{proof}
670
See~\cite[\S4]{mazur:rational} when~$A$ has dimension~$1$,
671
and \cite{agashe-stein:manin} in general.
672
\end{proof}
673
674
We now give the proof of Theorem~\ref{thm:ratpart}.
675
\begin{proof}[Proof of Theorem~\ref{thm:ratpart}]
676
Let $H=H_1(X_0(N),\Z)$ and $S=S_2(\Gamma_0(N))$.
677
There is a perfect pairing $\T \cross S \ra \Z$ given by
678
$\langle T_n, f\rangle = a_n(f)$, which
679
induces a canonical isomorphism of rings $\T\isom \Hom_\Z(S,\Z)$,
680
where $\Hom_\Z(S,\Z)$ is a ring under multiplication of functions.
681
The subring $W=\Hom_\Z(S[I],\Z)$ of $\Hom_\Z(S[I],\R)$ is
682
isomorphic to $\T/I$, since $S[I]$ is saturated in~$S$.
683
Thus
684
\begin{eqnarray*}
685
[W : P(\{0,\infty\})W] &=& [W:P(\T \{0,\infty\})]\\
686
&=& [W:P(H)^+] \cdot [P(H)^+ : P(\T \{0,\infty\})].
687
\end{eqnarray*}
688
To complete the proof, observe that
689
that $\Omega_A = [W:P(H)^+] \cdot c_\infty\cdot c_A$ and observe
690
that multiplication by $P(\{0,\infty\})$ has determinant
691
$\prod_{i=1}^d 2\pi i \int_{\{0,\infty\}} f^{(i)} = \pm L(A,1)$.
692
\end{proof}
693
694
\subsection{Emerton's work}
695
When~$N$ is prime, M.~Emerton has proved in \cite{emerton:myconj}
696
that $\#A_f(\Q)$ and $c_p(A_f)$ divide the numerator of $(N-1)/12$.
697
698
\section{Visibility tables}\label{sec:tables}
699
The tables in this section guide and motivate the conjectures
700
and questions of Section~\ref{sec:conj}.
701
702
In {\bf Table~\ref{table:invisible}}, we list each of the $8$ invisible odd
703
Shafarevich-Tate groups found in \cite{cremona-mazur}, and
704
prove\footnote{This computation is currently only partially complete.} that
705
they are visible in some $J_0(Nq)$.
706
707
{\bf Table~\ref{table:prime}} lists every quotient $A_f$ of $J_0(p)$
708
with $p\leq 2593$ and $L(A_f,1)\neq 0$ such that the BSD conjecture
709
predicts that $\#\Sha(\Adual_f)$ is divisible by an odd
710
prime. In addition, the table contains data that can frequently be
711
used in conjuction with Theorem~\ref{thm:shaexists} to deduce that
712
there are visible elements of $\Sha(\Adual_f)$. When the {\bf B}
713
column is labeled NONE then there is definitely nothing in
714
$\Sha(\Adual_f)$ of the predicted order. When the {\bf B} column
715
contains an elliptic curve, its rank has been computed and is~$2$, so
716
there are visible elements of $\Sha(\Adual_f)$. When the {\bf B}
717
column contains an abelian variety of dimension greater than~$1$, we
718
have verified that $L(B,1)=0$, so the BSD conjecture predicts that
719
$B(\Q)$ is infinite; however, we have not proved that $B(\Q)$ is
720
infinite. If we assume that $B(\Q)$ is infinite, it follows in these cases that
721
$\Sha(A)$ is visible in $J_0(p)$. Note that~$B$ has rank~$2$ over the
722
Hecke algebra here, so the results of \cite{gross-zagier} say
723
nothing about $B(\Q)$.
724
725
{\bf Table~\ref{table:prime2}} continues the computations of Table~\ref{table:prime}
726
up to level $5647$. For each prime~$p$ between $2609$ and $5674$, we computed each
727
factor $A$ such that $L(A,1)\neq 0$ and the odd part of
728
$\#\Shaan(A)$ is nontrivial. We then found all factors~$B$ such that
729
$L(B,1)=0$ and there is a mod~$\ell$ congruence between~$A$ and~$B$,
730
where $\ell\mid \#\Shaan(A)$. The column labeled~$N$ gives the level,
731
the column labeled $d(A)$ gives the dimension of~$A$, the column labeled
732
$d(B)$ gives the dimension of~$B$, and the column labeled ``cong'' gives
733
the odd part of $\gcd(\# A\intersect B, \#\Shaan(A))$.
734
735
{\bf Table~\ref{table:mordell-weil}}
736
lists every quotient $A_f$ of $J_0(N)$ with $N\leq 1642$ such
737
that $L(A_f,1)=0$ but the sign in the functional equation for~$f$ is $+1$.
738
For each such $A_f$, we looked for an abelian variety~$B$ such that~$B$
739
has rank~$0$ and $A_f^{\vee}$ probably gives rise to odd visible
740
elements of $\Sha(B)$. This table contains initial data towards the idea of constructing
741
points on high-rank abelian varieties by constructing visible elements
742
of Shafarevich-Tate groups using, e.g., Euler system methods.
743
For example, to prove that $A=\mathbf{1061B}$ really has positive rank,
744
we consider the variety $B=\mathbf{1061D}$.
745
To prove that $A(\Q)\neq 0$, it suffices to construct an appropriate element of $\Sha(B)$
746
and show that this element is visible in $A+B\subset J_0(1061)$.
747
748
{\bf Table~\ref{table:motive}} suggests a first tenuous step towards a
749
computational theory of motives attached to modular forms of weight
750
greater than two. This table is organized like
751
Table~\ref{table:prime}, except that the abelian varieties are
752
replaced by motives attached to weight~$4$ modular forms. For
753
example, at prime level~$127$ there is a $17$-dimensional motive $\cM$
754
such that $\Sha(\cM)(2)$ seems to contain elements of order~$43$. The
755
computations used to suggest this conclusion were carried out using
756
algorithms for higher weight modular symbols as described in
757
\cite{merel:1585}, \cite{stein:phd}, and \cite{stein-verrill:periods}.
