CoCalc Public Fileswww / nsf / project_description_old.tex
Author: William A. Stein
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7
8\section{Introduction}
9My research reflects the essential interplay of abstract theory
10with explicit machine computation, which is illustrated by the
11following quote of Bryan Birch~\cite{birch:bsd} about computations
12that led to a central conjecture in number theory:
13\begin{quote}
14I want to describe some computations undertaken by myself and
15Swinnerton-Dyer on EDSAC by which we have calculated the
16zeta-functions of certain elliptic curves. As a result of these
17computations we have found an analogue for an elliptic curve of
18the Tamagawa number of an algebraic group; and conjectures (due to
19ourselves, due to Tate, and due to others) have proliferated.
20\end{quote}
21
22
23I am primarily interested in abelian varieties attached to modular
24forms via Shimura's construction \cite{shimura:factors}.
25Let~$f=\sum a_n q^n$ be a weight~$2$ newform on $\Gamma_1(N)$.  We
26may view~$f$ as a differential on the modular curve $X_1(N)$,
27which is a curve whose affine points over~$\C$ correspond to
28isomorphism classes of pairs $(E,P)$, where~$E$ is an elliptic
29curve and $P\in E$ is a point of order~$N$.  We view the Hecke
30algebra
31$\T=\Z[T_1,T_2,T_3,\ldots]$
32 as a subring of the endomorphism ring of the Jacobian $J_1(N)$
33of $X_1(N)$. Let $I_f$ be the annihilator of~$f$ in $\T$, and
34attach to~$f$ the quotient $$A_f=J_1(N)/I_f J_1(N).$$ Then $A_f$
35is an abelian variety over~$\Q$ of dimension equal to the degree
36of the field $\Q(a_1,a_2,a_3,\ldots)$ generated by the
37coefficients of~$f$.
38
39The abelian varieties $A_f$ attached to newforms are important.
40For example,the celebrated modularity theorem of C.~Breuil,
41B.~Conrad, F.~Diamond, Taylor, and Wiles
42\cite{breuil-conrad-diamond-taylor} asserts that every elliptic
43curve over~$\Q$ is isogenous to some $A_f$.  Also, Serre
44conjectures that up to twist every two-dimensional odd irreducible
45Galois representation appears in the torsion points on some $A_f$.
46
47My investigations into modular abelian varieties are inspired by
48the following special case of the Birch and Swinnerton-Dyer
49conjecture (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}
57Here $L(A_f,s)$ is the canonical $L$-series attached to $A_f$, the
58real volume $\Omega_{A_f}$ is the measure of $A_f(\R)$ with
59respect 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
66Shafarevich-Tate group of $A_f$.  When $A_f(\Q)$ is infinite, the
67right hand side should be interpreted as~$0$, so, in particular,
68the conjecture asserts that $L(A_f,1)=0$ if and only if $A_f(\Q)$
69is infinite. Birch and Swinnerton-Dyer also conjectured that the
70order 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
76The PI proposes to continue developing algorithms and making
77available tools for computing with modular forms, modular abelian
78varieties, and motives attached to modular forms. This includes
79finishing a major new {\sc Magma} \cite{magma} package for
80computing directly with modular abelian varieties over number
81fields, extending the Modular Forms Database \cite{mfd}, and
82searching for algorithms for computing the quantities appearing in
83the Birch and Swinnerton-Dyer for modular abelian varieties and
84the Bloch-Kato conjecture for modular motives.
85
86
87
88\subsection{The Modular Forms Database}%
89The Modular Forms Database \cite{mfd} is a freely-available
90collection of data about objects attached to cuspidal modular
91forms, that is a much used resource for number theorists.  It is
92analogous to Sloane's tables of integer sequences, and extends
93Cremona's tables to dimension bigger than one and weight bigger
94than two (\cite{cremona-tables} contain more precise data about
95elliptic curves than \cite{mfd}).
