Sharedwww / nsf / project_description.texOpen in CoCalc
Author: William A. Stein
1
\documentclass[11pt]{article}
2
\newcommand{\thisdocument}{Project Description}
3
\include{macros}
4
5
\begin{document}
6
7
\section{Background}
8
The proposed project reflects the interplay of abstract theory
9
with explicit machine computation, as illustrated by the following
10
quote of Bryan Birch~\cite{birch:bsd}:
11
\begin{quote}
12
I want to describe some computations undertaken by myself and
13
Swin-nerton-Dyer on EDSAC by which we have calculated the
14
zeta-functions of certain elliptic curves. As a result of these
15
computations we have found an analogue for an elliptic curve of
16
the Tamagawa number of an algebraic group; and conjectures (due to
17
ourselves, due to Tate, and due to others) have proliferated.
18
\end{quote}
19
20
21
The PI is primarily interested in abelian varieties attached to
22
modular forms via Shimura's construction \cite{shimura:factors},
23
which we now recall. Let~$f=\sum a_n q^n$ be a weight~$2$ newform
24
on $\Gamma_1(N)$. Then~$f$ corresponds to a differential on the
25
modular curve $X_1(N)$, which is a curve whose affine points
26
over~$\C$ correspond to isomorphism classes of pairs $(E,P)$,
27
where~$E$ is an elliptic curve and $P\in E$ is a point of
28
order~$N$. We view the Hecke algebra
29
\[\T=\Z[T_1,T_2,T_3,\ldots]\]
30
as a subring of the endomorphism ring of the Jacobian $J_1(N)$
31
of $X_1(N)$. Let $I_f$ be the kernel of the homomorphism $\T\to
32
\Z[a_1,a_2,a_3]$ that sends $T_n$ to $a_n$, and attach to~$f$ the
33
quotient $$A_f=J_1(N)/I_f J_1(N).$$ Then $A_f$ is a simple abelian
34
variety over~$\Q$ of dimension equal to the degree of the field
35
$\Q(a_1,a_2,a_3,\ldots)$ generated by the coefficients of~$f$. We
36
also sometimes consider a similar construction with $J_1(N)$
37
replaced by the Jacobian $J_0(N)$ of the modular curve $X_0(N)$
38
that parametrizes isomorphism classes of pairs $(E,C)$, where $C$
39
is a cyclic subgroup of $E$ of order~$N$.
40
41
\begin{definition}[Modular abelian variety]
42
A {\em newform abelian variety} is an abelian variety over~$\Q$ of
43
the form $A_f$. An abelian variety over a number field is a {\em
44
modular abelian varieties} if it is a quotient of $J_1(N)$ for
45
some~$N$.
46
\end{definition}
47
Over $\Q$, newform abelian varieties are simple and every modular
48
abelian variety is isogenous to a product of copies of newform
49
abelian varieties. Newform abelian varieties are typically not
50
absolutely simple.
51
52
Newform abelian varieties $A_f$ are important. For example,the
53
celebrated modularity theorem of C.~Breuil, B.~Conrad, F.~Diamond,
54
R.~Taylor, and A.~Wiles \cite{breuil-conrad-diamond-taylor}
55
asserts that every elliptic curve over~$\Q$ is isogenous to some
56
$A_f$. Also, J-P.~Serre conjectures that, up to twist, every
57
two-dimensional odd irreducible mod~$p$ Galois representation
58
appears in the torsion points on some $A_f$.
59
60
Much of this research proposal is inspired by the following
61
special case of the Birch and Swinnerton-Dyer conjecture:
62
\begin{conjecture}[BSD Conjecture (special case)]\label{conj:bsd}
63
Let $A$ be a modular abelian variety over~$\Q$.
64
\begin{enumerate}%
65
\item $L(A,1)=0$ if and
66
only if $A(\Q)$ is infinite.%
67
\item If $L(A,1)\neq 0$, then
68
\[
69
\frac{L(A,1)}{\Omega_{A}} =%
70
\frac{\prod c_p \cdot \#\Sha(A)}%
71
{\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}},
72
\]
73
where the objects and notation in this formula are discussed
74
below.