758
759
760
\subsection{Odd invisible $\BigSha$ in \cite{cremona-mazur}}
761
\label{table:invisible}
762
$$
763
\begin{array}{lccll}
764
\mbox{\rm\bf E}&\#\Sha(E)& \text{mod deg}(E) & \mbox{\rm\bf F}
765
& \text{Where $\Sha(E)$ is visible}\\
766
& & & & \vspace{-3ex} \\
767
\mbox{\rm\bf 2849A}& 3^2 &2^5\cdot 5\cdot 61&\mbox{\rm\bf NONE}& \text{visible using an elliptic curve at level $3\cdot 2849$}\\
768
\mbox{\rm\bf 3364C}& 7^2 &2^6\cdot3^2\cdot5^2\cdot7 &\mbox{\rm\bf none}& \text{visible using a $3$-dimensional $F$ at level $3364$}\\
769
\mbox{\rm\bf 4229A}& 3^2 &2^3\cdot3\cdot7\cdot13 &\mbox{\rm\bf none}& \text{not visible at level $4299$,}\\
770
&&&&\text{???}\\
771
\mbox{\rm\bf 4343B}& 3^2 &2^4\cdot1583 &\mbox{\rm\bf NONE}& ???\\
772
\mbox{\rm\bf 4914N}& 3^2 &2^4\cdot 3^5 &\mbox{\rm\bf none}& ??? \\
773
\mbox{\rm\bf 5054C}& 3^2 &2^3\cdot 3^3\cdot 11&\mbox{\rm\bf none}& ???\\
774
\mbox{\rm\bf 5073D}& 3^2 &2^5\cdot 3\cdot 5\cdot7\cdot23
775
&\mbox{\rm\bf none}& ???\\
776
\mbox{\rm\bf 5389A}& 3^2 &2^2\cdot 2333 &\mbox{\rm\bf NONE}& \text{visible using an elliptic curve at level $7\cdot 5389$}\\
777
\end{array}
778
$$
779
780
\comment{\subsubsection*{Remarks}
781
The elliptic curve~$E$ denoted {\bf 3364C} is labeled ``none'' because there is no
782
{\em elliptic curve} that satisfies an appropriate $7$-congruence with
783
{\bf 3364C}. However, the modular degree is divisible by~$7$, so there
784
must be some abelian subvariety that satisfies a $7$-congruence with~$E$.
785
Computing, we find a $3$-dimensional abelian variety~$A$ such that $T_2$, $T_3$,
786
and $T_5$ satisfies the polynomials $x^3$, $x^3 + 5x^2 + 6x + 1$, and
787
$x^3 + 5x^2 + 6x + 1$ on~$A$, respectively. Furthermore, $L(A,1)=0$, so
788
the BSD conjecture strongly suggests that there are elements of $\Sha(E)$
789
of order~$7$ that are visible in $E+A \subset J_0(3364)$.
790
}
791
792
\newpage
793
\subsection{Visibility of $\BigSha$ at prime level}\label{table:prime}
794
The entries in the columns ``mod deg'' and ``$\Shaan$'' are only really
795
the odd parts of ``mod deg'' and ``$\Shaan$''. Theorem~\ref{thm:shaexists} does not
796
apply to the two entries marked with a $*$.\vspace{-.25ex}
797
{\small
798
$$
799
\begin{array}{lccclcc}
800
\mbox{\rm\bf A}& \mbox{\rm dim}& \Shaan(A) & \mbox{\rm mod deg}(A) & \mbox{\rm\bf B} & \mbox{\rm dim} & \mbox{\rm mod deg} (B)\\
801
& & & & & & \vspace{-3ex} \\
802
\mbox{\rm\bf 389E}& 20 &5^{2}&5&\mbox{\rm\bf 389A}& 1 &5\\
803
\mbox{\rm\bf 433D}& 16 &7^{2}&3\cdot7\cdot37&\mbox{\rm\bf 433A}& 1 &7\\
804
\mbox{\rm\bf 563E}& 31 &13^{2}&13&\mbox{\rm\bf 563A}& 1 &13\\
805
\mbox{\rm\bf 571D}& 2 &3^{2}&3^{2}\cdot127&\mbox{\rm\bf 571B}& 1 &3\\
806
\mbox{\rm\bf 709C}& 30 &11^{2}&11&\mbox{\rm\bf 709A}& 1 &11\\
807
\mbox{\rm\bf 997H}& 42 &3^{4}&3^{2}&\mbox{\rm\bf 997B}& 1 &3\\
808
\mbox{\rm\bf 1061D}& 46 &151^{2}&61\cdot151\cdot179&\mbox{\rm\bf 1061B}& 2 &151\\
809
\mbox{\rm\bf 1091C}& 62 &7^{2}&1&\mbox{\rm NONE} & & \\
810
\mbox{\rm\bf 1171D}& 53 &11^{2}&3^{4}\cdot11&\mbox{\rm\bf 1171A}& 1 &11\\
811
\mbox{\rm\bf 1283C}& 62 &5^{2}&5\cdot41\cdot59&\mbox{\rm NONE} & & \\
812
\mbox{\rm\bf 1429B}& 64 &5^{2}&1&\mbox{\rm NONE} & & \\
813
\mbox{\rm\bf 1481C}& 71 &13^{2}&5^{2}\cdot2833&\mbox{\rm NONE} & & \\
814
\mbox{\rm\bf 1483D}& 67 &3^{2}\cdot5^{2}&3\cdot5&\mbox{\rm\bf 1483A}& 1 &3\cdot5\\
815
\mbox{\rm\bf 1531D}*& 73 &3^{2}&3&\mbox{\rm\bf 1531A}& 1 &3\\
816
\mbox{\rm\bf 1559B}& 90 &11^{2}&1&\mbox{\rm NONE} & & \\
817
\mbox{\rm\bf 1567D}& 69 &7^{2}\cdot41^{2}&7\cdot41&\mbox{\rm\bf 1567B}& 3 &7\cdot41\\
818
\mbox{\rm\bf 1613D}& 75 &5^{2}&5\cdot19&\mbox{\rm\bf 1613A}& 1 &5\\
819
\mbox{\rm\bf 1621C}& 70 &17^{2}&17&\mbox{\rm\bf 1621A}& 1 &17\\
820
\mbox{\rm\bf 1627C}& 73 &3^{4}&3^{2}&\mbox{\rm\bf 1627A}& 1 &3^{2}\\
821
\mbox{\rm\bf 1693C}& 72 &1301^{2}&1301&\mbox{\rm\bf 1693A}& 3 &1301\\
822
\mbox{\rm\bf 1811D}& 98 &31^{2}&1&\mbox{\rm NONE} & & \\
823
\mbox{\rm\bf 1847B}& 98 &3^{6}&1&\mbox{\rm NONE} & & \\
824
\mbox{\rm\bf 1871C}& 98 &19^{2}&14699&\mbox{\rm NONE} & & \\
825
\mbox{\rm\bf 1877B}& 86 &7^{2}&1&\mbox{\rm NONE} & & \\
826
