Sharedwww / nsf / project_description_old.texOpen in CoCalc
Author: William A. Stein
1
\documentclass[11pt]{article}
2
\newcommand{\thisdocument}{Project Description}
3
\include{macros}
4
%\textheight=0.8\textheight
5
6
\begin{document}
7
8
\section{Introduction}
9
My research reflects the essential interplay of abstract theory
10
with explicit machine computation, which is illustrated by the
11
following quote of Bryan Birch~\cite{birch:bsd} about computations
12
that led to a central conjecture in number theory:
13
\begin{quote}
14
I want to describe some computations undertaken by myself and
15
Swinnerton-Dyer on EDSAC by which we have calculated the
16
zeta-functions of certain elliptic curves. As a result of these
17
computations we have found an analogue for an elliptic curve of
18
the Tamagawa number of an algebraic group; and conjectures (due to
19
ourselves, due to Tate, and due to others) have proliferated.
20
\end{quote}
21
22
23
I am primarily interested in abelian varieties attached to modular
24
forms via Shimura's construction \cite{shimura:factors}.
25
Let~$f=\sum a_n q^n$ be a weight~$2$ newform on $\Gamma_1(N)$. We
26
may view~$f$ as a differential on the modular curve $X_1(N)$,
27
which is a curve whose affine points over~$\C$ correspond to
28
isomorphism classes of pairs $(E,P)$, where~$E$ is an elliptic
29
curve and $P\in E$ is a point of order~$N$. We view the Hecke
30
algebra
31
\[\T=\Z[T_1,T_2,T_3,\ldots]\]
32
as a subring of the endomorphism ring of the Jacobian $J_1(N)$
33
of $X_1(N)$. Let $I_f$ be the annihilator of~$f$ in $\T$, and
34
attach to~$f$ the quotient $$A_f=J_1(N)/I_f J_1(N).$$ Then $A_f$
35
is an abelian variety over~$\Q$ of dimension equal to the degree
36
of the field $\Q(a_1,a_2,a_3,\ldots)$ generated by the
37
coefficients of~$f$.
38
39
The abelian varieties $A_f$ attached to newforms are important.
40
For example,the celebrated modularity theorem of C.~Breuil,
41
B.~Conrad, F.~Diamond, Taylor, and Wiles
42
\cite{breuil-conrad-diamond-taylor} asserts that every elliptic
43
curve over~$\Q$ is isogenous to some $A_f$. Also, Serre
44
conjectures that up to twist every two-dimensional odd irreducible
45
Galois representation appears in the torsion points on some $A_f$.
46
47
My investigations into modular abelian varieties are inspired by
48
the following special case of the Birch and Swinnerton-Dyer
49
conjecture (see \cite{tate:bsd, lang:nt3}):
50
\begin{conjecture}[BSD Conjecture]
51
\[
52
\frac{L(A_f,1)}{\Omega_{A_f}} =%
53
\frac{\prod c_p \cdot \#\Sha(A_f/\Q)}%
54
{\#A_f(\Q)\cdot \#A_f^{\vee}(\Q)}.
55
\]
56
\end{conjecture}
57
Here $L(A_f,s)$ is the canonical $L$-series attached to $A_f$, the
58
real volume $\Omega_{A_f}$ is the measure of $A_f(\R)$ with
59
respect to a basis of differentials for the N\'eron model of
60
$A_f$, the $c_p$ are the Tamagawa numbers of $A_f$, the dual of
61
$A_f$ is denoted $A_f^{\vee}$, and
62
\[
63
\Sha(A_f/\Q) = \ker\left(\H^1(\Q,A_f) \to \bigoplus_{p\leq
64
\infty} \H^1(\Q_p,A_f)\right)
65
\] is the
66
Shafarevich-Tate group of $A_f$. When $A_f(\Q)$ is infinite, the
67
right hand side should be interpreted as~$0$, so, in particular,
68
the conjecture asserts that $L(A_f,1)=0$ if and only if $A_f(\Q)$
69
is infinite. Birch and Swinnerton-Dyer also conjectured that the
70
order of vanishing of $L(A_f,s)$ at $s=1$ equals the rank of
71
$A_f(\Q)$.
72
73
74
\section{Computing with modular abelian varieties}
75
76
The PI proposes to continue developing algorithms and making
77
available tools for computing with modular forms, modular abelian
78
varieties, and motives attached to modular forms. This includes
79
finishing a major new {\sc Magma} \cite{magma} package for
80
computing directly with modular abelian varieties over number
81
fields, extending the Modular Forms Database \cite{mfd}, and
82
searching for algorithms for computing the quantities appearing in
83
the Birch and Swinnerton-Dyer for modular abelian varieties and
84
the Bloch-Kato conjecture for modular motives.
85
86
87
88
\subsection{The Modular Forms Database}%
89
The Modular Forms Database \cite{mfd} is a freely-available
90
collection of data about objects attached to cuspidal modular
91
forms, that is a much used resource for number theorists. It is
92
analogous to Sloane's tables of integer sequences, and extends
93
Cremona's tables to dimension bigger than one and weight bigger
94
than two (\cite{cremona-tables} contain more precise data about
95
elliptic curves than \cite{mfd}).
96
97
The PI proposes to greatly expand the database. The major
98
challenge is that data about modular abelian varieties of large
99
dimension takes a huge amount of space to store. For example, the
100
database currently occupies 40GB disk space. He proposes to find
101
better method for storing information about modular abelian so
102
that the database can grow larger, and to investigate methods used
103
by astronomers or the human genome project to see how they cope
104
with a torrent of data while making it available to their
105
colleagues.
106
107
The PI implemented the current database using PostgreSQL coupled
108
with a Python web interface. To speed access and improve
109
efficiency, the PI is considering rewriting key portions of the
110
database using MySQL and PHP. Also, the database currently runs
111
on a three-year-old 933Mhz Pentium III, so the PI is requesting
112
more modern hardware.
113
114
\subsubsection{M{\small AGMA} package for modular abelian varieties}%
115
{\sc Magma} is a nonprofit computer algebra system developed
116
primarily at the University of Sydney, which is supported mostly
117
by grant money from organizations such as the US National Security
118
Agency. {\sc Magma} is considered by many to be the most
119
comprehensive tool for research in number theory, finite group
120
theory, and cryptography, and it is widely distributed. The PI
121
has already written over 400 pages (26000 lines) of modular forms
122
code and extensive documentation that is distributed with {\sc
123
Magma}, and intends to ``publish'' future work in {\sc Magma}.
124
125
An abelian variety $A$ over a number field is {\em modular} if it
126
is a quotient of $J_1(N)$ for some $N$. Modular abelian varieties
127
were studied intensively by Ken Ribet, Barry Mazur, and others
128
during recent decades, and studying them is popular because
129
results about them often yield surprising insight into number
130
theoretic questions. Computation with modular abelian varieties
131
is popular because most of them are easily described by giving a
132
level and the first few coefficients of a modular form, and the
133
$L$-functions of modular abelian varieties are particularly well
134
understood. This is in sharp contrast to the case of general
135
abelian varieties which, in general, can only be described by
136
unwieldy systems of polynomial equations, and whose $L$-functions
137
are very mysterious.
138
139
The PI recently designed and partially implemented a general
140
package for computing with modular abelian varieties over number
141
fields. Several crucial algorithms still need to be developed or
142
refined. When available, this package will likely be useful for
143
people working with modular abelian varieties. The following
144
major problems arose in work on this package, and they must be
145
resolved in order to have a completely satisfactory system for
146
computing with modular abelian varieties:
147
\begin{itemize}
148
\item {\em Given a modular abelian variety $A$, efficiently
149
compute the endomorphism ring $\End(A)$ as a ring of matrices
150
acting $\H_1(A,\Z)$.} The PI has found a modular symbols solution
151
that draws on work of Ribet \cite{ribet:twistsendoalg} and Shimura
152
\cite{shimura:factors}, but it is too slow to be really useful in
153
practice. In \cite{merel:1585}, Merel uses Herbrand matrices and
154
Manin symbols to give efficient algorithms for computing with
155
Hecke operators. The PI intends to carry over Merel's method to
156
give an efficient algorithm to compute $\End(A)$.%
157
\item {\em Given $\End(A)\otimes\Q$, compute an isogeny
158
decomposition of $A$ as a product of simple abelian varieties.}
159
This is a standard and difficult problem in general, but it might
160
be possible to combine work of Allan Steel on his
161
``characteristic~$0$ Meataxe'' with special features of modular
162
abelian varieties to solve it in practice. It is absolutely {\em
163
essential} to solve this problem in order to explicitly enumerate
164
all modular abelian varieties over number fields of given level.