75
\end{enumerate}
76
\end{conjecture}
77
Here $L(A,s)$ is the $L$-series attached to $A$, which is entire
78
because~$A$ is modular, so $L(A,1)$ makes sense. The real volume
79
$\Omega_{A}$ is the measure of $A(\R)$ with respect to a basis of
80
differentials for the N\'eron model of $A$. For each prime $p\mid
81
N$, the integer $c_p=\#\Phi_{A,p}(\F_p)$ is the {\em Tamagawa
82
number} of~$A$ at~$p$, where $\Phi_{A,p}$ denotes the component
83
group of the N\'eron model of~$A$ at $p$. The dual of $A$ is
84
denoted $A^{\vee}$, and in the conjecture $A(\Q)_{\tor}$ and
85
$A^{\vee}(\Q)_{\tor}$ are the torsion subgroups. The {\em
86
Shafarevich-Tate group} of $A$ is
87
\[
88
\Sha(A) = \Ker\left(\H^1(\Q,A) \to \bigoplus_{p\leq
89
\infty} \H^1(\Q_p,A)\right),
90
\]
91
which is a group that measures the failure of a local-to-global
92
principle. When $L(A,1)\neq 0$, Kato proved in \cite{kato:secret}
93
that $\Sha(A)$ and $A(\Q)$ are finite, so $\#\Sha(A)$ makes sense
94
and one implication of part 1 of the conjecture is known.
95
96
\begin{remark}
97
The general Birch and Swinnerton-Dyer conjecture (see
98
\cite{tate:bsd, lang:nt3}) is a conjecture about any abelian
99
variety $A$ over a global field~$K$. It asserts that the order of
100
vanishing of $L(A,s)$ at $s=1$ equals the free rank of $A(K)$, and
101
gives a formula for the leading coefficient of the Taylor
102
expansion of $L(A,s)$ about $s=1$.
103
\end{remark}
104
105
The rest of this proposal is divided into two parts. The first is
106
about computing with modular forms and abelian varieties, and
107
making the results of these computations available to the
108
mathematical community.
109
The second is about visibility of Mordell-Weil and Shafarevich-Tate groups, the
110
ultimate goal being to obtain relationships between parts 1 and 2
111
of Conjecture~\ref{conj:bsd}.
112
113
\section{Computing with modular forms}
114
115
The PI proposes to continue developing algorithms and making
116
available tools for computing with modular forms, modular abelian
117
varieties, and motives attached to modular forms. This includes
118
finishing a major new {\sc Magma} \cite{magma} package for
119
computing directly with modular abelian varieties over number
120
fields, extending the Modular Forms Database \cite{mfd}, and
121
searching for algorithms for computing the quantities appearing in
122
Conjecture~\ref{conj:bsd} and in the Bloch-Kato conjecture for
123
modular motives.
124
125
126
127
\subsection{The Modular Forms Database}%
128
The Modular Forms Database \cite{mfd} is a freely-available
129
collection of data about objects attached to cuspidal modular
130
forms. It is analogous to Sloane's tables of integer sequences,
131
and extends Cremona's tables \cite{cremona:onlinetables} to
132
dimension bigger than one and weight bigger than two. Cremona's
133
tables contain more refined data about elliptic curves than
134
\cite{mfd}, but the PI intends to work with Cremona to make the
135
modular forms database a superset of \cite{cremona:onlinetables}.
136
137
The database is used world-wide by prominent number theorists,
138
including Noam Elkies, Matthias Flach, Dorian Goldfeld, Benedict
139
Gross, Ken Ono, and Don Zagier.
140
141
The PI proposes to greatly expand the database. A major challenge
142
is that data about modular abelian varieties of large dimension
143
takes a huge amount of space to store. For example, the database
144
currently occupies 40GB disk space. He proposes to find and
145
implement a better method for storing information about modular
146
abelian varieties so that the database will be more useful. He
147
has found a method whereby a certain eigenvector is computed by
148
the database server, which may (or may not!) enable storing
149
coefficients of modular forms far more efficiently; however, he
150
has not yet tried to implement it and study its properties.