\mbox{\rm\bf 1907D}& 90 &7^{2}&3\cdot5\cdot7\cdot11&\mbox{\rm\bf 1907A}& 1 &7\\
827
\mbox{\rm\bf 1913B}& 1 &3^{2}&3\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\
828
\mbox{\rm\bf 1913E}& 84 &5^{4}\cdot61^{2}&5^{2}\cdot61\cdot103&\mbox{\rm\bf 1913A,C}& 1,2 &3\cdot5^{2}, 5^2\cdot 61\\
829
\mbox{\rm\bf 1933C}*& 83 &3^{2}\cdot7^{2}&3\cdot7&\mbox{\rm\bf 1933A}& 1 &3\cdot7\\
830
\mbox{\rm\bf 1997C}& 93 &17^{2}&1&\mbox{\rm NONE} & & \\
831
\mbox{\rm\bf 2027C}& 94 &29^{2}&29&\mbox{\rm\bf 2027A}& 1 &29\\
832
\mbox{\rm\bf 2029C}& 90 &5^{2}\cdot269^{2}&5\cdot269&\mbox{\rm\bf 2029A}& 2 &5\cdot269\\
833
\mbox{\rm\bf 2039F}& 99 &3^{4}\cdot5^{2}&19\cdot29\cdot7759\cdot3214201&\mbox{\rm NONE} & & \\
834
\mbox{\rm\bf 2063C}& 106 &13^{2}&61\cdot139&\mbox{\rm NONE} & & \\
835
\mbox{\rm\bf 2089J}& 91 &11^{2}&3\cdot5\cdot11\cdot19\cdot73\cdot139&\mbox{\rm\bf 2089B}& 1 &11\\
836
\mbox{\rm\bf 2099B}& 106 &3^{2}&1&\mbox{\rm NONE} & & \\
837
\mbox{\rm\bf 2111B}& 112 &211^{2}&1&\mbox{\rm NONE} & & \\
838
\mbox{\rm\bf 2113B}& 91 &7^{2}&1&\mbox{\rm NONE} & & \\
839
\mbox{\rm\bf 2161C}& 98 &23^{2}&1&\mbox{\rm NONE} & & \\
840
\mbox{\rm\bf 2213C}& 101 & 3^4 & ? & \mbox{\rm NONE} & & \\
841
\mbox{\rm\bf 2239B}& 110 & 11^4 & 1 & \mbox{\rm NONE} & & \\
842
\mbox{\rm\bf 2251E}& 99 & 37^2 & 37 & \mbox{\rm\bf 2251A} & 1 & 37\\
843
\mbox{\rm\bf 2273C}& 105 & 7^2 & ? & \mbox{\rm NONE}& & \\
844
\mbox{\rm\bf 2287B}& 109 & 71^2 & 1 & \mbox{\rm NONE}& & \\
845
\mbox{\rm\bf 2293C}& 96 & 479^2& 479 & \mbox{\rm\bf 2293A} & 2 & 479\\
846
\mbox{\rm\bf 2311B}& 110 & 5^2 & 1 & \mbox{\rm NONE}& & \\
847
\mbox{\rm\bf 2333C}& 101 &83341^{2}&83341&\mbox{\rm\bf 2333A}& 4 &83341\\
848
\mbox{\rm\bf 2339C}& 114 &3^{8}&6791&\mbox{\rm NONE} & & \\
849
\mbox{\rm\bf 2411B}& 123 &11^{2}&1&\mbox{\rm NONE} & & \\
850
\mbox{\rm\bf 2593C}& 109 &67^2\cdot 2213^2 & 67 \cdot 2213&\mbox {\bf 2593A}& 4
851
& 67 \cdot 2213\\
852
\end{array}
853
$$
854
}
855
856
\subsection{More $\Sha$ at prime level}\label{table:prime2}
857
Only odd parts of $\Shaan$ and congruences are given.
858
Observe that $\Shaan$ is only visible roughly 10 percent of the time!
859
As the level gets large, we find that there is almost always some
860
nontrivial $\Sha$ in a large-dimensional factor of $J_0(p)$, and that
861
this $\Sha$ is invisible.
862
(Warning: In making this table, $53$ primes below $5647$ were not analyzed.)
863
{\tiny
864
$$
865
\hspace{-6em}\begin{array}{lcccc|}
866
N & d(A) & \Shaan & d(B) & cong\\
867
\mathbf{2609} & 127 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\
868
\mathbf{2617} & 114 & 11^{2}\cdot19^{2} & 2 & 11\cdot19 \\
869
\mathbf{2647} & 117 & 13^{2} & \text{NONE} & \\
870
\mathbf{2659} & 123 & 53^{2} & \text{NONE} & \\
871
\mathbf{2663} & 132 & 43^{2} & \text{NONE} & \\
872
\mathbf{2671} & 122 & 37^{2} & \text{NONE} & \\
873
\mathbf{2677} & 115 & 3^{2} & 1 & 3 \\
874
\mathbf{2693} & 122 & 3^{4} & \text{NONE} & \\
875
\mathbf{2699} & 125 & 19^{2} & \text{NONE} & \\
876
\mathbf{2707} & 119 & 5^{2} & \text{NONE} & \\
877
\mathbf{2713} & 118 & 19^{2} & \text{NONE} & \\
878
\mathbf{2731} & 124 & 53^{2} & \text{NONE} & \\
879
\mathbf{2749} & 124 & 7^{2} & \text{NONE} & \\
880
\mathbf{2767} & 125 & 5^{2} & \text{NONE} & \\
881
\mathbf{2789} & 136 & 83^{2} & \text{NONE} & \\
882
\mathbf{2791} & 135 & 29^{2} & \text{NONE} & \\
883
\mathbf{2797} & 119 & 11^{2} & 1 & 11 \\
884
\mathbf{2819} & 138 & 13^{2} & \text{NONE} & \\
885
\mathbf{2837} & 128 & 23^{2} & 1 & 23 \\
886
\mathbf{2843} & 129 & 3^{6}\cdot587^{2} & \text{NONE} & \\
887
\mathbf{2851} & 129 & 7^{2} & \text{NONE} & \\
888
\mathbf{2861} & 133 & 11^{4}\cdot61^{2} & 2 & 11\cdot61 \\
889
\mathbf{2879} & 148 & 97^{2} & \text{NONE} & \\
890
\mathbf{2903} & 150 & 643^{2} & \text{NONE} & \\
891
\mathbf{2939} & 150 & 17^{2}\cdot19^{2} & \text{NONE} & \\
892
\mathbf{2953} & 127 & 29^{2} & 1 & 29 \\
893
\mathbf{2963} & 134 & 5^{2}\cdot31^{2}\cdot61^{2} & 2 & 31\cdot61
894
\\
895
\mathbf{2969} & 136 & 103^{2} & \text{NONE} & \\
896
\mathbf{2999} & 161 & 1459^{2} & \text{NONE} & \\
897
\mathbf{3001} & 132 & 3^{4} & \text{NONE} & \\