96
97The PI proposes to greatly expand the database.  The major
98challenge is that data about modular abelian varieties of large
99dimension takes a huge amount of space to store.  For example, the
100database currently occupies 40GB disk space.   He proposes to find
101better method for storing information about modular abelian so
102that the database can grow larger, and to investigate methods used
103by astronomers or the human genome project to see how they cope
104with a torrent of data while making it available to their
105colleagues.
106
107The PI implemented the current database using PostgreSQL coupled
108with a Python web interface.   To speed access and improve
109efficiency, the PI is considering rewriting key portions of the
110database using MySQL and PHP.  Also, the database currently runs
111on a three-year-old 933Mhz Pentium III, so the PI is requesting
112more modern hardware.
113
114\subsubsection{M{\small AGMA} package for modular abelian varieties}%
115{\sc Magma} is a nonprofit computer algebra system developed
116primarily at the University of Sydney, which is supported mostly
117by grant money from organizations such as the US National Security
118Agency. {\sc Magma} is considered by many to be the most
119comprehensive tool for research in number theory, finite group
120theory, and cryptography, and it is widely distributed.  The PI
121has already written over 400 pages (26000 lines) of modular forms
122code and extensive documentation that is distributed with {\sc
123Magma}, and intends to publish'' future work in {\sc Magma}.
124
125An abelian variety $A$ over a number field is {\em modular} if it
126is a quotient of $J_1(N)$ for some $N$.  Modular abelian varieties
127were studied intensively by Ken Ribet, Barry Mazur, and others
128during recent decades, and studying them is popular because
129results about them often yield surprising insight into number
130theoretic questions.  Computation with modular abelian varieties
131is popular because most of them are easily described by giving a
132level and the first few coefficients of a modular form, and the
133$L$-functions of modular abelian varieties are particularly well
134understood.   This is in sharp contrast to the case of general
135abelian varieties which, in general, can only be described by
136unwieldy systems of polynomial equations, and whose $L$-functions
137are very mysterious.
138
139The PI recently designed and partially implemented a general
140package for computing with modular abelian varieties over number
141fields. Several crucial algorithms still need to be developed or
142refined.  When available, this package will likely be useful for
143people working with modular abelian varieties.   The following
144major problems arose in work on this package, and they must be
145resolved in order to have a completely satisfactory system for
146computing with modular abelian varieties:
147\begin{itemize}
148\item {\em Given a modular abelian variety $A$, efficiently
149compute the endomorphism ring $\End(A)$ as a ring of matrices
150acting $\H_1(A,\Z)$.} The PI has found a modular symbols solution
151that draws on work of Ribet \cite{ribet:twistsendoalg} and Shimura
152\cite{shimura:factors}, but it is too slow to be really useful in
153practice.  In \cite{merel:1585}, Merel uses Herbrand matrices and
154Manin symbols to give efficient algorithms for computing with
155Hecke operators. The PI intends to carry over Merel's method to
156give an efficient algorithm to compute $\End(A)$.%
157\item {\em Given $\End(A)\otimes\Q$, compute an isogeny
158decomposition of $A$ as a product of simple abelian varieties.}
159This is a standard and difficult problem in general, but it might
160be possible to combine work of Allan Steel on his
161characteristic~$0$ Meataxe'' with special features of modular
162abelian varieties to solve it in practice.  It is absolutely {\em
163essential} to solve this problem in order to explicitly enumerate
164all modular abelian varieties over number fields of given level.
165Such an enumeration would be a major step towards the ultimate
166possible generalization of Cremona's tables \cite{cremona:tables}
167to 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
170the endomorphism ring of each abelian variety is known and both
171are simple, it is possible to reduce this problem to mostly
172well-studied norm equations.  This problem is analogous to the
173problem of testing isomorphism for modules over a fixed ring,
174which has been solved with much effort for many classes of rings.
175One application of isomorphism testing is that it could be used to
176prove that an abelian variety is not principally polarized.%
177\end{itemize}
178
179We finish be describing recent work with $J_1(p)$ that was
180partially inspired by  PI's modular forms software. This
181conjecture 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}
185Let $p$ be a prime. The torsion subgroup of $J_1(p)(\Q)$ is the
186group generated by the cusps on $X_1(p)$ that lie over $\infty\in 187X_0(p)$.