151
152
The PI proposes to improve the usability of the database. It is
153
currently implemented using a PostgreSQL database coupled with a
154
Python web interface. To speed access and improve efficiency, he
155
is considering rewriting key portions of the database using MySQL
156
and PHP. He hopes to rewrite key portions of the database in
157
response to user feedback that he has received. The database
158
currently runs on a three-year-old 933Mhz Pentium III, which has
159
unduly limited disk space and no offsite backup, so the PI is
160
requesting a powerful modern computer with a large hard drive
161
array and external hard drives for offsite backups.
162
163
\subsubsection{M{\small AGMA} package for modular abelian varieties}\label{sec:magma}%
164
The PI's software is published as part of the non-commercial {\sc
165
Magma} computer algebra system. The core of {\sc Magma} is
166
developed by a group of academics at the University of Sydney, who
167
are supported mostly by grant money. {\sc Magma} is considered by
168
many to be the most comprehensive tool for research in number
169
theory, finite group theory, and cryptography, and is widely
170
distributed. The PI has already written over 400 pages (26000
171
lines) of modular forms code and extensive documentation that is
172
distributed with {\sc Magma}, and intends to ``publish'' future
173
work in {\sc Magma}.
174
175
As mentioned above, an abelian variety $A$ over a number field~$K$
176
is {\em modular} if it is a quotient of $J_1(N)$ for some $N$.
177
Modular abelian varieties were studied intensively by Ken Ribet,
178
Barry Mazur, and others during recent decades, and studying them
179
is popular because results about them often yield surprising
180
insight into number theoretic questions. Computation with modular
181
abelian varieties is attractive because they are much easier to
182
describe than arbitrary abelian varieties, and their $L$-functions
183
are reasonably well understood when~$K$ is an abelian extension
184
of~$\Q$.
185
186
The PI recently designed and partially implemented a general
187
system for computing with modular abelian varieties over number
188
fields. He hopes to develop and refine several crucial components
189
of the system. For example, three major problems arose, and the
190
PI intends to resolve them in order to have a completely
191
satisfactory system for computing with modular abelian varieties.
192
\begin{enumerate}
193
\item {\em Given a modular abelian variety $A$, efficiently
194
compute the endomorphism ring $\End(A)$ as a ring of matrices
195
acting on $\H_1(A,\Z)$.} The PI has found a modular symbols
196
solution that draws on work of Ribet \cite{ribet:twistsendoalg}
197
and Shimura \cite{shimura:factors}, but it is too slow to be
198
really useful in practice. In \cite{merel:1585}, Merel uses
199
Herbrand matrices and Manin symbols to give efficient algorithms
200
for computing with Hecke operators. The PI intends to carry over
201
Merel's method to
202
give an efficient algorithm to compute $\End(A)$.%
203
\item {\em Given $\End(A)\otimes\Q$, compute an isogeny
204
decomposition of $A$ as a product of simple abelian varieties.}
205
This is a standard and difficult problem in general, but it might
206
be possible to combine work of Allan Steel on his
207
``characteristic~$0$ Meataxe'' with special features of modular
208
abelian varieties to solve it in practice. It is absolutely {\em
209
essential} to solve this problem in order to explicitly enumerate
210
all modular abelian varieties over $\Qbar$ of given level~$N$.
211
Such an enumeration would be a major step towards the ultimate
212
possible generalization of Cremona's tables
213
\cite{cremona:onlinetables} to modular abelian varieties.
214
Computation of a decomposition is also crucial to other
215
algorithms, e.g., computing
216
complements and duals of abelian subvarieties.%
217
\item {\em Given two modular abelian varieties over a number field
218
$K$, decide whether there is an isomorphism between them.} When
219
the endomorphism ring of each abelian variety is known and both
220
are simple, it is possible to reduce this problem to the solution
221
of a norm equation, which has been studied extensively in many
222
cases. This problem is analogous to the problem of testing
223
isomorphism for modules over a fixed ring, which has been solved
224
with much effort for many classes of rings. One application is
225
to proving that specific abelian varieties can not be principally polarized.%
226
\end{enumerate}
227
228
\subsection{An Example: The Arithmetic of $J_1(p)$}
229
We finish be describing recent work of the PI on the modular
230
Jacobian $J_1(p)$, where~$p$ is a prime, that was partly inspired
231
by computation. The following conjecture generalizes a conjecture
232
of Ogg, which asserts that $J_0(p)(\Q)_{\tor}$ is cyclic of order
233
the numerator of $(p-1)/12$, a fact that Mazur proved in
234
\cite{mazur:eisenstein}.