898
\mathbf{3011} & 146 & 5^{2}\cdot101^{2} & \text{NONE} & \\
899
\mathbf{3019} & 130 & 3259^{2} & 2 & 3259 \\
900
\mathbf{3041} & 147 & 103^{2} & \text{NONE} & \\
901
\mathbf{3067} & 134 & 5^{4} & \text{NONE} & \\
902
\mathbf{3079} & 148 & 131^{2} & \text{NONE} & \\
903
\mathbf{3083} & 141 & 179^{2} & \text{NONE} & \\
904
\mathbf{3089} & 135 & 5^{2}\cdot131^{2} & 2 & 5\cdot131 \\
905
\mathbf{3109} & 136 & 5^{2} & \text{NONE} & \\
906
\mathbf{3119} & 164 & 11^{2}\cdot59^{2} & \text{NONE} & \\
907
\mathbf{3181} & 144 & 43^{2} & \text{NONE} & \\
908
\mathbf{3187} & 139 & 3^{4} & \text{NONE} & \\
909
\mathbf{3191} & 167 & 53^{2} & \text{NONE} & \\
910
\mathbf{3203} & 143 & 13^{2} & \text{NONE} & \\
911
\mathbf{3221} & 149 & 7^{2}\cdot41^{2} & \text{NONE} & \\
912
\mathbf{3229} & 142 & 3^{2} & \text{NONE} & \\
913
\mathbf{3251} & 166 & 3^{4} & \text{NONE} & \\
914
\mathbf{3257} & 143 & 13^{2} & \text{NONE} & \\
915
\mathbf{3271} & 146 & 7^{4}\cdot43^{2}\cdot71^{2} & 3 &
916
7\cdot43\cdot71 \\
917
\mathbf{3299} & 164 & 6131^{2} & \text{NONE} & \\
918
\mathbf{3301} & 145 & 5^{2} & \text{NONE} & \\
919
\mathbf{3319} & 158 & 5^{4} & \text{NONE} & \\
920
\mathbf{3323} & 155 & 179^{2} & \text{NONE} & \\
921
\mathbf{3329} & 157 & 83^{2} & \text{NONE} & \\
922
\mathbf{3331} & 152 & 937^{2} & \text{NONE} & \\
923
\mathbf{3343} & 148 & 7^{2}\cdot53^{2} & \text{NONE} & \\
924
\mathbf{3347} & 150 & 139^{2} & \text{NONE} & \\
925
\mathbf{3359} & 174 & 67^{4} & \text{NONE} & \\
926
\mathbf{3371} & 159 & 1259^{2} & \text{NONE} & \\
927
\mathbf{3391} & 159 & 29^{2} & \text{NONE} & \\
928
\mathbf{3407} & 170 & 499^{2} & \text{NONE} & \\
929
\mathbf{3433} & 148 & 5^{4}\cdot7^{2} & \text{NONE} & \\
930
\mathbf{3449} & 168 & 107^{2} & \text{NONE} & \\
931
\mathbf{3461} & 167 & 83^{2} & \text{NONE} & \\
932
\mathbf{3463} & 151 & 199^{2} & 2 & 199 \\
933
\mathbf{3467} & 162 & 5^{4} & \text{NONE} & \\
934
\mathbf{3469} & 151 & 47^{2} & \text{NONE} & \\
935
\mathbf{3491} & 168 & 67^{2} & \text{NONE} & \\
936
\mathbf{3511} & 166 & 37^{2} & \text{NONE} & \\
937
\mathbf{3527} & 179 & 659^{2} & \text{NONE} & \\
938
\mathbf{3529} & 153 & 79^{2} & \text{NONE} & \\
939
\mathbf{3533} & 164 & 3^{4} & \text{NONE} & \\
940
\mathbf{3539} & 170 & 1871^{2} & \text{NONE} & \\
941
\mathbf{3541} & 156 & 5^{4} & \text{NONE} & \\
942
\mathbf{3557} & 156 & 229^{2} & \text{NONE} & \\
943
\mathbf{3559} & 170 & 1109^{2} & \text{NONE} & \\
944
\end{array}
945
\begin{array}{|lcccc|}
946
N & d(A) & \Shaan & d(B) & cong\\
947
\mathbf{3571} & 163 & 67^{2} & \text{NONE} & \\
948
\mathbf{3583} & 161 & 3319^{2} & 2 & 3319 \\
949
\mathbf{3607} & 159 & 7^{4}\cdot19^{2} & \text{NONE} & \\
950
\mathbf{3613} & 156 & 7^{2} & \text{NONE} & \\
951
\mathbf{3617} & 165 & 3^{2} & \text{NONE} & \\
952
\mathbf{3623} & 172 & 3^{6} & \text{NONE} & \\
953
\mathbf{3631} & 172 & 433^{2} & \text{NONE} & \\
954
\mathbf{3643} & 160 & 5^{2} & \text{NONE} & \\
955
\mathbf{3659} & 181 & 3^{2}\cdot11^{4} & \text{NONE} & \\
956
\mathbf{3671} & 193 & 509^{2} & \text{NONE} & \\
957
\mathbf{3691} & 166 & 353^{2} & \text{NONE} & \\
958
\mathbf{3701} & 174 & 3^{4}\cdot281^{2} & 2 & 3^{2}\cdot281 \\
959
\mathbf{3709} & 164 & 3^{12} & \text{NONE} & \\
960
\mathbf{3719} & 188 & 13^{2}\cdot977^{2} & \text{NONE} & \\
961
\mathbf{3739} & 166 & 83^{2} & \text{NONE} & \\
962
\mathbf{3761} & 176 & 677^{2} & \text{NONE} & \\
963
\mathbf{3769} & 168 & 13^{2} & \text{NONE} & \\
964
\mathbf{3779} & 187 & 73^{2}\cdot149^{2} & 1 & 73 \\
965
\mathbf{3797} & 172 & 19^{2} & \text{NONE} & \\
966
\mathbf{3803} & 171 & 2531^{2} & \text{NONE} & \\
967
\mathbf{3821} & 182 & 307^{2} & \text{NONE} & \\
968
\mathbf{3823} & 173 & 7^{2} & \text{NONE} & \\
969
\mathbf{3863} & 191 & 11^{2}\cdot23^{2}\cdot311^{2} &
970
\text{NONE} & \\
971
\mathbf{3907} & 168 & 3^{4} & \text{NONE} & \\
972
\mathbf{3919} & 182 & 71^{2} & \text{NONE} & \\
973
\mathbf{3929} & 185 & 877^{2} & \text{NONE} & \\
974
\mathbf{3931} & 174 & 31^{2} & \text{NONE} & \\
975
\mathbf{3943} & 173 & 2479319^{2} & 4 & 2479319 \\
976
\mathbf{3967} & 180 & 3^{6}\cdot13^{2} & 1 & 3\cdot13 \\