165
Such an enumeration would be a major step towards the ultimate
166
possible generalization of Cremona's tables \cite{cremona:tables}
167
to modular abelian varieties. %
168
\item {\em Given two modular abelian varieties over a number field
169
$K$, decide whether there is an isomorphism between them.} When
170
the endomorphism ring of each abelian variety is known and both
171
are simple, it is possible to reduce this problem to mostly
172
well-studied norm equations. This problem is analogous to the
173
problem of testing isomorphism for modules over a fixed ring,
174
which has been solved with much effort for many classes of rings.
175
One application of isomorphism testing is that it could be used to
176
prove that an abelian variety is not principally polarized.%
177
\end{itemize}
178
179
We finish be describing recent work with $J_1(p)$ that was
180
partially inspired by PI's modular forms software. This
181
conjecture generalizes a conjecture of Ogg, which asserted that
182
$J_0(p)(\Q)_{\tor}$ is cyclic of order the numerator of
183
$(p-1)/12$, a fact that Mazur proved in \cite{mazur:eisenstein}.
184
\begin{conjecture}[Stein]\label{conj:tor}
185
Let $p$ be a prime. The torsion subgroup of $J_1(p)(\Q)$ is the
186
group generated by the cusps on $X_1(p)$ that lie over $\infty\in
187
X_0(p)$.
188
\end{conjecture}
189
Significant numerical evidence for this conjecture is given in
190
\cite{j1p}, and cuspidal subgroups of $J_1(p)$ are considered in
191
detail in \cite{kubert-lang}, where, e.g., they compute orders of
192
such groups in terms of Bernoulli numbers.
193
194
Mazur's proof of Ogg's conjecture for $J_0(p)$ is deep, though the
195
proof for the odd part of $J_0(p)(\Q)_{\tor}$ is much easier. The
196
PI intends explore whether or not it is possible to build on
197
Mazur's method and prove results towards
198
Conjecture~\ref{conj:tor}. The PI also intends to develop his
199
computational methods for computing torsion subgroups in order to
200
answer, at least conjecturally, the following question.
201
\begin{question}
202
If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
203
natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective? If
204
so, is the product of the orders of all $A_f(\Q)_{\tor}$ over all
205
classes of newforms $f$ equal to $\# J_1(p)(\Q)_{\tor}$?
206
\end{question}
207
The PI conjectured that the analogous questions for $J_0(p)$
208
should have ``yes'' answers, and in \cite{emerton:optimal}
209
M.~Emerton proved this conjecture.
210
211
In \cite{j1p}, it is proved that $J_1(p)$ has trivial component
212
group (component groups are closely related to the Tamagawa
213
numbers $c_p$, which appear in the Birch and Swinnerton-Dyer
214
conjecture).
215
\begin{question}
216
If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
217
component group of $A_f$ trivial?
218
\end{question}
219
Even assuming the full Birch and Swinnerton-Dyer conjecture,
220
extensive computations by the PI have not produced a conjectural
221
answer to this question. However, he and Bjorn Poonen formulated
222
a strategy to answer this question in some interesting cases by
223
using an explicit decomposition of $\End({A_f}_{\Qbar})$ to obtain
224
a curve whose Jacobian is $A_f$.
225
226
\section{Visibility of Shafarevich-Tate and Mordell-Weil Groups}
227
\subsection{Computational problems}
228
Before describing work the PI proposes to do on
229
visibility and its uses in obtaining evidence for the Birch and
230
Swinnerton-Dyer conjecture, we recall the precise definition of
231
visibility over~$\Q$ (the definition makes sense over any global
232
field, but we restrict to $\Q$ below for simplicity).
233
234
\begin{definition}[Visibility of Shafarevich-Tate Groups]
235
Suppose that $\iota:A\hra J$ is an inclusion of abelian varieties
236
over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$ with respect
237
to $\iota$ is the kernel of the induced map $\H^1(\Q,A)\to
238
\H^1(\Q,J)$. The {\em visible subgroup} of $\Sha(A/\Q)$ is the
239
intersection of $\Sha(A/\Q)$ with the visible subgroup of
240
$\H^1(\Q,A)$; equivalently, it is the kernel of the induced map
241
$\Sha(A/\Q)\to \Sha(J/\Q)$.
242
\end{definition}
243
244
Before discussing theoretical questions about visibility, we
245
describe computational evidence for the Birch and Swinnerton-Dyer
246
conjecture for modular abelian varieties (and motives) that we
247
obtained using a theorem inspired by the definition of visibility.
248
In \cite{agashe-stein:visibility}, the PI and Agashe prove a
249
theorem that makes it possible to use abelian varieties of large
250
rank to explicitly construct subgroups of Shafarevich-Tate groups
251
of abelian varieties. The main theorem is that if $A$ and $B$
252
are abelian subvarieties of an abelian variety $J$, and
253
$B[p]\subset A$, then, under certain hypothesis, there is an
254
injection from $B(\Q)/p B(\Q)$ into the visible subgroup of
255
$\Sha(A/\Q)$. The paper concludes with the first ever example of
256
an abelian variety $A_f$ attached to a newform, of large dimension
257
($20$), whose Shafarevich-Tate group is provably nontrivial.
258
259
In \cite{agashe-stein:bsd}, the PI and Agashe describe algorithms
260
that we found and I implemented in {\sc Magma} for computing with
261
modular abelian varieties. We then compute an odd divisor and
262
even multiple of the BSD conjectural order of $\Sha(A/\Q)$ for the
263
over ten thousand quotients $A=A_f$ of $J_0(N)$ with $L(A,1)\neq
264
0$. For over a hundred of these, our lower bound on the
265
conjectural order of $\Sha(A/\Q)$ is is divisible by an odd prime;
266
for a quarter of these we prove, using the main theorem of
267
\cite{agashe-stein:visibility}, that if $n$ is the conjectural
268
divisor of $\#\Sha(A/\Q)$, then there are at least~$n$ elements of
269
$\Sha(A/\Q)$ that are visible in $J_0(N)$. \emph{The PI intends
270
to investigate the remaining 75\% of the examples with $n>1$ by
271
considering the image of $A$ in $J_0(NM)$ for small integer $M$.}
272
Information about which primes $\ell$ to choose can be extracted
273
from Ribet's level raising theorem (see \cite{ribet:raising}). As
274
a test, the PI tried the first example not already done, which is
275
an $18$ dimension abelian variety of level $551$, whose
276
Shafarevich-Tate group conjecturally contains elements of order
277
$3$; these are not visible in $J_0(551)$, but he shows in
278
\cite{stein:bsdmagma} that they are visible in $J_(551\cdot 2)$.
279
280
One objective of the PI's past NSF supported research was to
281
generalize visibility to the context of modular motives.
282
Fortunately, Neil Dummigan, Mark Watkins, and the PI did
283
significant work in this direction in
284
\cite{dummigan-stein-watkins:motives}. There we prove a theorem
285
that can sometimes be used to deduce the existence of visible
286
Shafarevich-Tate groups in motives attached to modular forms,
287
assuming a conjecture of Beilinson about ranks of Chow groups.
288
However, we give several pages of tables that suggest that
289
Shafarevich-Tate groups of modular motives of level~$N$ are very
290
rarely visible in the higher weight motivic analogue of $J_0(N)$,
291
much more rarely than for weight~$2$. Just as above, the question
292
remains to decide whether one expects these groups to be visible
293
in $J_0(N M)$ for some integer~$M$. It would be relatively
294
straightforward for the PI to do computations in this direction,
295
but none have been done yet. The PI intends to do such
296
computations.
297
298
Before moving on to theoretical questions about visibility, we
299
pause to emphasize that the above computational investigations
300
into the Birch and Swinnerton-Dyer conjecture motivated the PI and
301
others to develop new algorithms for computing with modular
302
abelian varieties. For example, in \cite{stein:compgroup}, Conrad
303
and the PI use Grothendieck's monodromy pairing to give a complete
304
proof of correctness of an algorithm the PI found for computing
305
the order of the component group of certain purely toric abelian
306
varieties. This algorithm makes it practical to compute component
307
groups of quotients of $A_f$ of $J_0(N)$ at primes~$p$ that
308
exactly divide $N$; without such an algorithm it would probably be
309
difficult to get anywhere in computational investigations into the
310
Birch and Swinnerton-Dyer conjecture for abelian varieties;
311
indeed, the only other paper in this direction is \cite{evidence},
312
which restricts to the case of dimension~$2$ Jacobians.
313
314
315
\subsection{Theoretical problems}
316
Suppose $A_f$ is a quotient of $J_1(N)$ attached to a newform and
317
let $A=A_f^{\vee}\subset J_1(N)$ be its dual. One expects that
318
most of $\Sha(A)$ is {\em not} visible in $J_1(N)$. The following
319
conjecture then arises.
320
\begin{conjecture}\label{conj:allvis}
321
For each $x\in \Sha(A/\Q)$, there is an integer $M$ and a morphism
322
$f:A\to J_1(NM)$ of finite degree coprime to the order of~$x$ such
323
that the image of~$x$ in $\Sha(f(A)/\Q)$ is visible in $J_1(NM)$.