235
\begin{conjecture}[Stein]\label{conj:tor}
236
Let $p$ be a prime. The torsion subgroup of $J_1(p)(\Q)$ is the
237
group generated by the cusps on $X_1(p)$ that lie over $\infty\in
238
X_0(p)$.
239
\end{conjecture}
240
The PI gives significant numerical evidence for this conjecture in
241
\cite{conrad-edixhoven-stein:j1p}, and cuspidal subgroups of
242
$J_1(p)$ are considered in detail in \cite{kubert-lang}, where,
243
e.g., they compute orders of such groups in terms of Bernoulli
244
numbers.
245
246
Mazur's proof of Ogg's conjecture for $J_0(p)$ is deep, though the
247
proof for the odd part of $J_0(p)(\Q)_{\tor}$ is much easier. The
248
PI intends to explore whether or not it is possible to build on
249
Mazur's method and prove results towards
250
Conjecture~\ref{conj:tor}. The PI also intends to develop his
251
computational methods for computing torsion subgroups in order to
252
answer, at least conjecturally, the following question.
253
\begin{question}
254
If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
255
natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective? Is
256
the product of the orders of all $A_f(\Q)_{\tor}$ over all classes
257
of newforms $f$ equal to $\# J_1(p)(\Q)_{\tor}$?
258
\end{question}
259
The PI conjectured that the analogous questions for $J_0(p)$
260
should have ``yes'' answers, and in \cite{emerton:optimal}
261
M.~Emerton proved this conjecture. There he also proved that the
262
natural map from the component group of $J_0(p)$ to that of $A_f$
263
is surjective. By \cite{conrad-edixhoven-stein:j1p}, the
264
component group of $J_1(p)$ is trivial, which suggests the
265
following question.
266
\begin{question}\label{ques:comp}
267
If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
268
component group of $A_f$ trivial?
269
\end{question}
270
Even assuming the full BSD conjecture (and a conjecture about a
271
Manin constant), the PI has not yet produced enough data to give a
272
conjectural answer to this question. He has many examples in
273
which the conjecture predicts that either $\Sha(A_f)$ is
274
nontrivial or the component group of $A_f$ is nontrivial. He and
275
B.~Poonen formulated, and hope to carry out, a strategy to decide
276
which of these two is nontrivial by using an explicit description
277
of $\End({A_f/\Qbar})$ to obtain a curve whose Jacobian is $A_f$.
278
Note that computing $\End(A_f/\Qbar)$ in general is the second
279
problem in Section~\ref{sec:magma} above.
280
281
\section{Visibility}
282
The underlying motivation for this part of the proposal is to
283
prove implications between the two parts of
284
Conjecture~\ref{conj:bsd}, in examples and eventually in some
285
generality. That is, we link information about the first part of
286
the BSD conjecture for an abelian variety~$B$ to information about
287
the second part of the conjecture for a related abelian
288
variety~$A$. Visibility provides a conceptual framework in which
289
to organize our ideas.
290
291
\subsection{Computational problems}\label{sec:compprob}
292
Barry Mazur introduced visibility in order to unify various
293
constructions of Shafarevich-Tate groups.
294
\begin{definition}[Visibility of Shafarevich-Tate Groups]
295
Suppose that $$\iota:A\hra J$$ is an inclusion of abelian
296
varieties over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$
297
with respect to $J$ is
298
\[
299
\Vis_J\H^1(\Q,A) := \Ker(\H^1(\Q,A)\to \H^1(\Q,J)).
300
\]
301
The {\em visible subgroup} of $\Sha(A)$ in~$J$ is the intersection
302
of $\Sha(A)$ with $\Vis_J\H^1(\Q,A)$; equivalently,
303
\[
304
\Vis_J\Sha(A) := \Ker(\Sha(A)\to \Sha(J)).