977
\mathbf{4007} & 195 & 7321^{2} & \text{NONE} & \\
978
\mathbf{4013} & 176 & 61^{2} & \text{NONE} & \\
979
\mathbf{4019} & 186 & 3^{4}\cdot5^{2}\cdot7^{4} & \text{NONE} & \\
980
\mathbf{4021} & 182 & 5^{4}\cdot71^{2} & \text{NONE} & \\
981
\mathbf{4027} & 174 & 29^{2}\cdot79^{2} & 2 & 29\cdot79 \\
982
\mathbf{4049} & 186 & 5^{2}\cdot3491^{2} & \text{NONE} & \\
983
\mathbf{4057} & 173 & 103^{2} & \text{NONE} & \\
984
\mathbf{4079} & 212 & 5^{2}\cdot157^{2}\cdot179^{2} &
985
\text{NONE} & \\
986
\mathbf{4091} & 203 & 7^{4} & \text{NONE} & \\
987
\mathbf{4093} & 174 & 3^{2}\cdot89^{4} & 2 & 89^{2} \\
988
\mathbf{4099} & 185 & 3^{4}\cdot19^{2} & \text{NONE} & \\
989
\mathbf{4111} & 190 & 229^{2} & \text{NONE} & \\
990
\mathbf{4139} & 188 & 29^{2}\cdot67^{2} & 1 & 67 \\
991
\mathbf{4153} & 177 & 7^{2} & \text{NONE} & \\
992
\mathbf{4157} & 193 & 373^{2} & \text{NONE} & \\
993
\mathbf{4159} & 188 & 997^{2} & \text{NONE} & \\
994
\mathbf{4177} & 183 & 3^{2}\cdot17^{2} & \text{NONE} & \\
995
\mathbf{4217} & 186 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\
996
\mathbf{4219} & 190 & 71^{2} & \text{NONE} & \\
997
\mathbf{4229} & 1 & 3^{2} & \text{NONE} & \\
998
\mathbf{4229} & 194 & 3^{4} & \text{NONE} & \\
999
\mathbf{4231} & 201 & 3^{6} & \text{NONE} & \\
1000
\mathbf{4253} & 184 & 3^{6}\cdot2843^{2} & 3 & 3^{3}\cdot2843 \\
1001
\mathbf{4261} & 185 & 5^{2} & \text{NONE} & \\
1002
\mathbf{4271} & 210 & 163^{2}\cdot853^{2} & \text{NONE} & \\
1003
\mathbf{4273} & 183 & 181^{2} & \text{NONE} & \\
1004
\mathbf{4283} & 198 & 683^{2} & \text{NONE} & \\
1005
\mathbf{4289} & 205 & 8807^{2} & \text{NONE} & \\
1006
\mathbf{4339} & 196 & 17^{2} & \text{NONE} & \\
1007
\mathbf{4349} & 191 & 127^{2} & \text{NONE} & \\
1008
\mathbf{4357} & 187 & 7^{2}\cdot13^{2}\cdot17^{2} & 1 & 7\cdot13
1009
\\
1010
\mathbf{4373} & 199 & 3^{12}\cdot29^{2} & \text{NONE} & \\
1011
\mathbf{4391} & 222 & 5^{4}\cdot372037^{2} & \text{NONE} & \\
1012
\mathbf{4409} & 200 & 157^{2} & \text{NONE} & \\
1013
\mathbf{4421} & 206 & 1523^{2} & \text{NONE} & \\
1014
\mathbf{4423} & 200 & 3^{6}\cdot587^{2} & \text{NONE} & \\
1015
\mathbf{4441} & 198 & 59^{2}\cdot101^{2} & \text{NONE} & \\
1016
\mathbf{4451} & 213 & 809^{2} & \text{NONE} & \\
1017
\mathbf{4457} & 199 & 337^{2} & \text{NONE} & \\
1018
\mathbf{4463} & 213 & 8951^{2} & \text{NONE} & \\
1019
\mathbf{4483} & 193 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\
1020
\mathbf{4517} & 201 & 181^{2} & \text{NONE} & \\
1021
\mathbf{4519} & 202 & 2503^{2} & \text{NONE} & \\
1022
\mathbf{4547} & 205 & 73^{2} & 1 & 73 \\
1023
\mathbf{4549} & 203 & 19^{2}\cdot53^{2} & \text{NONE} & \\
1024
\mathbf{4591} & 215 & 6317^{2} & \text{NONE} & \\
1025
\end{array}
1026
\begin{array}{|lcccc}
1027
N & d(A) & \Shaan & d(B) & cong\\
1028
\mathbf{4597} & 200 & 7^{2}\cdot17^{2} & \text{NONE} & \\
1029
\mathbf{4603} & 198 & 829^{2} & \text{NONE} & \\
1030
\mathbf{4621} & 196 & 13^{2} & \text{NONE} & \\
1031
\mathbf{4639} & 218 & 89^{2} & \text{NONE} & \\
1032
\mathbf{4649} & 215 & 751^{2} & \text{NONE} & \\
1033
\mathbf{4651} & 210 & 13^{4} & \text{NONE} & \\
1034
\mathbf{4673} & 207 & 11^{2}\cdot197^{2} & \text{NONE} & \\
1035
\mathbf{4691} & 216 & 43^{2} & \text{NONE} & \\
1036
\mathbf{4729} & 204 & 673^{2} & \text{NONE} & \\
1037
\mathbf{4733} & 210 & 17^{2} & \text{NONE} & \\
1038
\mathbf{4783} & 210 & 797^{2} & \text{NONE} & \\
1039
\mathbf{4789} & 206 & 13^{4} & \text{NONE} & \\
1040
\mathbf{4799} & 230 & 3^{2}\cdot7^{2}\cdot12203^{2} & 1 & 3\cdot7
1041
\\
1042
\mathbf{4801} & 213 & 60271^{2} & \text{NONE} & \\
1043
\mathbf{4813} & 207 & 3^{2}\cdot6883^{2} & \text{NONE} & \\
1044
\mathbf{4817} & 214 & 283^{2} & \text{NONE} & \\
1045
\mathbf{4831} & 217 & 1151^{2} & \text{NONE} & \\
1046
\mathbf{4861} & 216 & 204749^{2} & \text{NONE} & \\
1047
\mathbf{4877} & 219 & 3^{4}\cdot103^{2} & \text{NONE} & \\
1048
\mathbf{4931} & 240 & 17^{2}\cdot37^{2}\cdot43^{2} &
1049
\text{NONE} & \\
1050
\mathbf{4933} & 211 & 239^{2} & \text{NONE} & \\
1051
\mathbf{4957} & 212 & 5^{2} & \text{NONE} & \\
1052