324
\end{conjecture}
325
In \cite{agashe-stein:visibility}, the PI proved that if $x\in
326
\H^1(\Q,A)$ then there is an abelian variety $B$ and an inclusion
327
$\iota:A\to B$ such that $x$ is visible in $B$; moreover, $B$ is a
328
quotient of $J_1(NM)$ for some $M$. The PI hopes to prove
329
Conjecture~\ref{conj:allvis} by understanding much more precisely
330
how $A$, $B$, and $J_1(NM)$ are related.
331
332
A more analytical approach to Conjecture~\ref{conj:allvis} is to
333
assume the rank statement of the Birch and Swinnerton-Dyer
334
conjecture and relate when elements of $\Sha(A/\Q)$ becoming
335
visible at level $NM$ to when there is a congruence between $f$
336
and a newform $g$ of level $NM$ with $L(g,1)=0$. Such an approach
337
leads one to wish to formulate a refinement of Ribet's level
338
raising theorem that includes a statement about the behavior of
339
the value at $1$ of the $L$-function attached to the form at
340
higher level. The PI intends to do further computations in order
341
to give a good conjectural refinement of Ribet's theorem.
342
343
344
The Gross-Zagier theorem asserts that points on elliptic curves of
345
rank $1$ come from Heegner points, and that points on curves of
346
rank bigger than one do not. Over fifteen years later, it still
347
seems mysterious to give an interpretation of points on elliptic
348
curves of rank higher than~$1$. The PI introduced the following
349
definition, in hopes of eventually creating a framework for giving
350
a conjectural explanation.
351
352
\begin{definition}[Visibility of Mordell-Weil Groups]
353
Suppose that $\pi : J\to A$ is a surjective morphism of abelian
354
varieties with connected kernel~$C$. Let $\delta : A(\Q)\to
355
\H^1(\Q,C)$ be the connecting homomorphism of Galois cohomology.
356
An element $x\in A(\Q)$ is \emph{$n$-visible in $\H^1$} (with
357
respect to $\pi$) if $\delta(x)$ has order divisible by~$n$, and
358
$x$ is \emph{$n$-visible in $\Sha$} if moreover $\delta(x)\in
359
\Sha(C/\Q)$.
360
\end{definition}
361
362
The following theorem is not difficult to prove by combining
363
Kato's powerful results towards the Birch and Swinnerton-Dyer
364
conjecture (see \cite{kato:secret,rubin:kato}) with a nonvanishing
365
theorem of Rohrlich \cite{rohrlich:cyclo}.
366
\begin{theorem}[Stein]\label{thm:allmwvis}
367
If~$A$ is a modular abelian variety and $x\in A(\Q)$ has infinite
368
order, then for every integer~$n$ there is a covering $J\to A$
369
with connected kernel such that $x$ is $n$-visible in $\H^1$.
370
\end{theorem}
371
%The key idea of the proof is that if $p$ is any prime and
372
%$\Q_\infty$ is the cyclotomic $\Z_p$-extension of $\Q$, then by a
373
%nonvanishing theorem of Rohrlich and Kato's theorem , the group
374
%$A(\Q_\infty)$ is finitely generated. From this we deduce that
375
%there is an (abelian) extension $K$ of $\Q$ such that $n$ divides
376
%the order of the image of $x$ in $A(\Q)/\Tr_{K/\Q}(A(K))$. Trace
377
%defines a morphism from the restriction of scalars
378
%$\pi:J=\Res_{K/\Q}(A_K)$ to $A$ with connected kernel. Then $x$
379
%is $n$-visible with respect to $\pi$.
380
381
\begin{conjecture}[Stein]
382
If~$A$ be a modular abelian variety and $x\in A(\Q)$ has infinite
383
order, then for every integer~$n$, there is an abelian variety $B$
384
and a surjective morphism $\pi:B\to A$ with connected kernel such
385
that $x$ is $n$-visible in $\Sha$ with respect to~$\pi$.
386
\end{conjecture}
387
388
The PI proved partial results towards this conjecture in
389
\cite{stein:nonsquaresha}. Suppose $E$ is an elliptic curve over
390
$\Q$ with conductor $N$, and let $f$ be the newform attached to
391
$E$. Fix a prime~$p\nmid N \prod c_p$ and such that the mod~$p$
392
Galois representation attached to $E$ is surjective. Suppose
393
$\chi:(\Z/\ell\Z)\to\mu_p$ is a Dirichlet character with
394
$\ell\nmid N$ such that
395
\[
396
L(E,\chi,1)\neq 0 \qquad\text{and}\qquad
397
a_{\ell}(E) \not\con \ell+1 \pmod{p},
398
\]
399
and let $K$ be the corresponding abelian extension. In
400
\cite{stein:nonsquaresha}, the PI uses Kato's theorem and
401
restriction of scalars to construct abelian varieties $A$ and $J$
402
and an exact sequence
403
\[
404
0 \to E(\Q)/ p E(\Q) \to \Sha(A/\Q) \to \Sha(J/\Q) \to \Sha(E/\Q)
405
\to 0.
406
\]
407
Here $A$ is an abelian variety of rank $0$ with $L(A,1)\neq 0$.
408
Thus all of $E(\Q)$ is $p$-visible in $\Sha$ and one can interpret
409
$E(\Q)/ p E(\Q)$ as a visible subgroup of a rank~$0$ abelian
410
variety~$A$.
411
412
There are two problems with this picture. First, the PI does not
413
know a proof that such a $\chi$ exists (he has verified existence
414
of $\chi$ in thousands of examples). Second, even if the existence
415
of $\chi$ were known it seems unlikely that visibility of
416
Mordell-Weil groups in an abstract restriction of scalars abelian
417
variety is likely to yield a satisfactory interpretation of
418
$E(\Q)$ when $E$ has large rank. To answer these problems, the PI
419
intends to study the system of all possible abelian varieties $B$
420
in which $E(\Q)$ induces visible elements of Shafarevich-Tate
421
groups, and moreover to try to keep track of how these abelian
422
varieties $B$ are related to $J_1(NM)$ for various $M$. This
423
will involve a combination of computer computation of examples
424
followed hopefully by the development of a concise notation for
425
keeping track of all relevant data, and eventually perhaps new
426
ideas that clarify our understanding of elliptic curves of rank
427
bigger than one.
428
429
\bibliographystyle{amsalpha}
430
\bibliography{biblio}
431
\end{document}
432
433
434
435
%%%%%%%%%%%
436
437
438
439
\begin{conjecture}[Stein]\label{conj:nonvanishchi}
440
Assume~$p$ is an odd prime and the mod~$p$ Galois representation
441
attached to $E$ is surjective. Then there are infinitely many
442
Dirichlet characters $\chi$ as above that satisfies the following
443
hypothesis:
444
\[
445
L(E,\chi,1)\neq 0 \qquad\text{and}\qquad
446
a_{\ell}(E) \not\con \ell+1 \pmod{p}.
447
\]
448
\end{conjecture}
449
I have verified this conjecture numerically in thousands of
450
examples, and I hope to prove something about it by assuming it is
451
false and constructing many relations between modular symbols.
452
Analytic methods involving averaging special values of
453
$L$-functions seem incapable of handling twists of high degree.
454
455
\begin{definition}[Modularity of Mordell-Weil]\label{defn:modmw}
456
If $A$ is a modular abelian variety, and $n$ is an integer, we say
457
that the Mordell-Weil group of $A$ is {\em $n$-modular of level
458
$M$} if there is a quotient $\pi:J_1(M)\to A'$, with connected
459
kernel, such that $A'$ is isogenous to $A$ and%
460
$\pi(J_1(M))\subset n A'(\Q)$. We say that the Mordell-Weil group
461
of $A$ is {\em modular} if it is $n$-modular for every
462
integer~$n$.
463
\end{definition}
464
465
I think the following conjecture is within reach.
466
\begin{conjecture}[Stein]\label{conj:mwallmodular}
467
The Mordell-Weil group $A(\Q)$ of every modular abelian variety is
468
itself modular, in the sense of Definition~\ref{defn:modmw}.
469
\end{conjecture}
470
This is closely related to Theorem~\ref{thm:allmwvis} since the
471
restriction of scalars of a modular abelian variety is again
472
modular. However it is unclear exactly how $A$, $\Res_{K/\Q}(A)$,
473
and $J_1(M)$ all fit together, and it is essential to understand
474
{\em exactly} how they fit together in order to verify the
475
conjecture. Also, for a given $n$, it would be interesting to
476
decide if the Mordell-Weil group is $n$-visible of level $M$ for
477
some naturally defined~$M$.