305
\]
306
\end{definition}
307
The terminology ``visible'' arises from the fact that if
308
$x\in\Sha(A)$ is visible in~$J$, then a principal homogenous
309
space~$X$ corresponding to~$x$ can be realized as a subvariety
310
of~$J$.
311
312
Before discussing theoretical questions about visibility, we
313
describe computational evidence for the Birch and Swinnerton-Dyer
314
conjecture for modular abelian varieties (and motives) that the PI
315
and A.~Agashe obtained using theorems inspired by the definition
316
of visibility. In \cite{agashe-stein:visibility}, the PI and
317
Agashe prove a theorem that makes it possible to use abelian
318
varieties of positive rank to explicitly construct subgroups of
319
Shafarevich-Tate groups of other abelian varieties. The main
320
theorem is that if $A$ and $B$ are abelian subvarieties of an
321
abelian variety $J$, and $B[p]\subset A$, then, under certain
322
hypothesis, there is an injection
323
\[ B(\Q)/p B(\Q) \hra \Vis_J\Sha(A).\]
324
The paper concludes with the first ever example of an abelian
325
variety $A_f$ attached to a newform, of large dimension ($20$),
326
whose Shafarevich-Tate group has order that is provably divisible
327
by an odd prime ($5$).
328
329
The PI has used the result described above to give evidence for
330
the BSD conjecture for many $A\subset J_0(N)$, where $A$ is
331
attached to a newform of level~$N\leq 2333$. The PI proposes to
332
give similar evidence using visibility in $J_0(NM)$ for small~$M$.
333
More precisely, \cite{agashe-stein:bsd} describes the computation
334
of an odd divisor of the BSD conjectural order of $\Sha(A)$ for
335
over ten thousand $A\subset J_0(N)$ with $L(A,1)\neq 0$ (these are
336
{\em all} simple $A$ with $N\leq 2333$ and $L(A,1)\neq 0$). For
337
over a hundred of these, the divisor of the conjectural order of
338
$\Sha(A)$ is divisible by an odd prime; for a quarter of these the
339
PI and Agashe prove that if~$n$ is the conjectural divisor of the
340
order of $\Sha(A)$, then there are at least~$n$ elements of
341
$\Sha(A)$ that are visible in $J_0(N)$.
342
343
The PI intends to investigate the remaining 75\% of the $A$ with
344
$n>1$ by considering the image of $A$ in $J_0(NM)$ for small
345
integers~$M$. Information about which~$M$ to choose can be
346
extracted from Ribet's level raising theorem (see
347
\cite{ribet:raising}). As a test, the PI recently tried the first
348
example with conjectural odd $\Sha(A)$ that is not visible in
349
$J_0(N)$ (this is an $18$ dimensional abelian variety $A$ of level
350
$551$ such that $9\mid \#\Sha(A)$). He showed in
351
\cite{stein:bsdmagma} that there are elements of order~$3$ in
352
$\Sha(A)$ that are visible in $J_0(551\cdot 2)$. Since the
353
dimension of $J_0(NM)$ grows very quickly, a huge amount of
354
computer memory will be required to investigate visibility at
355
higher level. Fortunately, the PI recently received a grant from
356
Sun Microsystems for a \$67,000 computer that contains 22GB of
357
contiguously addressable RAM (the processors are relatively slow
358
and the hard drive is small, making this computer less suitable as
359
a platform for the modular forms database, which requires a large
360
hard drive but not so much RAM).
361
362
Some of these ideas generalize to the context of Grothendieck
363
motives, which A.~Scholl attached to newforms of weight greater
364
than two. N.~Dummigan, M.~Watkins, and the PI did work in this
365
direction in \cite{dummigan-stein-watkins:motives}. There we prove
366
a theorem that can sometimes be used to deduce the existence of
367
visible Shafarevich-Tate groups in motives attached to modular
368
forms, assuming a conjecture of Beilinson about ranks of Chow
369
groups. However, we give several pages of tables that suggest that
370
Shafarevich-Tate groups of modular motives of level~$N$ are rarely
371
visible in the higher-weight motivic analogue of $J_0(N)$, much
372
more rarely than for weight~$2$. Just as above, the question
373
remains to decide whether one expects these groups to be visible
374
in the analogue of $J_0(N M)$ for some integer~$M$. It would be
375
relatively straightforward for the PI to do computations in this
376
direction, and he intends to do so.