\mathbf{4967} & 236 & 7^{2}\cdot53881^{2} & \text{NONE} & \\
1053
\mathbf{4969} & 220 & 11^{4} & \text{NONE} & \\
1054
\mathbf{4973} & 223 & 5^{2}\cdot11^{2} & \text{NONE} & \\
1055
\mathbf{4993} & 215 & 4013^{2} & \text{NONE} & \\
1056
\mathbf{4999} & 224 & 985121^{2} & \text{NONE} & \\
1057
\mathbf{5003} & 220 & 97^{2}\cdot1861^{2} & 3 & 97\cdot1861 \\
1058
\mathbf{5009} & 223 & 23^{2}\cdot977^{2} & \text{NONE} & \\
1059
\mathbf{5011} & 229 & 11^{4} & \text{NONE} & \\
1060
\mathbf{5021} & 225 & 1609^{2} & \text{NONE} & \\
1061
\mathbf{5023} & 221 & 51431^{2} & \text{NONE} & \\
1062
\mathbf{5039} & 251 & 166363^{2} & \text{NONE} & \\
1063
\mathbf{5051} & 239 & 13^{2}\cdot2633^{2} & \text{NONE} & \\
1064
\mathbf{5059} & 229 & 5^{2}\cdot13^{2}\cdot31^{2} & \text{NONE} & \\
1065
\mathbf{5077} & 216 & 283^{2} & \text{NONE} & \\
1066
\mathbf{5081} & 240 & 19^{2}\cdot149^{2} & \text{NONE} & \\
1067
\mathbf{5099} & 251 & 7^{4}\cdot11^{2}\cdot461^{2} &
1068
\text{NONE} & \\
1069
\mathbf{5113} & 223 & 19^{2}\cdot61^{2} & \text{NONE} & \\
1070
\mathbf{5119} & 232 & 53^{2}\cdot103^{2} & \text{NONE} & \\
1071
\mathbf{5153} & 223 & 3^{4}\cdot41^{2} & \text{NONE} & \\
1072
\mathbf{5167} & 231 & 367^{2} & \text{NONE} & \\
1073
\mathbf{5171} & 249 & 73^{2}\cdot773^{2} & 1 & 73 \\
1074
\mathbf{5179} & 226 & 7^{2}\cdot13^{2} & \text{NONE} & \\
1075
\mathbf{5189} & 240 & 83^{2} & \text{NONE} & \\
1076
\mathbf{5197} & 223 & 37^{2} & \text{NONE} & \\
1077
\mathbf{5209} & 227 & 181^{2}\cdot1471^{2} & \text{NONE} & \\
1078
\mathbf{5227} & 232 & 3^{2}\cdot7717^{2} & \text{NONE} & \\
1079
\mathbf{5231} & 255 & 4507^{2} & \text{NONE} & \\
1080
\mathbf{5233} & 223 & 163^{2} & \text{NONE} & \\
1081
\mathbf{5237} & 229 & 7^{2} & \text{NONE} & \\
1082
\mathbf{5261} & 239 & 24103^{2} & \text{NONE} & \\
1083
\mathbf{5273} & 227 & 17389^{2} & \text{NONE} & \\
1084
\mathbf{5279} & 263 & 120431^{2} & \text{NONE} & \\
1085
\mathbf{5281} & 232 & 67^{2} & \text{NONE} & \\
1086
\mathbf{5297} & 238 & 397^{2} & \text{NONE} & \\
1087
\mathbf{5303} & 247 & 13^{2}\cdot73^{2}\cdot15467^{2} &
1088
\text{NONE} & \\
1089
\mathbf{5309} & 247 & 1822693^{2} & \text{NONE} & \\
1090
\mathbf{5323} & 233 & 3^{4}\cdot120563^{2} & 3 & 120563 \\
1091
\mathbf{5333} & 237 & 967^{2} & \text{NONE} & \\
1092
\mathbf{5347} & 231 & 3643^{2} & \text{NONE} & \\
1093
\mathbf{5501} & 250 & 163^{2} & \text{NONE} & \\
1094
\mathbf{5503} & 241 & 7^{2}\cdot17^{2} & \text{NONE} & \\
1095
\mathbf{5507} & 252 & 103^{2}\cdot233^{2} & \text{NONE} & \\
1096
\mathbf{5519} & 278 & 61^{2}\cdot211469^{2} & \text{NONE} & \\
1097
\mathbf{5521} & 244 & 5^{4} & \text{NONE} & \\
1098
\mathbf{5531} & 253 & 977^{2} & \text{NONE} & \\
1099
\mathbf{5563} & 246 & 3^{4}\cdot1213^{2} & \text{NONE} & \\
1100
\mathbf{5569} & 239 & 3^{4}\cdot5^{2}\cdot13^{2} & \text{NONE} & \\
1101
\mathbf{5573} & 247 & 9901^{2} & \text{NONE} & \\
1102
\mathbf{5581} & 242 & 28927^{2} & \text{NONE} & \\
1103
\mathbf{5591} & 282 & 3^{2}\cdot13^{4}\cdot1061^{2} &
1104
\text{NONE} & \\
1105
\mathbf{5639} & 278 & 229717^{2} & \text{NONE} & \\
1106
\mathbf{5641} & 244 & 41^{2}\cdot431^{2} & \text{NONE} & \\
1107
\mathbf{5647} & 245 & 4463^{2} & \text{NONE} & \\
1108
\end{array}
1109
$$
1110
}
1111
1112
\subsection{Mordell-Weil groups of positive even rank and the $\BigSha$ they probably induce}
1113
\label{table:mordell-weil}
1114
1115
\noindent
1116
$$\begin{array}{lclccc}
1117
L(1)=0 & $d$ &
1118
\hspace{-.5em}L(1)\neq 0 &
1119
\hspace{-.6em}$d$ &
1120
\hspace{-.8em}\text{cong} &
1121
\hspace{-.7em}
1122
L(1)/\Omega \cdot 2^*\\
1123
& & & & & \vspace{-2ex}\\
1124
\mbox{\bf 389A} & 1 & \mbox{\bf 389E} & 20 & 5 & 25/97\\
1125
\mbox{\bf 433A} & 1 & \mbox{\bf 433D} & 16 & 7 & 49/9\\
1126
\mbox{\bf 446B} & 1 & \mbox{\bf 446F} & 8 & 11 & 121/3\\
1127
\mbox{\bf 563A} & 1 & \mbox{\bf 563E} & 31 & 13 & 169/281\\
1128
\mbox{\bf 571B} & 1 & \mbox{\bf 571D} & 2 & 3 & 9\\
1129
\mbox{\bf 643A} & 1 & \text{NONE} & & & \\
1130
\mbox{\bf 655A} & 1 & \mbox{\bf 655D} & 13 & 9 & 81\\
1131
\mbox{\bf 664A} & 1 & \mbox{\bf 664F} & 8 & 5 & 25\\
1132