478
479
480
481
482
483
484
\section{OLD STUFF}
485
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
486
487
488
489
490
[summary/intro/the point]
491
492
493
\begin{definition}[Visibility]
494
Suppose that $\iota:A\hra J$ is an inclusion of abelian varieties
495
over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$ with respect
496
to $\iota$ is the kernel of the induced map $\H^1(\Q,A)\to
497
\H^1(\Q,J)$. The visible subgroup of $\Sha(A/\Q)$ is the
498
intersection of $\Sha(A/\Q)$ with the visible subgroup of
499
$\H^1(\Q,A)$, or equivalently, the kernel of the induced map
500
$\Sha(A/\Q)\to \Sha(J/\Q)$.
501
\end{definition}
502
503
The paper \cite{agashe-stein:visibility} extends work of Cremona
504
and Mazur \cite{mazur:visord3, cremona-mazur} to lay the
505
foundations for studying visibility of Shafarevich-Tate groups of
506
abelian varieties. In it, I use a restriction of scalars
507
construction to prove that if $A$ is an abelian variety and $x\in
508
\Sha(A/\Q)$ then there is an inclusion $A\hra B$ such that $x$ is
509
visible in $B$. We then prove that if $A$ and $B$ are abelian
510
subvarieties of an abelian variety $J$, and $B[p]\subset A$, then,
511
under certain hypothesis, there is an injection from $B(\Q)/p
512
B(\Q)$ into the visible subgroup of $\Sha(A/\Q)$. We apply this
513
theorem to prove that $25$ divides the order of the visible
514
subgroup of the Shafarevich-Tate group of an abelian variety of
515
dimension $20$ and level $389$. We also give the first explicit
516
example of an element of the Shafarevich-Tate group of an elliptic
517
curve that only becomes visible at higher level.
518
519
\subsubsection{Computational evidence for the BSD conjecture}
520
In \cite{agashe-stein:bsd}, Agashe and I
521
describe a number of algorithms that we found and I implemented in
522
{\sc Magma} for computing with modular abelian varieties. We then
523
compute a divisor and multiple of the BSD conjectural order of
524
$\Sha(A/\Q)$ for the 10360 (optimal) abelian variety quotients
525
$A=A_f$ of $J_0(N)$ with $L(A,1)\neq 0$ and $f$ of level $N\leq
526
2333$. For $168$ of these $A$ our divisor is divisible by an odd
527
prime, and for $37$ of these $168$, we prove using visibility that
528
if $n$ is the prime--to-$2$ part of the conjectural order of
529
$\Sha(A/\Q)$, then there are $n$ elements of $\Sha(A/\Q)$ that are
530
visible in $J_0(N)$. The challenge remains to use other
531
techniques, e.g., visibility at higher level, to show that there
532
are $n$ elements in $\Sha(A/\Q)$ in the remaining $131$ cases. For
533
example, in \cite{stein:bsdmagma} we do this for the first
534
example, which has level $551$ and whose $\Sha$ becomes visible at
535
level $2\cdot 551$.
536
537
\subsubsection{Visibility for modular motives} One objective of my
538
past NSF supported research was to generalize visibility theory to
539
the context of modular motives. Fortunately, Neil Dummigan, Mark
540
Watkins, and I succeeded in carrying out such a program in
541
\cite{dummigan-stein-watkins:motives}. There we prove a general
542
theorem that can be used in many cases to deduce the existence of
543
visible Shafarevich-Tate groups in motives attached to modular
544
forms, assuming a conjecture of Beilinson about ranks of Chow
545
groups. We give several pages of tables that suggest that
546
Shafarevich-Tate groups of modular motives of level~$N$ are very
547
rarely visible in the higher weight motivic analogue of $J_0(N)$,
548
much more rarely than for weight~$2$.
549
550
551
\subsubsection{Nonsquare Shafarevich-Tate groups} In the paper
552
\cite{stein:nonsquaresha} I give a surprising application of
553
visibility to understanding the possibilities for the orders of
554
Shafarevich-Tate groups. Before \cite{stein:nonsquaresha} no
555
examples of Shafarevich-Tate groups of order $p\cdot n^2$ were
556
known for any odd prime $p$, and the literature even suggested
557
such examples do not exist ([swinnerton-dyer]). In
558
\cite{poonen-stoll}, Poonen and Stoll give the first examples of
559
Shafarevich-Tate groups of order $2\cdot n^2$, which inspired me
560
to look for examples of order $3\cdot n^2$. Using an adaptation
561
of the ideas from visibility along with a deep theorem of Kato on
562
the Birch and Swinnerton-Dyer conjecture, I was able to construct,
563
under suitable hypothesis, an abelian variety $A$ with
564
$\#\Sha(A/\Q)=p\cdot n^2$; I then used a computer to verify the
565
hypothesis for all odd primes $p<25000$. The $p$-part of
566
$\Sha(A/\Q)$ is visible in an abelian variety that is isogenous to
567
$A\times E$, where $E$ is an elliptic curve. To remove the
568
hypothesis on~$p$ would require proving a nonvanishing result
569
about twists of~$L$ functions (see Section~\ref{conj:nonvanishchi}
570
below).
571
572
\subsection{The Arithmetic of modular abelian varieties}
573
574
\subsubsection{$J_1(p)$ has connected fibers} In the paper \cite{conrad-edixhoven-stein:j1p},
575
Conrad, S.~Edixhoven, and I prove a remarkable uniformity result
576
for the component group of the N\'eron model of $J_1(p)$: {\em It
577
has order~$1$ for all primes $p$!} We do this by determining the
578
closed fiber at~$p$ of a model for $X_1(p)$, then do intersection
579
theory computations to find a regular model for $X_1(p)$ over
580
$\Z$. In this paper, I use theorems of Mazur, Kato, and a
581
computation to determine the primes~$p$ such that $J_1(p)$ has
582
rank~$0$. I also do significant computations of many of the
583
invariants appearing in the BSD conjecture for each of the simple
584
factors of $J_1(p)$. The results of these numerical computations
585
combined with the main theoretical result of the paper on
586
component groups suggest several questions, which I intend to
587
address (see Section~\ref{sec:j1pques}).
588
589
\subsubsection{Component groups of purely toric abelian varieties}
590
In \cite{stein:compgroup}, Conrad and I use Grothendieck's
591
monodromy pairing to give a complete proof of correctness of an
592
algorithm I found for computing the order of the component group
593
of certain purely toric abelian varieties. I found this algorithm
594
after reading a letter from Ribet to Mestre, which addressed
595
certain numerical relations for elliptic curves in terms of
596
Mestre's method of graphs. In \cite{kohel-stein:ants4}, D.~Kohel
597
and I explain how to calculate Tamagawa numbers (the $c_p$ in the
598
BSD conjecture, which are orders of component groups) for purely
599
toric modular abelian varieties using this algorithm. In general,
600
we only obtain the Tamagawa number up to a bounded power of $2$, a
601
shortcoming I intend to remedy with further work. We also do not
602
determine the structure of the underlying component groups, which
603
is something I hope to do.
604
605
\subsubsection{The BSD conjecture for Jacobians of genus two
606
curves} The paper \cite{empirical} is about the BSD conjecture for
607
$32$ modular Jacobians of genus~$2$ curves. I learned of an early
608
version of \cite{empirical} before it was published, and was
609
shocked by the table of orders of Shafarevich-Tate groups that it
610
contained. I used the equation-free algorithms I developed in
611
\cite{stein:phd} to do the computations in a new way, and found
612
that the most striking example in the paper, a Shafarevich-Tate
613
group of order $49$, was incorrect. I was made a coauthor and
614
wrote a section of the paper describing my methods.
615
616
\subsection{Other research}
617
\subsubsection{Application of Kolyvagin's Euler system}%
618
In \cite{stein:index} I give an innovative application of
619
Kolyvagin's Euler system to an old question of E.~Artin, S.~Lang,
620
and Tate (see \cite{lang-tate}).
621
622
Let $X$ be a curve over $\Q$ (say) of
623
genus~$g$. The {\em index} of $X$ is the greatest common divisor
624
of the degrees of the extensions of $\Q$ in which $X$ has a
625
rational point. Then the canonical divisor has degree $2g-2$, so
626
the index of $X$ divides $2g-2$. When $g=1$ this is no condition
627
at all. {\bf Question:} {\em For every integer $n$, is there a
628
genus one curve with index exactly~$n$?}
629
630
In \cite{lang-tate}, Lang and Tate prove that for each $n$ there
631
is a genus one curve~$X$ over some number field~$K$ (which depends
632
on $n$) such that~$X$ has index~$n$. In \cite{stein:index}, I
633
prove that if~$K$ is a fixed number field, then for any~$n$ not
634
divisible by~$8$ there is a genus one curve~$X$ over~$K$ of
635
index~$n$. The proof involves reinterpreting genus one curves and
636
the notion of index in terms of Galois cohomology, then finding
637
nontrivial Galois cohomology classes with the requisite properties
638
in the Euler system of Heegner points on $X_0(17)$.