377
378
Before moving on to theoretical questions about visibility, we
379
pause to emphasize that the above computational investigations
380
into the Birch and Swinnerton-Dyer conjecture motivated the PI and
381
others to develop new algorithms for computing with modular
382
abelian varieties. For example, in \cite{conrad-stein:compgroup},
383
B.~Conrad and the PI use Grothendieck's monodromy pairing to give
384
an algorithm for computing orders of component groups of certain
385
purely toric abelian varieties. This algorithm makes it practical
386
to compute component groups of quotients $A_f$ of $J_0(N)$ at
387
primes~$p$ that exactly divide $N$. Without such an algorithm it
388
would probably be difficult to get very far in computational
389
investigations into the Birch and Swinnerton-Dyer conjecture for
390
abelian varieties; indeed, the only other paper in this direction
391
is \cite{empirical}, which restricts to the case of Jacobians of
392
genus~$2$ curves.
393
394
395
\subsection{Theoretical problems}
396
\subsubsection{Visibility at higher level}
397
Suppose $A_f$ is a quotient of $J_1(N)$ attached to a newform and
398
let $A=A_f^{\vee}\subset J_1(N)$ be its dual. One expects that
399
most of $\Sha(A)$ is {\em not} visible in $J_1(N)$. The following
400
conjecture then arises.
401
\begin{conjecture}[Stein]\label{conj:allvis} For each $x\in \Sha(A)$, there is an integer
402
$M$ and a morphism $f:A\to J_1(NM)$, of finite degree and coprime
403
to the order of~$x$, such that the image of $x$ in $\Sha(f(A))$ is
404
visible in $J_1(NM)$.
405
\end{conjecture}
406
In \cite{agashe-stein:visibility}, the PI proved that if $x\in
407
\H^1(\Q,A)$, then there is an abelian variety~$B$ and an inclusion
408
$\iota:A\to B$ such that~$x$ is visible in~$B$; moreover,~$B$ is a
409
quotient of $J_1(NM)$ for some~$M$. This theorem is the main
410
reason why the PI makes Conjecture~\ref{conj:allvis}. The PI hopes
411
to prove Conjecture~\ref{conj:allvis} by understanding the precise
412
relationship between $A$, $B$, and $J_1(NM)$. First he will
413
investigate explicitly the example with $N=551$ described in
414
Section~\ref{sec:compprob} above.
415
416
A more analytical, and possibly deeper, approach to
417
Conjecture~\ref{conj:allvis} is to assume the rank statement of
418
the Birch and Swinnerton-Dyer conjecture and relate when elements
419
of $\Sha(A)$ becoming visible at level $NM$ to when there is a
420
congruence between $f$ and a newform $g$ of level $NM$ with
421
$L(g,1)=0$. Such an approach leads one to try to formulate a
422
refinement of Ribet's level raising theorem that includes a
423
statement about the behavior of the value at $1$ of the
424
$L$-function attached to the form at higher level. The PI intends
425
to do further computations in the hopes of finding a satisfactory
426
conjectural refinement of Ribet's theorem, which he then hopes to
427
subsequently prove.
428
429
The PI also proposes to investigate whether there is an~$M$ that
430
is minimal with respect to some property, such that every element
431
of $\Sha(A)$ is simultaneously visible in $J_1(NM)$. This is well
432
worth looking into, since the payoffs could be huge---the
433
existence of such an~$M$ would imply finiteness of $\Sha(A)$,
434
since $\Vis_J(\Sha(A))$ is always finite. Finiteness of $\Sha(A)$
435
is a mysterious open problem when $L(A,1)=0$ and $A$ is not a
436
quotient of $J_0(N)$ with $\ord_{s=1}L(A,s)=\dim A$.