\mbox{\bf 681C} & 1 & \mbox{\bf 681B} & 1 & 3 & 9\\
1133
\mbox{\bf 707A} & 1 & \mbox{\bf 707G} & 15 & 13 & 169\\
1134
\mbox{\bf 718B} & 1 & \mbox{\bf 718F} & 7 & 7 & 49\\
1135
\mbox{\bf 794A} & 1 & \mbox{\bf 794G} & 12 & 11 & 121/3\\
1136
\mbox{\bf 817A} & 1 & \mbox{\bf 817E} & 15 & 7 & 49/5\\
1137
\mbox{\bf 916C} & 1 & \mbox{\bf 916G} & 9 & 11 & 121\\
1138
\mbox{\bf 944E} & 1 & \mbox{\bf 944O} & 6 & 7 & 49\\
1139
\mbox{\bf 997B} & 1 & \mbox{\bf 997H} & 42 & 3 & 81/83\\
1140
\mbox{\bf 997C} & 1 & \mbox{\bf 997H} & 42 & 3 & 81/83\\
1141
\mbox{\bf 1001C} & 1 & \mbox{\bf 1001F} & 3 & 3 & 27\\
1142
\mbox{\bf 1001C} & 1 & \mbox{\bf 1001L} & 7 & 7 & 49\\
1143
\mbox{\bf 1028A} & 1 & \mbox{\bf 1028E} & 14 & 11 & 3267\\
1144
\mbox{\bf 1034A} & 1 & \text{NONE} & & & \\
1145
\mbox{\bf 1041B} & 2 & \mbox{\bf 1041E} & 4 & 5 & 25\\
1146
\mbox{\bf 1041B} & 2 & \mbox{\bf 1041J} & 13 & 25 & 625\\
1147
\mbox{\bf 1058C} & 1 & \mbox{\bf 1058D} & 1 & 5 & 25\\
1148
\mbox{\bf 1061B} & 2 & \mbox{\bf 1061D} & 46 & 151 & 22801/265\\
1149
\mbox{\bf 1070A} & 1 & \mbox{\bf 1070M} & 7 & 15 & 75\\
1150
\mbox{\bf 1073A} & 1 & \text{NONE} & & & \\
1151
\mbox{\bf 1077A} & 1 & \mbox{\bf 1077J} & 15 & 9 & 81\\
1152
\mbox{\bf 1088J} & 1 & \mbox{\bf 1088R} & 2 & 3 & 9\\
1153
\mbox{\bf 1094A} & 1 & \mbox{\bf 1094F} & 13 & 11 & 121/3\\
1154
\mbox{\bf 1102A} & 1 & \mbox{\bf 1102K} & 4 & 3 & 9\\
1155
\mbox{\bf 1126A} & 1 & \mbox{\bf 1126F} & 11 & 11 & 121\\
1156
\mbox{\bf 1132A} & 1 & \mbox{\bf 1132F} & 12 & 5 & 225\\
1157
\mbox{\bf 1137A} & 1 & \mbox{\bf 1137C} & 14 & 9 & 81\\
1158
\mbox{\bf 1141A} & 1 & \mbox{\bf 1141I} & 22 & 7 & 1524537/41\\
1159
\end{array}
1160
\begin{array}{lclccc}
1161
& & & & & \\
1162
& & & & & \vspace{-2ex}\\
1163
\mbox{\bf 1143C} & 1 & \mbox{\bf 1143J} & 9 & 11 & 121\\
1164
\mbox{\bf 1147A} & 1 & \mbox{\bf 1147H} & 23 & 5 & 225/19\\
1165
\mbox{\bf 1171A} & 1 & \mbox{\bf 1171D} & 53 & 11 & 121/195\\
1166
\mbox{\bf 1246C} & 1 & \mbox{\bf 1246B} & 1 & 5 & 25\\
1167
\mbox{\bf 1309B} & 1 & \text{NONE} & & & \\
1168
\mbox{\bf 1324A} & 1 & \mbox{\bf 1324E} & 14 & 9 & 6561\\
1169
\mbox{\bf 1325E} & 1 & \mbox{\bf 1325T} & 11 & 9 & 2187\\
1170
\mbox{\bf 1363B} & 2 & \mbox{\bf 1363F} & 25 & 31 & 961/5\\
1171
\mbox{\bf 1431A} & 1 & \mbox{\bf 1431L} & 14 & 3 & 9\\
1172
\mbox{\bf 1436A} & 1 & \text{NONE} & & & \\
1173
\mbox{\bf 1443C} & 1 & \mbox{\bf 1443G} & 5 & 7 & 49\\
1174
\mbox{\bf 1446A} & 1 & \mbox{\bf 1446N} & 7 & 3 & 9\\
1175
\mbox{\bf 1466B} & 1 & \mbox{\bf 1466H} & 23 & 13 & \hspace{-2em}\mbox{{\tiny 4331806939187/367}}\\
1176
\mbox{\bf 1477A} & 1 & \mbox{\bf 1477C} & 24 & 13 & 169\\
1177
\mbox{\bf 1480A} & 1 & \mbox{\bf 1480G} & 5 & 7 & 49\\
1178
\mbox{\bf 1483A} & 1 & \mbox{\bf 1483D} & 67 & 15 & 225/247\\
1179
\mbox{\bf 1525C} & 1 & \mbox{\bf 1525O} & 16 & 7 & 49\\
1180
\mbox{\bf 1531A} & 1 & \mbox{\bf 1531D} & 73 & 3 & 3/85\\
1181
\mbox{\bf 1534B} & 1 & \mbox{\bf 1534J} & 6 & 3 & 3\\
1182
\mbox{\bf 1567B} & 3 & \mbox{\bf 1567D} & 69 & 287 & 82369/261\\
1183
\mbox{\bf 1570B} & 1 & \mbox{\bf 1570J} & 6 & 11 & 121\\
1184
\mbox{\bf 1576A} & 1 & \mbox{\bf 1576E} & 14 & 11 & 121\\
1185
\mbox{\bf 1591A} & 1 & \mbox{\bf 1591F} & 35 & 31 & 6727/19\\
1186
\mbox{\bf 1594A} & 1 & \mbox{\bf 1594J} & 17 & 3 & 3370648239/19\\
1187
\mbox{\bf 1608A} & 1 & \mbox{\bf 1608J} & 6 & 13 & 169\\
1188
\mbox{\bf 1611D} & 1 & \mbox{\bf 1611O} & 11 & 9 & 81\\
1189
\mbox{\bf 1613A} & 1 & \mbox{\bf 1613D} & 75 & 5 & 25/403\\
1190
\mbox{\bf 1615A} & 1 & \mbox{\bf 1615J} & 13 & 9 & 102141\\
1191
\mbox{\bf 1621A} & 1 & \mbox{\bf 1621C} & 70 & 17 & 289/135\\
1192
\mbox{\bf 1627A} & 1 & \mbox{\bf 1627C} & 73 & 9 & 81/271\\
1193
\mbox{\bf 1633A} & 3 & \mbox{\bf 1633D} & 27 & 189 & 35721\\
1194
\mbox{\bf 1639B} & 1 & \mbox{\bf 1639G} & 34 & 17 & 680017/25\\
1195
\mbox{\bf 1641B} & 1 & \mbox{\bf 1641J} & 24 & 23 & 529\\
1196
\mbox{\bf 1642A} & 1 & \mbox{\bf 1642D} & 14 & 7 & 49\\
1197
& & & & & \\
1198
\end{array}$$
1199
1200
\mbox{}\par\noindent
1201
{{\bf 643A}, {\bf 1034A}, {\bf 1073A}, {\bf 1309B} all have modular degree
1202
a power of~$2$; {\bf 1436A} has modular degree divisible by~$3$.