188\end{conjecture}
189Significant numerical evidence for this conjecture is given in
190\cite{j1p}, and cuspidal subgroups of $J_1(p)$ are considered in
191detail in \cite{kubert-lang}, where, e.g.,  they compute orders of
192such groups in terms of Bernoulli numbers.
193
194Mazur's proof of Ogg's conjecture for $J_0(p)$ is deep, though the
195proof for the odd part of $J_0(p)(\Q)_{\tor}$ is much easier.  The
196PI intends explore whether or not it is possible to build on
197Mazur's method and prove results towards
198Conjecture~\ref{conj:tor}.  The PI also intends to develop his
199computational methods for computing torsion subgroups in order to
200answer, at least conjecturally, the following question.
201\begin{question}
202If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
203natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective?  If
204so, is the product of the orders of all $A_f(\Q)_{\tor}$ over all
205classes of newforms $f$ equal to $\# J_1(p)(\Q)_{\tor}$?
206\end{question}
207The PI conjectured that the analogous questions for $J_0(p)$
208should have yes'' answers, and in \cite{emerton:optimal}
209M.~Emerton proved this conjecture.
210
211In \cite{j1p}, it is proved that $J_1(p)$ has trivial component
212group (component groups are closely related to the Tamagawa
213numbers $c_p$, which appear in the Birch and Swinnerton-Dyer
214conjecture).
215\begin{question}
216If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
217component group of $A_f$ trivial?
218\end{question}
219Even assuming the full Birch and Swinnerton-Dyer conjecture,
220extensive computations by the PI have not produced a conjectural
221answer to this question.  However, he and Bjorn Poonen formulated
222a strategy to answer this question in some interesting cases by
223using an explicit decomposition of $\End({A_f}_{\Qbar})$ to obtain
224a 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
229visibility and its uses in obtaining evidence for the Birch and
230Swinnerton-Dyer conjecture, we recall the precise definition of
231visibility over~$\Q$ (the definition makes sense over any global
232field, but we restrict to $\Q$ below for simplicity).
233
234\begin{definition}[Visibility of Shafarevich-Tate Groups]
235Suppose that $\iota:A\hra J$ is an inclusion of abelian varieties
236over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$ with respect
237to $\iota$ 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
239intersection 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
244Before discussing theoretical questions about visibility, we
245describe computational evidence for the Birch and Swinnerton-Dyer
246conjecture for modular abelian varieties (and motives) that we
247obtained using a theorem inspired by the definition of visibility.
248In \cite{agashe-stein:visibility}, the PI and Agashe prove a
249theorem that makes it possible to use abelian varieties of large
250rank to explicitly construct subgroups of Shafarevich-Tate groups
251of abelian varieties.   The main theorem is that if $A$ and $B$
252are abelian subvarieties of an abelian variety $J$, and
253$B[p]\subset A$, then, under certain hypothesis, there is an
254injection from $B(\Q)/p B(\Q)$ into the visible subgroup of
255$\Sha(A/\Q)$.  The paper concludes with the first ever example of
256an abelian variety $A_f$ attached to a newform, of large dimension
257($20$), whose Shafarevich-Tate group is provably nontrivial.
258
259In  \cite{agashe-stein:bsd}, the PI and Agashe describe algorithms
260that we found and I implemented in {\sc Magma} for computing with
261modular abelian varieties.  We then compute an odd divisor and
262even multiple of the BSD conjectural order of $\Sha(A/\Q)$ for the
263over ten thousand quotients $A=A_f$ of $J_0(N)$ with $L(A,1)\neq 2640$. For over a hundred of these, our lower bound on the
265conjectural order of $\Sha(A/\Q)$ is is divisible by an odd prime;
266for a quarter of these we prove, using the main theorem of
267\cite{agashe-stein:visibility}, that if $n$ is the conjectural
268divisor 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
270to investigate the remaining 75\% of the examples with $n>1$ by
271considering the image of $A$ in $J_0(NM)$ for small integer $M$.}
272Information about which primes $\ell$ to choose can be extracted
273from Ribet's level raising theorem (see \cite{ribet:raising}).  As
274a test, the PI tried the first example not already done, which is
275an $18$ dimension abelian variety of level $551$, whose
276Shafarevich-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
280One objective of the PI's past NSF supported research was to
281generalize visibility to the context of modular motives.