437
438
\subsubsection{Visibility of Mordell-Weil groups}
439
The Gross-Zagier theorem asserts that points on elliptic curves of
440
rank $1$ come from Heegner points, and that points on curves of
441
rank bigger than one do not. It seems difficult to describe
442
where points on elliptic curves of rank bigger than~$1$ ``come
443
from''. The PI introduced the following definition, in hopes of
444
eventually creating a framework for giving a conjectural
445
explanation.
446
447
\begin{definition}[Visibility of Mordell-Weil Groups]
448
Suppose that $\pi : J\to A$ is a surjective morphism of abelian
449
varieties with connected kernel. The {\em visible quotient of
450
$A(\Q)$} with respect to~$J$ (and $\pi$) is
451
\[
452
\Vis^J(A(\Q)) := \Coker(J(\Q)\to A(\Q)).
453
\]
454
\end{definition}
455
456
Visibility of Mordell-Weil groups is closely connected to
457
visibility of Shafarevich-Tate groups. If $C$ is the kernel
458
of~$\pi$ and $\delta : A(\Q)\to \H^1(\Q,C)$ is the connecting
459
homomorphism of Galois cohomology, then $\delta$ induces an
460
isomorphism
461
\[
462
\tilde{\delta}: \Vis^J(A(\Q)) \isom \Vis_J(\H^1(\Q,C)).
463
\]
464
Note that this implies $\Vis^J(A(\Q))$ is finite. Let
465
\[
466
\Vis^J_{\scriptsize\Sha}(A(\Q)) := \tilde{\delta}^{-1}(\Vis_J(\Sha(C))).
467
\]
468
469
470
Though we have introduced nothing fundamentally new, this
471
different point of view suggested questions that seemed unnatural
472
before, which inspired the following theorem and conjecture (the
473
proof of the theorem relies on \cite{kato:secret,rubin:kato} and
474
\cite{rohrlich:cyclo}):
475
\begin{theorem}[Stein]\label{thm:allmwvis}
476
Let $A$ be an elliptic curve. If $x\in A(\Q)$ has order~$n$ (set
477
$n=0$ if $x$ has infinite order), then for every divisor $d$ of
478
$n$, there is surjective morphism $J\to A$, with connected kernel,
479
such that the image of~$x$ in $\Vis^J(A(\Q))$ has order~$d$.
480
\end{theorem}
481
482
\begin{conjecture}[Stein]
483
Suppose~$A$ is a modular abelian variety and $x\in A(\Q)$ has
484
order~$n$. For every divisor~$d$ of $\,n$ there is a surjective
485
morphism $J\to A$, with connected kernel, such that the image of
486
$\,x$ in $\Vis^J(A(\Q))$ lies in $\Vis_{\scriptsize \Sha}^J(A(\Q))$ and
487
has order~$d$.
488
\end{conjecture}
489
490
We now describe partial results about this conjecture that the PI
491
proved in \cite{stein:nonsquaresha}. Suppose $E$ is an elliptic
492
curve over $\Q$ with conductor $N$, and let $f$ be the newform
493
attached to~$E$. Fix a prime~$p\nmid 2 N \prod c_p$ such that the
494
Galois representation $\Gal(\Qbar)\to \Aut(E[p])$ is surjective.
495
\begin{conjecture}[Stein]\label{conj:nonvanishtwist}
496
There is a prime~$\ell\nmid N$ and a surjective Dirichlet
497
character $\chi:(\Z/\ell\Z)^*\to\mu_p$ such that
498
\[
499
L(E,\chi,1)\neq 0 \qquad\text{and}\qquad
500
a_{\ell}(E) \not\con \ell+1 \pmod{p}.
501
\]
502
\end{conjecture}
503
According to Sarnak and Kowalski, this conjecture does not seem
504
amenable to standard analytic averaging arguments. The PI has
505
verified this conjecture for the elliptic curve of rank~$1$ and
506
conductor~$37$ and all $p\leq 25000$. In almost all cases, the
507
smallest $\ell\nmid N$ such that $a_{\ell}(E) \not \con
508
\ell+1\pmod{p}$ and $\ell\con 1\pmod{p}$ satisfies the conjecture.