639
640
\subsubsection{Elliptic curves with full torsion} In \cite{merel-stein}, L.~Merel
641
and I investigate a natural question about fields of definition
642
that is connected with points on modular curves. Let $p$ be a
643
prime. Suppose $E$ is an elliptic curve over a number field~$K$
644
and all of the $p$ torsion on $E$ is defined over $K$. Properties
645
of the Weil pairing imply that the field $\Q(\zeta_p)$ of $p$th
646
roots of unity is contained in $K$. {\bf Question.} {\em Is there
647
an elliptic curve defined over $\Q(\zeta_p)$ all of whose
648
$p$-torsion is also defined over $\Q(\zeta_p)$?} By combining
649
the significant theory developed in \cite{merel:cyclo} with a
650
nontrivial modular symbols computation, we show that the question
651
has a ``no'' answer for all $p<1000$, except $p=2,3,5,13$. (A
652
student of Merel showed that $13$ also has a ``no'' answer.)
653
654
655
\subsubsection{Modularity of icosahedral Galois representations} In \cite{buzzard-stein:artin},
656
K.~Buzzard and I prove $8$ new cases of the Artin conjecture about
657
modularity of icosahedral Galois representations, only $3$ of
658
which are covered by the subsequent landmark work of Taylor which
659
gave infinitely many new examples. Buzzard and I push through an
660
explicit application of \cite{buzzard-taylor} by combining various
661
theorems with significant modular symbols computations over the
662
finite field of order~$5$.
663
664
665
\subsubsection{Approximating $p$-adic modular forms} In
666
\cite{coleman-stein:padicapprox}, R.~Coleman and I consider from a
667
theoretical and computational point of view questions about
668
$p$-adic approximation of infinite slope modular eigenforms by
669
modular eigenforms of finite slope. The slope of an eigenform
670
$f=\sum a_n q^n$ is the $p$-adic valuation of $a_p$, so an
671
eigenform has infinite slope precisely when $a_p=0$. When~$f$ is
672
an eigenform having infinite slope, Naomi Jochnowitz asked if for
673
every~$n$ there is an eigenform~$g$ of finite slope such that
674
$f\equiv g\pmod{p^n}$. We show that the answer in general is no,
675
but prove that if~$f$ is a twist of a finite slope eigenform,
676
then~$f$ can be approximated. We also investigate computationally
677
which forms can be approximated and how the weight of~$g$ grows as
678
a function of~$n$. These computations lead to intriguing
679
unanswered questions.
680
681
\section{Project Proposal}
682
683
\subsection{Visibility of Shafarevich-Tate groups at higher level}
684
685
The following conjecture is the central open problem in visibility
686
theory.
687
\begin{conjecture}[Stein]\label{conj:allvis}
688
Let~$A$ be a modular abelian variety.%
689
\begin{enumerate}%
690
\item Then there is an integer~$N$ and a morphism $f:A\to J_1(N)$
691
such that every element of $\Sha(f(A))$ is visible in $J_1(N)$.%
692
\item The level $N$ should be determined in some natural way in
693
terms of properties of~$A$. (Part of the conjecture is to give a
694
reasonable interpretation of natural.)
695
\end{enumerate}
696
\end{conjecture}
697
698
If true, Conjecture~\ref{conj:allvis} would imply finiteness of
699
the Shafarevich-Tate group of~$A$, which would massively
700
strengthen many current results towards the Birch and
701
Swinnerton-Dyer conjecture. In \cite{agashe-stein:visibility}, I
702
proved that each element of the Shafarevich-Tate group of~$A$ is
703
visible in some modular abelian variety~$B$, but in this
704
construction~$B$ depends on the element. As a first step toward
705
Conjecture~\ref{conj:allvis}, I hope to use my result to prove
706
that if $\Sha(A)$ is finite then part 1 of the conjecture is true.
707
The main obstruction is that it is unclear how $A$, $B$ and
708
$J_1(N)$ all fit together, and in order to prove the conjecture it
709
is essential to know {\em exactly} how these abelian varieties fit
710
together. I strongly believe resolving this difficulty is within
711
reach and will lead to new ideas. (See
712
Conjecture~\ref{conj:mwallmodular} below for a similar situation.)
713
714
The first part of Conjecture~\ref{conj:allvis} for a single
715
element of $\Sha(A)$ is analogous to the easy-to-prove assertion
716
that each ideal class in the ring of integers of a number field
717
becomes principal in a suitable extension field, where the
718
extension depends on the ideal class. The second part of the
719
conjecture is reminiscent of the existence of the Hilbert class
720
field of a number field, and deeper investigation into it may
721
prove crucial to understanding Shafarevich-Tate groups.
722
723
I intend to revisit the computations of \cite{agashe-stein:bsd}
724
and see how far visibility at higher level goes toward
725
constructing the odd part of $\Sha(A/\Q)$ in the remaining $131$
726
cases not already covered.
727
728
Suppose $A=A_f$ with $f\in S_2(\Gamma_1(N))$ and $L(A_f,1)\neq 0$.
729
The following discussion illustrates one way in which ideas from
730
visibility have vague unexplored implications for the BSD
731
conjecture, namely for the assertion that if $p\mid \#\Sha(A)$
732
then $p$ divides the conjectural order of $\Sha(A)$. This is
733
only one of many similar ideas.
734
735
Suppose $x\in \Sha(A/\Q)[p]$ is an element of prime order $p$ that
736
is visible in $J_1(NM)$ for some $M$. Then in most cases there
737
should be a factor $A_g$ of $J_1(NM)$ that has positive
738
Mordell-Weil rank such that $x$ is in the image of $A_g(\Q)$ under
739
some map. Usually this should imply that~$g$ that is congruent
740
to~$f$ modulo a prime of characteristic~$p$; then by Kato's
741
theorem \cite{kato:secret, rubin:kato} we must have $L(g,1)=0$,
742
since otherwise $A_g$ would have rank $0$. Because congruences
743
between eigenforms usually induce congruences between special
744
values of $L$ functions, this will often imply that
745
\[
746
L(A_f,1)/\Omega_{A_f} \con L(A_g,1)/\omega = 0 \pmod{p}.
747
\]
748
749
%%%% I just commented this out because isn't it trivially true by switching parity in functional equation
750
%This discussion also motivates the following conjecture, which may
751
%be viewed as an analytic shadow of visibility:
752
%\begin{conjecture}[Stein]
753
%Suppose $f$ is a newform and $p$ is a prime such that
754
%$L(A_f,1)/\Omega_{A_f}\con 0\pmod{p}$. Then there exists a
755
%newform $g$ that is congruent to $f$ modulo~$p$ such that
756
%$L(g,1)=0$.
757
%\end{conjecture}
758
%Note that the conjecture is trivially true in case $L(f,1)=0$,
759
%since we just take $g=f$.
760
761
762
763
\subsection{Visibility of Mordell-Weil Groups of abelian varieties}
764
Turning Mazur's visibility idea on its head, I introduced the
765
notion of visibility of Mordell-Weil groups.
766
767
\begin{definition}[Visibility of Mordell-Weil]
768
Suppose that $\pi : J\to A$ is a surjective morphism of abelian
769
varieties with connected kernel~$C$. Let $\delta : A(\Q)\to
770
\H^1(\Q,C)$ be the connecting homomorphism. An element $x\in
771
A(\Q)$ is \emph{$n$-visible} with respect to $\pi$ if $\delta(x)$
772
has order divisible by~$n$, and $x$ is \emph{$n$-visible in
773
$\Sha$} if moreover $\delta(x)\in \Sha(C/\Q)$.
774
\end{definition}
775
776
\begin{theorem}\label{thm:allmwvis}
777
If~$A$ is a modular abelian variety and $x\in A(\Q)$, then for
778
every integer~$n$ there is a covering $J\to A$ with connected
779
kernel such that $x$ is $n$-visible in $\H^1(\Q,J)$.
780
\end{theorem}
781
The key idea of the proof is that if $p$ is any prime and
782
$\Q_\infty$ is the cyclotomic $\Z_p$-extension of $\Q$, then by a
783
nonvanishing theorem of Rohrlich \cite{rohrlich:cyclo} and Kato's
784
theorem \cite{kato:secret,rubin:kato}, the group $A(\Q_\infty)$ is
785
finitely generated. From this we deduce that there is an (abelian)
786
extension $K$ of $\Q$ such that $n$ divides the order of the image
787
of $x$ in $A(\Q)/\Tr_{K/\Q}(A(K))$. Trace defines a morphism from
788
the restriction of scalars $\pi:J=\Res_{K/\Q}(A_K)$ to $A$ with
789
connected kernel. Then $x$ is $n$-visible with respect to $\pi$.