282Fortunately, Neil Dummigan, Mark Watkins, and the PI did
283significant work in this direction in
284\cite{dummigan-stein-watkins:motives}.  There we prove a theorem
285that can sometimes be used to deduce the existence of visible
286Shafarevich-Tate groups in motives attached to modular forms,
287assuming a conjecture of Beilinson about ranks of Chow groups.
288However, we give several pages of tables that suggest that
289Shafarevich-Tate groups of modular motives of level~$N$ are very
290rarely visible in the higher weight motivic analogue of $J_0(N)$,
291much more rarely than for weight~$2$.  Just as above, the question
292remains to decide whether one expects these groups to be visible
293in $J_0(N M)$ for some integer~$M$.  It would be relatively
294straightforward for the PI to do computations in this direction,
295but none have been done yet.  The PI intends to do such
296computations.
297
298Before moving on to theoretical questions about visibility, we
299pause to emphasize that the above computational investigations
300into the Birch and Swinnerton-Dyer conjecture motivated the PI and
301others to develop new algorithms for computing with modular
302abelian varieties.  For example, in \cite{stein:compgroup}, Conrad
303and the PI use Grothendieck's monodromy pairing to give a complete
304proof of correctness of an algorithm the PI found for computing
305the order of the component group of certain purely toric abelian
306varieties.  This algorithm makes it practical to compute component
307groups of quotients of $A_f$ of $J_0(N)$ at primes~$p$ that
308exactly divide $N$; without such an algorithm it would probably be
309difficult to get anywhere in computational investigations into the
310Birch and Swinnerton-Dyer conjecture for abelian varieties;
311indeed, the only other paper in this direction is \cite{evidence},
312which restricts to the case of dimension~$2$ Jacobians.
313
314
315\subsection{Theoretical problems}
316Suppose $A_f$ is a quotient of $J_1(N)$ attached to a newform and
317let $A=A_f^{\vee}\subset J_1(N)$ be its dual.  One expects that
318most of $\Sha(A)$ is {\em not} visible in $J_1(N)$.  The following
319conjecture then arises.
320\begin{conjecture}\label{conj:allvis}
321For 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
323that the image of~$x$ in $\Sha(f(A)/\Q)$ is visible in $J_1(NM)$.
324\end{conjecture}
325In \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
328quotient of $J_1(NM)$ for some $M$.  The PI hopes to prove
329Conjecture~\ref{conj:allvis} by understanding much more precisely
330how $A$, $B$, and $J_1(NM)$ are related.
331
332A more analytical approach to Conjecture~\ref{conj:allvis} is to
333assume the rank statement of the Birch and Swinnerton-Dyer
334conjecture and relate when elements of $\Sha(A/\Q)$ becoming
335visible at level $NM$ to when there is a congruence between $f$
336and a newform $g$ of level $NM$ with $L(g,1)=0$.  Such an approach
337leads one to wish to formulate a refinement of Ribet's level
338raising theorem that includes a statement about the behavior of
339the value at $1$ of the $L$-function attached to the form at
340higher level.  The PI intends to do further computations in order
341to give a good conjectural refinement of Ribet's theorem.
342
343
344The Gross-Zagier theorem asserts that points on elliptic curves of
345rank $1$ come from Heegner points, and that points on curves of
346rank bigger than one do not.   Over fifteen years later, it still
347seems mysterious to give an interpretation of points on elliptic
348curves of rank higher than~$1$.  The PI introduced the following
349definition, in hopes of eventually creating a framework for giving
350a conjectural explanation.