509
510
The PI proved the following theorem in \cite{stein:nonsquaresha}.
511
\begin{theorem}[Stein]\label{thm:exact}
512
Let $E$ be an elliptic curve over~$\Q$ and suppose~$p$ and $\chi$
513
are as in Conjecture~\ref{conj:nonvanishtwist} above. Then there
514
is an exact sequence $0\to A\to J\to E\to 0$ that induces an exact
515
sequence
516
\[
517
0 \to E(\Q)/ p E(\Q) \to \Sha(A) \to \Sha(J) \to \Sha(E) \to 0.
518
\]
519
In particular,
520
\[
521
E(\Q)/p E(\Q) \isom \Vis_{\scriptsize\Sha}^J(E(\Q)) \isom \Vis_J(\Sha(A)).
522
\]
523
\end{theorem}
524
%When the hypothesis of the theorem are satisfied, the conclusion
525
%explains $E(\Q)\otimes\F_p$ in terms of the Shafarevich-Tate group
526
%of an abelian variety with analytic rank~$0$. It thus links parts
527
%1 and 2 of Conjecture~\ref{conj:bsd}. The PI proposes to explore
528
%the significance of this further.
529
530
%There are two problems with this picture. First, the PI does not
531
%know a proof of Conjecture~\ref{conj:nonvanishtwist}, and is
532
%having difficulty finding one. Second, even if
533
%Conjecture~\ref{conj:nonvanishtwist} were known, it is unclear
534
%what implications the isomorphism
535
%\[
536
%E(\Q)/p E(\Q) \isom \Vis_J(\Sha(A))
537
%\]
538
%has for Conjecture~\ref{conj:bsd}.
539
%
540
541
We finish this research proposal by explaining how
542
Theorem~\ref{thm:exact} may lead to a link between the two parts
543
of the BSD Conjecture (Conjecture~\ref{conj:bsd}). Suppose $E$ is
544
an elliptic curve over~$\Q$ and $L(E,1)=0$. Then part 1 of
545
Conjecture~\ref{conj:bsd} asserts that $E(\Q)$ is infinite. Under
546
our hypothesis that $L(E,1)=0$, a standard argument shows that
547
$$\frac{L(A,1)}{\Omega_A} \con 0\pmod{p},$$ where $A$ is as in
548
Theorem~\ref{thm:exact}. If part 2 of Conjecture~\ref{conj:bsd}
549
were true, there would be an element $x\in \Sha(A)$ of order~$p$
550
(the proof of Theorem~\ref{thm:exact} rules out the possibility
551
that~$p$ divides a Tamagawa number). If, in addition,~$x$ were
552
visible in $J$, then $E(\Q)$ would be infinite, since $E(\Q)$ has
553
no elements of order~$p$. Part 2 of Conjecture~\ref{conj:bsd} does
554
not assert that $x$ is visible in~$J$, so one can only hope that a
555
close examination of an eventual proof of part 2 of
556
Conjecture~\ref{conj:bsd} would yield some insight into whether or
557
not $x$ is visible. Alternatively, one could try to replace the
558
isomorphism $E(\Q)/p E(\Q) \isom \Vis_J(\Sha(A)) $ by an
559
isomorphism
560
\[
561
\mbox{\rm Sel}^{(p)}(E)\isom \Sha(A)[I]
562
\]
563
where $I$ is an appropriate ideal in the ring $\Z[\mu_p]$ of
564
endomorphism of~$A$. Then an appropriate refinement of part 2 of
565
Conjecture~\ref{conj:bsd} might imply that $\Sha(A)[I]$ contains
566
an element of order~$p$, which would imply that either $E(\Q)$ is
567
infinite or $\Sha(E/\Q)[p]$ is nonzero.
568
569
One can also work orthogonally to the above approach by
570
investigating similar situations coming from level raising, where
571
isomorphisms like the ones above may arise. The PI intends to
572
investigate this cluster of ideas from various directions in hopes
573
of finding a new perspective on where points on elliptic curves of
574
rank bigger than one come from it.
575
576
\newpage
577
\bibliographystyle{amsalpha}
578
\bibliography{biblio}
579
\end{document}
580