790
791
\begin{conjecture}[Stein]
792
Let $A$ be a modular abelian variety, let $x\in A(\Q)$, and let
793
$n$ be a positive integer. Then there is a surjective morphism
794
$\pi:J\to A$ with connected kernel such that $x$ is $n$-visible in
795
$\Sha$ with respect to~$\pi$.
796
\end{conjecture}
797
798
My attempts so far to prove this conjecture led to the paper
799
\cite{stein:nonsquaresha}, the connection being as follows.
800
Suppose $E$ is an elliptic curve over $\Q$ with $E(\Q)=\Z{}x$, and
801
let $f$ be the newform attached to $f$. Fix a prime~$p$. Suppose
802
$\chi:(\Z/\ell\Z)\to\mu_p$ is a Dirichlet character that satisfies
803
several carefully chosen hypothesis, and let $K$ be the
804
corresponding abelian extension. By chasing the appropriate
805
diagrams and using results about \'etale cohomology and N\'eron
806
models, I show that if $J=\Res_{K/\Q}(E)$ then $x$ is $p$-visible
807
in $\Sha$ with respect to $J\to E$. This means that
808
\[
809
E(\Q)/pE(\Q)\isom \Z/p\Z\subset \Sha(\ker(J\to E)),
810
\]
811
which is where the nonsquare part of $\Sha$ comes from.
812
\begin{conjecture}[Stein]\label{conj:nonvanishchi}
813
Assume~$p$ is an odd prime and the mod~$p$ Galois representation
814
attached to $E$ is surjective. Then there are infinitely many
815
Dirichlet characters $\chi$ as above that satisfies the following
816
hypothesis:
817
\[
818
L(E,\chi,1)\neq 0 \qquad\text{and}\qquad
819
a_{\ell}(E) \not\con \ell+1 \pmod{p}.
820
\]
821
\end{conjecture}
822
I have verified this conjecture numerically in thousands of
823
examples, and I hope to prove something about it by assuming it is
824
false and constructing many relations between modular symbols.
825
Analytic methods involving averaging special values of
826
$L$-functions seem incapable of handling twists of high degree.
827
828
\begin{definition}[Modularity of Mordell-Weil]\label{defn:modmw}
829
If $A$ is a modular abelian variety, and $n$ is an integer, we say
830
that the Mordell-Weil group of $A$ is {\em $n$-modular of level
831
$M$} if there is a quotient $\pi:J_1(M)\to A'$, with connected
832
kernel, such that $A'$ is isogenous to $A$ and%
833
$\pi(J_1(M))\subset n A'(\Q)$. We say that the Mordell-Weil group
834
of $A$ is {\em modular} if it is $n$-modular for every
835
integer~$n$.
836
\end{definition}
837
838
I think the following conjecture is within reach.
839
\begin{conjecture}[Stein]\label{conj:mwallmodular}
840
The Mordell-Weil group $A(\Q)$ of every modular abelian variety is
841
itself modular, in the sense of Definition~\ref{defn:modmw}.
842
\end{conjecture}
843
This is closely related to Theorem~\ref{thm:allmwvis} since the
844
restriction of scalars of a modular abelian variety is again
845
modular. However it is unclear exactly how $A$, $\Res_{K/\Q}(A)$,
846
and $J_1(M)$ all fit together, and it is essential to understand
847
{\em exactly} how they fit together in order to verify the
848
conjecture. Also, for a given $n$, it would be interesting to
849
decide if the Mordell-Weil group is $n$-visible of level $M$ for
850
some naturally defined~$M$.
851
852
853
854
\subsection{Computing with modular abelian varieties}
855
Bryan Birch once commented to me in reference to computation that
856
``It is always a good idea to try to prove true theorems.'' To
857
this end, the author proposes to continue developing algorithms
858
and making available tools for computing with modular forms,
859
modular abelian varieties, and motives attached to modular forms.
860
This includes finishing a major new {\sc Magma} \cite{magma}
861
package for computing directly with modular abelian varieties over
862
number fields, extending the Modular Forms Database \cite{mfd},
863
and searching for algorithms for computing the quantities
864
appearing in the Birch and Swinnerton-Dyer for modular abelian
865
varieties and the Bloch-Kato conjecture for modular motives. The
866
results of this work should give an explicit picture of modular
867
abelian varieties that could never be obtained from general
868
theory.
869
870
871
\subsubsection{The Modular forms database} The modular forms
872
database \cite{mfd} contains a large collection of information
873
about objects attached to cuspidal eigenforms. Though greatly
874
appreciated by the many mathematicians who use it, the database
875
currently only scratches the surface of what it should contain.
876
877
The database is stored using the database system PostgreSQL, and I
878
wrote the web user interface in Python. During Summer 2003 the
879
Harvard undergraduate Dimitar Jetchev did extensive work on the
880
database, and this pointed out significant deficiencies in how it
881
is currently implemented. It is more difficult than it should be
882
to modify the web interface to the database, the data is not
883
compressed well, and there is no way to submit new data to the
884
database using the web page. I intend to completely rewrite the
885
database using MySQL and PHP, and investigate better algorithms
886
for storing $q$-expansions of modular forms much more efficiently.
887
Currently the limit on the database is not the difficulty of
888
computing modular forms, but the space and time used in storing
889
them. This could be partially remedied by moving the database to
890
a more modern computer (it currently runs on a three year old
891
Pentium III), something I am requesting in this grant.
892
893
\subsubsection{Example database queries that have not yet been done}
894
\begin{itemize}
895
\item Suppose $d=2,3,4,5$, say. Using the algorithm described in
896
\cite{agashe-stein:bsd}, compute a multiple of the order of the
897
torsion subgroup of $A_f(\Q)$ for each $d$-dimensional $A_f$ in
898
the database. What is the maximum number that occurs? After what
899
level do no new numbers appear? For small $d$ such a computation
900
may suggest a conjectural generalization to modular abelian
901
varieties of Mazur's theorem on torsion points on elliptic curves.%
902
903
\item Make a conjectural list of all number fields of degree~$d$
904
(for $d=2,3,4,5$, say) that arise as the field generated by the
905
eigenvalues of a newform in the database. Coleman has conjectured
906
that for each~$d$ only finitely many number fields of degree~$d$
907
appear. When do new $d$ seem to stop appearing?%
908
\end{itemize}
909
910
\subsubsection{M{\small AGMA} package for modular abelian varieties} I
911
wrote the modular forms and modular symbols packages that are part
912
of the {\sc Magma} computer algebra system \cite{magma}. I spent
913
June 2003 in Sydney, Australia and did exciting work on designing
914
and implementing a very general package for computing with modular
915
abelian varieties over number fields. Much work is left to be
916
done to finish this package, and several crucial algorithms still
917
need to be developed or refined. When available this package
918
will likely be greatly appreciated by anybody working with modular
919
abelian varieties.
920
921
The following are problems that arose in work on this package,
922
which must be resolved in order to have a satisfactory system for
923
computing with modular abelian varieties. I need to solve all of
924
these problems.
925
\begin{itemize}
926
\item {\bf Endomorphism ring over $\Qbar$:} {\em Giving a modular
927
abelian variety $A$, explicitly (and efficiently) compute the
928
endomorphism ring $\End(A)$ as a ring of matrices acting
929
$\H_1(A,\Z)$.} I have a modular symbols solution that draw on
930
work of Ribet \cite{ribet:twistsendoalg} and Shimura
931
\cite{shimura:factors} but is too slow to be really useful in
932
practice; however, similar Manin symbols methods must
933
exist and be very efficient.%
934
\item {\bf Decomposition:} {\em Given the endomorphism ring of an
935
abelian variety $A$, compute its decomposition as a product of
936
simple abelian varieties.} This is a standard and difficult
937
problem in general, but it might be possible to combine work of
938
Allan Steel on a ``characteristic~$0$ Meataxe'' with special
939
features of modular abelian varieties to solve it in practice. It
940
is absolutely {\em essential} to solve this problem in order to
941
explicitly enumerate all modular abelian varieties over number
942
fields of given level.%
943
\item {\bf Isomorphism testing:} {\em Given
944
two modular abelian varieties over a number field $K$, represented
945
as explicit quotients of Jacobians $J_1(N)$, decide whether there
946
is an isomorphism between them.} I have solved this problem when
947
the two modular abelian varieties are simple. There are analogues
948
of this problem in other categories, which I intend to investigate.%
949
\end{itemize}
950
951
\subsection{Some conjectures that were inspired by my computations}
952
953
\subsubsection{Questions about $J_1(p)$}\label{sec:j1pques} The following
954
conjecture generalizes a famous conjecture of Ogg that
955
$J_0(p)(\Q)_{\tor}$ is cyclic of order the numerator of
956
$(p-1)/12$, a fact that Mazur proved in \cite{mazur:eisenstein}.