1203
1204
\newpage
1205
\subsection{Conjecturally visible $\BigSha$ of modular motives of weight~$4$}
1206
\label{table:motive}
1207
Suppose $f$ and $g$ are elements of $S_4(\Gamma_0(N))$ such that $p^2 \mid L(\sM_f,2)/\Omega$
1208
and $L(\sM_g,2)=0$. If~$f$ and~$g$ satisfy a ``$p$-congruence'',
1209
does~$p$ then divide the ``visible part'' of $\Sha(\sM_f(2))$?
1210
1211
$$
1212
\begin{array}{lcclcl}
1213
%\hspace{2em}\chi\hspace{2em}\mbox{} &
1214
\sM_f \hspace{1em}\mbox{}&
1215
\text{dim}
1216
&
1217
\hspace{2em}p^2\hspace{2em}\mbox{}
1218
& \sM_g & \text{dim}\\
1219
& & & & & \vspace{-2ex} \\
1220
1221
%(1,1) & \mbox{\bf 99C} & 8 & 19^2 & \mbox{\bf 99B} & 2 \\
1222
\mbox{\bf 127k4C} & 17 & 43^2 & \mbox{\bf 127k4A} & 1\\
1223
\mbox{\bf 159k4E} & 8 & 23^2 & \mbox{\bf 159k4B} & 1\\
1224
%(0,0,1) & \mbox{\bf 200E} & 8 & 7^2 & \mbox{\bf 200G} & 2\\
1225
\mbox{\bf 365k4E} & 18 & 29^2 & \mbox{\bf 365k4A} & 1\\
1226
\mbox{\bf 369k4I} & 9 & 13^2 & \mbox{\bf 369k4A} & 1\\
1227
\mbox{\bf 453k4E} & 23 & 17^2 & \mbox{\bf 453k4A} & 1\\
1228
\mbox{\bf 465k4H} & 7 & 11^2 & \mbox{\bf 465k4A} & 1\\
1229
\mbox{\bf 477k4L} & 12 & 73^2 & \mbox{\bf 477k4A} & 1\\
1230
\mbox{\bf 567k4G} & 8 & 13^2, 23^2 & \mbox{\bf 567k4A} & 1\\
1231
\mbox{\bf 581k4E} & 34 & 19^2 & \mbox{\bf 581k4A} & 1
1232
\end{array}
1233
$$
1234
%Here~$\chi$ is the common nebentypus character of~$f$ and~$g$.
1235
1236
\section{Questions and conjectures}\label{sec:conj}
1237
The following questions and conjectures were motivated by the
1238
tables above and the computations that went into creating them.
1239
The first conjecture suggests a generalization of a result of
1240
Ribet on level raising. The second conjecture asserts that
1241
$\Sha$ is always visible in an appropriate modular Jacobian.
1242
The third question suggests a new approach to the long-standing
1243
open problem of constructing points on abelian varieties of
1244
analytic rank greater than~$1$ over the Hecke algebra.
1245
1246
\subsection{Level raising nonvanishing conjecture}
1247
Let $f\in S_2(\Gamma_0(N))$ be a newform such that the sign
1248
of the functional equation of $L(A_f,s)$ is equal to $+1$,
1249
and fix a prime~$\lambda$ such
1250
that the associated Galois representation $\rho_{f,\lambda}=A_f[\lambda]$
1251
is irreducible.
1252
For each prime~$q$ not dividing~$N$, let $\delta: J_0(N)\ra J_0(Nq)$ be
1253
the injection obtained from the sum of the two degeneracy maps.
1254
Ribet's construction in \cite{ribet:raising}
1255
produces infinitely many primes~$q$ and newforms
1256
$g\in S_2(\Gamma_0(qN))$ such that
1257
$$\delta(\Adual_f[\lambda]) \subset \delta(\Adual_f)\intersect \Adual_g$$
1258
and the Tamagawa number $c_q$ of $\Adual_g$ is a power of~$2$.
1259
\begin{conjecture}
1260
Fix~$f$ and~$\lambda$.
1261
\begin{enumerate}
1262
\item Then there is a~$g$ among those constructed by Ribet
1263
such that $L(A_g,1)\neq 0$.
1264
\item If~$\lambda$ is in the support of the $\T$-module
1265
$[P(H_1(X_0(N),\Z))^+: P(\T\{0,\infty\})]$
1266
(see Theorem~\ref{thm:ratpart}),
1267
then there is a~$g$ as above such that $L(A_g,1)=0$.
1268
\end{enumerate}
1269
\end{conjecture}
1270
1271
\subsection{Eventual visibility conjecture}
1272
Let~$S$ be the set of all square-free positive integers.
1273
If $M, N\in S$ with $M\mid N$ then there is
1274
a natural injection $J_0(M)\hookrightarrow J_0(N)$, and hence a map
1275
$\Sha(J_0(M))\ra \Sha(J_0(N))$.
1276
These maps are compatible, so the collection of groups $\Sha(J_0(N))$, with
1277
$N\in S$, forms a directed system. Let
1278
$\lim_{N\in S} \Sha(J_0(N))$
1279
be the direct limit of the $\Sha(J_0(N))$.
1280
\begin{conjecture}
1281
$\lim_{N\in S} \Sha(J_0(N))=0$
1282
\end{conjecture}
1283
If true, this would imply that if $A\subset J_0(N)$, then
1284
each element of $\Sha(A)$ is visible in some $J_0(N')$,
1285
for some multiple~$N'$ of~$N$.
1286
1287
\subsection{Euler systems}
1288
\label{sec:constructing}
1289
In \cite{kolyvagin:structureofsha} and \cite{mccallum:kolyvagin}
1290
one finds a construction using the Heegner point Euler system of
1291
Kolyvagin of the Shafarevich-Tate groups of certain abelian varieties.
1292
Under an unverified hypothesis on Heegner points, the construction
1293
gives much of $\Sha(A_f/K)$, where~$K$ is a suitable imaginary
1294
quadratic field. Is it possible to verify the unverified hypothesis,
1295
construct $\Vis_J(\Sha(A_f/K))$, and thus prove that $\Sha(\Adual_f/K)$
1296
contains visible elements, when the BSD conjecture suggests that it should? If so,
1297
it would follow that there is a congruent $B_f^{\vee}$ having positive
1298
algebraic rank, as predicted by the BSD conjecture. Thus a construction of
1299
visible elements of $\Sha(\Adual_f)$ also leads to a construction of points
1300
on abelian varieties of positive analytic rank.
1301
1302
\bibliography{biblio}
1303
1304
\end{document}
1305
1306