351
352\begin{definition}[Visibility of Mordell-Weil Groups]
353Suppose that $\pi : J\to A$ is a surjective morphism of abelian
354varieties with connected kernel~$C$.  Let $\delta : A(\Q)\to 355\H^1(\Q,C)$ be the connecting homomorphism of Galois cohomology.
356An element $x\in A(\Q)$ is \emph{$n$-visible in $\H^1$} (with
357respect 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
362The following theorem is not difficult to prove by combining
363Kato's powerful results towards the Birch and Swinnerton-Dyer
364conjecture (see \cite{kato:secret,rubin:kato}) with a nonvanishing
365theorem of Rohrlich \cite{rohrlich:cyclo}.
366\begin{theorem}[Stein]\label{thm:allmwvis}
367If~$A$ is a modular abelian variety and $x\in A(\Q)$ has infinite
368order, then for every integer~$n$ there is a covering $J\to A$
369with 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]
382If~$A$ be a modular abelian variety and $x\in A(\Q)$ has infinite
383order, then for every integer~$n$, there is an abelian variety $B$
384and a surjective morphism $\pi:B\to A$ with connected kernel such
385that $x$ is $n$-visible in $\Sha$ with respect to~$\pi$.
386\end{conjecture}
387
388The 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~$p$
392Galois 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$396 L(E,\chi,1)\neq 0 \qquad\text{and}\qquad 397 a_{\ell}(E) \not\con \ell+1 \pmod{p}, 398$
399and let $K$ be the corresponding abelian extension. In
400\cite{stein:nonsquaresha}, the PI uses Kato's theorem and
401restriction of scalars to construct abelian varieties $A$ and $J$
402and an exact sequence
403$4040 \to E(\Q)/ p E(\Q) \to \Sha(A/\Q) \to \Sha(J/\Q) \to \Sha(E/\Q) 405\to 0. 406$
407Here $A$ is an abelian variety of rank $0$ with $L(A,1)\neq 0$.
408Thus 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
410variety~$A$.
411
412There are two problems with this picture.  First, the PI does not
413know a proof that such a $\chi$ exists (he has verified existence
414of $\chi$ in thousands of examples). Second, even if the existence
415of $\chi$ were known it seems unlikely that visibility of
416Mordell-Weil groups in an abstract restriction of scalars abelian
417variety is likely to yield a satisfactory interpretation of
418$E(\Q)$ when $E$ has large rank.  To answer these problems, the PI
419intends to study the system of all possible abelian varieties $B$
420in which $E(\Q)$ induces visible elements of Shafarevich-Tate
421groups, and moreover to try to keep track of how these abelian
422varieties $B$ are related to $J_1(NM)$ for various $M$.   This
423will involve a combination of computer computation of examples
424followed hopefully by the development of a concise notation for
425keeping track of all relevant data, and eventually perhaps new
426ideas that clarify our understanding of elliptic curves of rank
427bigger 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}
440Assume~$p$ is an odd prime and the mod~$p$ Galois representation
441attached to $E$ is surjective.  Then there are infinitely many
442Dirichlet characters $\chi$ as above that satisfies the following
443hypothesis:
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}
449I have verified this conjecture numerically in thousands of
450examples, and I hope to prove something about it by assuming it is
451false and constructing many relations between modular symbols.
452Analytic 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}
456If $A$ is a modular abelian variety, and $n$ is an integer, we say
457that 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
459kernel, such that $A'$ is isogenous to $A$ and%
460$\pi(J_1(M))\subset n A'(\Q)$.  We say that the Mordell-Weil group
461of $A$ is {\em modular} if it is $n$-modular for every
462integer~$n$.
463\end{definition}
464
465I think the following conjecture is within reach.
466\begin{conjecture}[Stein]\label{conj:mwallmodular}
467The Mordell-Weil group $A(\Q)$ of every modular abelian variety is
468itself modular, in the sense of Definition~\ref{defn:modmw}.