957
\begin{conjecture}[Stein]
958
The torsion subgroup of $J_1(p)(\Q)$ is exactly the group
959
generated by the cusps on $X_1(p)$ that lie over $\infty\in
960
X_0(p)$. This is a group of order
961
\[
962
\frac{p}{2^{p-3}} \cdot\prod_{\eps\neq 1} B_{2,\eps}
963
\]
964
where the product is over the nontrivial even Dirichlet
965
characters~$\varepsilon$ of conductor dividing~$p$, and
966
$B_{2,\eps}$ is a generalized Bernoulli number.
967
\end{conjecture}
968
Mazur's complete proof of the analogue of this statement for
969
$J_0(p)$ is quite deep, though the proof for the prime-to-$2$ part
970
of $J_0(p)(\Q)_{\tor}$ is much easier. I hope to mimic Mazur's
971
method and prove the conjecture above for the prime-to-$2$ part of
972
$J_1(p)(\Q)_{\tor}$.
973
974
More generally, I would like to investigate the torsion in
975
quotients of $J_1(p)$.
976
\begin{question}
977
If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
978
natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective? If
979
so, is the product of the orders of all $A_f(\Q)_{\tor}$ over all
980
classes of newforms $f$ equal to $\# J_1(p)(\Q)_{\tor}$?
981
\end{question}
982
I conjectured that the analogous questions for $J_0(p)$ should have
983
``yes'' answers, and in \cite{emerton:optimal} M.~Emerton
984
subsequently proved this conjecture. It is still not clear if
985
one should make this conjecture for $J_1(p)$.
986
987
\subsubsection{Congruences between modular forms of prime level}
988
Ken Ribet enticed me into studying modular forms as a graduate
989
student by asking me the following question: ``Is there a prime
990
$p$ so that $p$ is ramified in the Hecke algebra $\T$ attached to
991
$S_2(\Gamma_0(p))$?'' I answered his question by showing that
992
$p=389$ is the only prime less than $50000$ that ramifies in the
993
associated Hecke algebra of level~$p$. The question remains
994
whether $p=389$ is the only such example, and this seems extremely
995
difficult to say anything useful about. However a related
996
question exhibits a shockingly clear pattern, and this related
997
question is the question Ribet was really interested in for his
998
application to images of Galois representations
999
\cite{ribet:torsion}.
1000
1001
Let $p$ be a prime and $k$ a positive even integer. Let $\T$ be
1002
the Hecke algebra attached to $S_k(\Gamma_0(p))$ and let $d(k,p)$
1003
be the valuation at $p$ of the index of $\T$ in its normalization.
1004
The following conjecture is backed up by significant numerical
1005
evidence, and was discovered by staring at tables and looking for
1006
a pattern.
1007
\begin{conjecture}[Stein, F.~Calegari]
1008
Suppose $p\geq k-1$. Then
1009
$$
1010
d(k,p) = \left\lfloor\frac{p}{12}\right\rfloor\cdot
1011
\binom{m}{2} + a(p,m),
1012
$$
1013
where the function $a$ only depends on $p$ modulo $12$ as
1014
follows:
1015
$$
1016
a(p,m) =
1017
\begin{cases}
1018
0 & \text{if $p\equiv 1\pmod{12}$,}\\
1019
3\cdot\displaystyle\binom{\lceil \frac{m}{3}\rceil}{2} & \text{if $p\equiv 5\pmod{12}$,}\\
1020
2\cdot\displaystyle\binom{\lceil \frac{m}{2}\rceil}{2} & \text{if $p\equiv 7\pmod{12}$,}\\
1021
a(5,m)+a(7,m) & \text{if $p\equiv 11\pmod{12}$.}
1022
\end{cases}
1023
$$
1024
\end{conjecture}
1025
1026
The situation of interest to Ribet is $k=2$, in which case the
1027
conjecture simply asserts that $\T\otimes\Z_p$ is normal, i.e.,
1028
{\em there or no congruences in characteristic $p$ between
1029
non-Galois conjugate newforms in $S_2(\Gamma_0(p))$.} Calegari
1030
has given a conjectural interpretation of some of the congruences
1031
that the conjecture asserts must exist, which I intend to study
1032
further.
1033
1034
\section{Summary}
1035
This research proposal depicts an intricate network of ongoing
1036
investigations into the arithmetic of modular abelian varieties,
1037
which unite a theoretical and computational point of view. The
1038
basic foundations of visibility theory are nearly complete, but
1039
solutions to the questions about visibility outlined in this
1040
proposal demand a new level of precision in our understanding of
1041
the web of modular abelian varieties. I am determined to advance
1042
our understanding in this direction.
1043
1044
My work has produced results and tools that are of use to other
1045
mathematicians who are exploring the world of modular forms. By
1046
supporting my research, you will assure the sustained development
1047
of this technology.
1048
1049
1050
1051
1052
1053
Let $p$ be a prime and $k$ a positive even integer. Let $\T$ be
1054
the Hecke algebra attached to the space $S_k(\Gamma_0(p))$ of cusp
1055
forms for $\Gamma_0(p)$ and let $d(k,p)$ be the valuation at $p$
1056
of the index of $\T$ in its normalization. The following
1057
conjecture is backed up by significant numerical evidence, and was
1058
discovered by staring at tables computed using the PIs {\sc Magma}
1059
code and looking for a pattern.
1060
\begin{conjecture}[Stein, F.~Calegari]
1061
Suppose $p\geq k-1$. Then
1062
$$
1063
d(k,p) = \left\lfloor\frac{p}{12}\right\rfloor\cdot
1064
\binom{m}{2} + a(p,m),
1065
$$
1066
where the function $a$ only depends on $p$ modulo $12$ as
1067
follows:
1068
$$
1069
a(p,m) =
1070
\begin{cases}
1071
0 & \text{if $p\equiv 1\pmod{12}$,}\\
1072
3\cdot\displaystyle\binom{\lceil \frac{m}{3}\rceil}{2} & \text{if $p\equiv 5\pmod{12}$,}\\
1073
2\cdot\displaystyle\binom{\lceil \frac{m}{2}\rceil}{2} & \text{if $p\equiv 7\pmod{12}$,}\\
1074
a(5,m)+a(7,m) & \text{if $p\equiv 11\pmod{12}$.}
1075
\end{cases}
1076
$$
1077
\end{conjecture}
1078
The conjecture is of interest to Ribet in the case $k=2$, because
1079
it is a hypotheses to the main argument of \cite{ribet:torsion}.
1080
For a long time it was unclear what to conjecture when $k=2$;
1081
finally, investigation into what happens at higher weight
1082
suggested the above conjectural formula, which specializes in
1083
weight $2$ to the assertion that $\T\otimes \Z_p$ is normal. The
1084
PI has no idea how to prove this conjecture when $k=2$, but
1085
intends to at least find similar conjectures when $\Gamma_0(p)$ is
1086
replaced by $\Gamma_1(p)$ and when~$p$ is replaced by a composite
1087
number.
1088
1089
1090
1091
\newpage
1092
1093
\newcommand{\etalchar}[1]{$^{#1}$}
1094
\providecommand{\bysame}{\leavevmode\hbox
1095
to3em{\hrulefill}\thinspace}
1096
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
1097
% \MRhref is called by the amsart/book/proc definition of \MR.
1098
\providecommand{\MRhref}[2]{%
1099
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
1100
} \providecommand{\href}[2]{#2}
1101
\begin{thebibliography}{BCDT01}
1102
1103
\bibitem[AS]{agashe-stein:bsd}
1104
A.~Agashe and W.\thinspace{}A. Stein, \emph{Visible {E}vidence for
1105
the {B}irch
1106
and {S}winnerton-{D}yer {C}onjecture for {M}odular {A}belian {V}arieties of
1107
{A}nalytic {R}ank~$0$}, To appear in Mathematics of Computation.
1108
1109
\bibitem[AS02]{agashe-stein:visibility}
1110
\bysame, \emph{Visibility of {S}hafarevich-{T}ate groups of
1111
abelian varieties},
1112
J. Number Theory \textbf{97} (2002), no.~1, 171--185.
1113
1114
\bibitem[BCDT01]{breuil-conrad-diamond-taylor}
1115
C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor, \emph{On the
1116
modularity of
1117
elliptic curves over {$\bold Q$}: wild 3-adic exercises}, J. Amer. Math. Soc.
1118
\textbf{14} (2001), no.~4, 843--939 (electronic).
1119
1120
\bibitem[BCP97]{magma}
1121
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra
1122
system. {I}.
1123
{T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
1124
235--265, Computational algebra and number theory (London, 1993).
1125
1126
\bibitem[Bir71]{birch:bsd}
1127
B.\thinspace{}J. Birch, \emph{Elliptic curves over
1128
\protect{${\mathbf{Q}}$:
1129
{A}} progress report}, 1969 Number Theory Institute (Proc. Sympos. Pure
1130
Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math.
1131
Soc., Providence, R.I., 1971, pp.~396--400.