469\end{conjecture}
470This is closely related to Theorem~\ref{thm:allmwvis} since the
471restriction of scalars of a modular abelian variety is again
472modular.  However it is unclear exactly how $A$, $\Res_{K/\Q}(A)$,
473and $J_1(M)$ all fit together, and it is essential to understand
474{\em exactly} how they fit together in order to verify the
475conjecture.   Also, for a given $n$, it would be interesting to
476decide if the Mordell-Weil group is $n$-visible of level $M$ for
477some 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]
494Suppose that $\iota:A\hra J$ is an inclusion of abelian varieties
495over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$ with respect
496to $\iota$ 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
498intersection 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
503The paper \cite{agashe-stein:visibility} extends work of Cremona
504and Mazur \cite{mazur:visord3, cremona-mazur} to lay the
505foundations for studying visibility of Shafarevich-Tate groups of
506abelian varieties.  In it, I use a restriction of scalars
507construction 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
509visible in $B$. We then prove that if $A$ and $B$ are abelian
510subvarieties of an abelian variety $J$, and $B[p]\subset A$, then,
511under certain hypothesis, there is an injection from $B(\Q)/p 512B(\Q)$ into the visible subgroup of $\Sha(A/\Q)$.  We apply this
513theorem to prove that $25$ divides the order of the visible
514subgroup of the Shafarevich-Tate group of an abelian variety of
515dimension $20$ and level $389$. We also give the first explicit
516example of an element of the Shafarevich-Tate group of an elliptic
517curve that only becomes visible at higher level.
518
519\subsubsection{Computational evidence for the BSD conjecture}
520In \cite{agashe-stein:bsd}, Agashe and I
521describe a number of algorithms that we found and I implemented in
522{\sc Magma} for computing with modular abelian varieties.  We then
523compute 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 $N\leq 5262333$. For $168$ of these $A$ our divisor is divisible by an odd
527prime, and for $37$ of these $168$, we prove using visibility that
528if $n$ 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
530visible in $J_0(N)$.   The challenge remains to use  other
531techniques, e.g., visibility at higher level, to show that there
532are $n$ elements in $\Sha(A/\Q)$ in the remaining $131$ cases. For
533example, in \cite{stein:bsdmagma} we do this for the first
534example, which has level $551$ and whose $\Sha$ becomes visible at
535level $2\cdot 551$.
536
537\subsubsection{Visibility for modular motives} One objective of my
538past NSF supported research was to generalize visibility theory to
539the context of modular motives. Fortunately, Neil Dummigan, Mark
540Watkins, and I succeeded in carrying out such a program in
541\cite{dummigan-stein-watkins:motives}. There we prove a general
542theorem that can be used in many cases to deduce the existence of
543visible Shafarevich-Tate groups in motives attached to modular
544forms, assuming a conjecture of Beilinson about ranks of Chow
545groups.  We give several pages of tables that suggest that
546Shafarevich-Tate groups of modular motives of level~$N$ are very
547rarely visible in the higher weight motivic analogue of $J_0(N)$,
548much 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
553visibility to understanding the possibilities for the orders of
554Shafarevich-Tate groups. Before \cite{stein:nonsquaresha} no
555examples of Shafarevich-Tate groups of order $p\cdot n^2$ were
556known for any odd prime $p$, and the literature even suggested
557such examples do not exist ([swinnerton-dyer]).  In
558\cite{poonen-stoll}, Poonen and Stoll give the first examples of
559Shafarevich-Tate groups of order $2\cdot n^2$, which inspired me
560to look for examples of order $3\cdot n^2$.   Using an adaptation
561of the ideas from visibility along with a deep theorem of Kato on
562the Birch and Swinnerton-Dyer conjecture, I was able to construct,
563under suitable hypothesis, an abelian variety $A$ with
564$\#\Sha(A/\Q)=p\cdot n^2$; I then used a computer to verify the
565hypothesis 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
568hypothesis on~$p$ would require proving a nonvanishing result
569about twists of~$L$ functions (see Section~\ref{conj:nonvanishchi}
570below).