1132
1133
\bibitem[BS02]{buzzard-stein:artin}
1134
K.~Buzzard and W.\thinspace{}A. Stein, \emph{A mod five approach
1135
to modularity
1136
of icosahedral {G}alois representations}, Pacific J. Math. \textbf{203}
1137
(2002), no.~2, 265--282.
1138
1139
\bibitem[BT99]{buzzard-taylor}
1140
K.~Buzzard and R.~Taylor, \emph{Companion forms and weight one
1141
forms}, Ann. of
1142
Math. (2) \textbf{149} (1999), no.~3, 905--919.
1143
1144
\bibitem[CES03]{conrad-edixhoven-stein:j1p}
1145
B.~Conrad, S.~Edixhoven, and W.\thinspace{}A. Stein,
1146
\emph{${J}_1(p)$ {H}as
1147
{C}onnected {F}ibers}, To appear in Documenta Mathematica (2003).
1148
1149
\bibitem[CM00]{cremona-mazur}
1150
J.\thinspace{}E. Cremona and B.~Mazur, \emph{Visualizing elements
1151
in the
1152
{S}hafarevich-{T}ate group}, Experiment. Math. \textbf{9} (2000), no.~1,
1153
13--28.
1154
1155
\bibitem[Col03]{coleman-stein:padicapprox}
1156
R.~Coleman, \emph{Approximation of infinite-slope modular
1157
eigenforms by
1158
finite-slope eigenforms}, To appear in the Dwork Proceedings (2003).
1159
1160
\bibitem[CS02]{stein:compgroup}
1161
B.~Conrad and W.\thinspace{}A. Stein, \emph{Component {G}roups of
1162
{P}urely
1163
{T}oric {Q}uotients}, To appear in Math Research Letters (2002).
1164
1165
\bibitem[DWS]{dummigan-stein-watkins:motives}
1166
N.~Dummigan, M.~Watkins, and W.\thinspace{}A. Stein,
1167
\emph{{Constructing
1168
Elements in Shafarevich-Tate Groups of Modular Motives}}, To appear in the
1169
Swinnerton-Dyer proceedings.
1170
1171
\bibitem[Eme01]{emerton:optimal}
1172
M.~Emerton, \emph{Optimal {Q}uotients of {M}odular {J}acobians},
1173
preprint
1174
(2001).
1175
1176
\bibitem[FpS{\etalchar{+}}01]{empirical}
1177
E.\thinspace{}V. Flynn, F.~\protect{Lepr\'{e}vost},
1178
E.\thinspace{}F. Schaefer,
1179
W.\thinspace{}A. Stein, M.~Stoll, and J.\thinspace{}L. Wetherell,
1180
\emph{Empirical evidence for the {B}irch and {S}winnerton-{D}yer conjectures
1181
for modular {J}acobians of genus 2 curves}, Math. Comp. \textbf{70} (2001),
1182
no.~236, 1675--1697 (electronic).
1183
1184
\bibitem[Kat]{kato:secret}
1185
K.~Kato, \emph{$p$-adic {H}odge theory and values of zeta
1186
functions of modular
1187
forms}, Preprint, 244 pages.
1188
1189
\bibitem[KS00]{kohel-stein:ants4}
1190
D.\thinspace{}R. Kohel and W.\thinspace{}A. Stein, \emph{Component
1191
{G}roups of
1192
{Q}uotients of \protect{$J_0(N)$}}, Proceedings of the 4th International
1193
Symposium (ANTS-IV), Leiden, Netherlands, July 2--7, 2000 (Berlin), Springer,
1194
2000.
1195
1196
\bibitem[Lan91]{lang:nt3}
1197
S.~Lang, \emph{Number theory. {I}{I}{I}}, Springer-Verlag, Berlin,
1198
1991,
1199
Diophantine geometry.
1200
1201
\bibitem[LT58]{lang-tate}
1202
S.~Lang and J.~Tate, \emph{Principal homogeneous spaces over
1203
abelian
1204
varieties}, Amer. J. Math. \textbf{80} (1958), 659--684.
1205
1206
\bibitem[Maz77]{mazur:eisenstein}
1207
B.~Mazur, \emph{Modular curves and the \protect{Eisenstein}
1208
ideal}, Inst.
1209
Hautes \'Etudes Sci. Publ. Math. (1977), no.~47, 33--186 (1978).
1210
1211
\bibitem[Maz99]{mazur:visord3}
1212
\bysame, \emph{Visualizing elements of order three in the
1213
{S}hafarevich-{T}ate
1214
group}, Asian J. Math. \textbf{3} (1999), no.~1, 221--232, Sir Michael
1215
Atiyah: a great mathematician of the twentieth century.
1216
1217
\bibitem[Mer01]{merel:cyclo}
1218
L.~Merel, \emph{Sur la nature non-cyclotomique des points d'ordre
1219
fini des
1220
courbes elliptiques}, Duke Math. J. \textbf{110} (2001), no.~1, 81--119, With
1221
an appendix by E. Kowalski and P. Michel.
1222
1223
\bibitem[MS01]{merel-stein}
1224
L.~Merel and W.\thinspace{}A. Stein, \emph{The field generated by
1225
the points of
1226
small prime order on an elliptic curve}, Internat. Math. Res. Notices (2001),
1227
no.~20, 1075--1082.
1228
1229
\bibitem[PS99]{poonen-stoll}
1230
B.~Poonen and M.~Stoll, \emph{The {C}assels-{T}ate pairing on
1231
polarized abelian
1232
varieties}, Ann. of Math. (2) \textbf{150} (1999), no.~3, 1109--1149.
1233
1234
1235
\bibitem[Rib80]{ribet:twistsendoalg}
1236
K.\thinspace{}A. Ribet, \emph{Twists of modular forms and
1237
endomorphisms of
1238
abelian varieties}, Math. Ann. \textbf{253} (1980), no.~1, 43--62.
1239
1240
1241
\bibitem[Rib99]{ribet:torsion}
1242
\bysame, \emph{Torsion points on ${J}\sb 0({N})$ and {G}alois
1243
representations},
1244
Arithmetic theory of elliptic curves (Cetraro, 1997), Springer, Berlin, 1999,
1245
pp.~145--166.
1246
1247
\bibitem[Roh84]{rohrlich:cyclo}
1248
D.\thinspace{}E. Rohrlich, \emph{On {$L$}-functions of elliptic
1249
curves and
1250
cyclotomic towers}, Invent. Math. \textbf{75} (1984), no.~3, 409--423.
1251
1252
\bibitem[Rub98]{rubin:kato}
1253
K.~Rubin, \emph{Euler systems and modular elliptic curves}, Galois
1254
representations in arithmetic algebraic geometry (Durham, 1996), Cambridge
1255
Univ. Press, Cambridge, 1998, pp.~351--367.
1256
1257
\bibitem[Shi73]{shimura:factors}
1258
G.~Shimura, \emph{On the factors of the jacobian variety of a
1259
modular function
1260
field}, J. Math. Soc. Japan \textbf{25} (1973), no.~3, 523--544.
1261
1262
\bibitem[Ste]{stein:nonsquaresha}
1263
W.\thinspace{}A. Stein, \emph{Shafarevich-tate groups of nonsquare
1264
order},
1265
Proceedings of MCAV 2002, Progress of Mathematics (to appear).
1266
1267
\bibitem[Ste00]{stein:phd}
1268
\bysame, \emph{Explicit approaches to modular abelian varieties},
1269
Ph.D. thesis,
1270
University of California, Berkeley (2000).
1271
1272
\bibitem[Ste02]{stein:index}
1273
\bysame, \emph{There are genus one curves over {$\mathbf{Q}$} of
1274
every odd
1275
index}, J. Reine Angew. Math. \textbf{547} (2002), 139--147.
1276
1277
\bibitem[Ste03a]{mfd}
1278
\bysame, \emph{The {M}odular {F}orms {D}atabase}, \newline{\tt
1279
http://modular.fas.harvard.edu/Tables} (2003).
1280
1281
\bibitem[Ste03b]{stein:bsdmagma}
1282
\bysame, \emph{Studying the {B}irch and {S}winnerton-{D}yer
1283
{C}onjecture for
1284
{M}odular {A}belian {V}arieties {U}sing {\sc Magma}}, To appear in J.~Cannon, ed.,
1285
{\em Computational Experiments in Algebra and Geometry}, Springer-Verlag
1286
(2003).
1287
1288
\bibitem[Tat95]{tate:bsd}
1289
J.~Tate, \emph{On the conjectures of {B}irch and
1290
{S}winnerton-{D}yer and a
1291
geometric analog}, S\'eminaire Bourbaki, Vol.\ 9, Soc. Math. France, Paris,
1292
1995, pp.~Exp.\ No.\ 306, 415--440.
1293
1294
\end{thebibliography}
1295
1296
1297
\end{document}
1298
1299
1300
1301
\end{document}
1302