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},
575Conrad, S.~Edixhoven, and I prove a remarkable uniformity result
576for the component group of the N\'eron model of $J_1(p)$: {\em It
577has order~$1$ for all primes $p$!}  We do this by determining the
578closed fiber at~$p$ of a model for $X_1(p)$, then do intersection
579theory computations to find a regular model for $X_1(p)$ over
580$\Z$.   In this paper, I use theorems of Mazur, Kato, and a
581computation to determine the primes~$p$ such that $J_1(p)$ has
582rank~$0$. I also do significant computations of many of the
583invariants appearing in the BSD conjecture for each of the simple
584factors of $J_1(p)$. The results of these numerical computations
585combined with the main theoretical result of the paper on
586component groups suggest several questions, which I intend to
587address (see Section~\ref{sec:j1pques}).
588
589\subsubsection{Component groups of purely toric abelian varieties}
590In \cite{stein:compgroup}, Conrad and I use Grothendieck's
591monodromy pairing to give a complete proof of correctness of an
592algorithm I found for computing the order of the component group
593of certain purely toric abelian varieties.  I found this algorithm
594after reading a letter from Ribet to Mestre, which addressed
595certain numerical relations for elliptic curves in terms of
596Mestre's method of graphs.   In \cite{kohel-stein:ants4}, D.~Kohel
597and I explain how to calculate Tamagawa numbers (the $c_p$ in the
598BSD conjecture, which are orders of component groups) for purely
599toric modular abelian varieties using this algorithm.  In general,
600we only obtain the Tamagawa number up to a bounded power of $2$, a
601shortcoming I intend to remedy with further work.   We also do not
602determine the structure of the underlying component groups, which
603is something I hope to do.
604
605\subsubsection{The BSD conjecture for Jacobians of genus two
606curves} The paper \cite{empirical} is about the BSD conjecture for
607$32$ modular Jacobians of genus~$2$ curves. I learned of an early
608version of \cite{empirical} before it was published, and was
609shocked by the table of orders of Shafarevich-Tate groups that it
610contained. I used the equation-free algorithms I developed in
611\cite{stein:phd} to do the computations in a new way, and found
612that the most striking example in the paper, a Shafarevich-Tate
613group of order $49$, was incorrect.  I was made a coauthor and
614wrote a section of the paper describing my methods.
615
616\subsection{Other research}
617\subsubsection{Application of Kolyvagin's Euler system}%
618In \cite{stein:index} I give an innovative application of
619Kolyvagin's Euler system to an old question of E.~Artin, S.~Lang,
620and Tate (see \cite{lang-tate}).
621
622 Let $X$ be a curve over $\Q$ (say) of
623genus~$g$. The {\em index} of $X$ is the greatest common divisor
624of the degrees of the extensions of $\Q$ in which $X$ has a
625rational point. Then the canonical divisor has degree $2g-2$, so
626the index of $X$ divides $2g-2$. When $g=1$ this is no condition
627at all. {\bf Question:} {\em For every integer $n$, is there a
628genus one curve with index exactly~$n$?}
629
630In \cite{lang-tate}, Lang and Tate prove that for each $n$ there
631is a genus one curve~$X$ over some number field~$K$ (which depends
632on $n$) such that~$X$ has index~$n$.  In \cite{stein:index}, I
633prove that if~$K$ is a fixed number field, then for any~$n$ not
634divisible by~$8$ there is a genus one curve~$X$ over~$K$ of
635index~$n$.  The proof involves reinterpreting genus one curves and
636the notion of index in terms of Galois cohomology, then finding
637nontrivial Galois cohomology classes with the requisite properties
638in the Euler system of Heegner points on $X_0(17)$.
639
640\subsubsection{Elliptic curves with full torsion} In \cite{merel-stein}, L.~Merel
641and I investigate a natural question about fields of definition
642that is connected with points on modular curves. Let $p$ be a
643prime. Suppose $E$ is an elliptic curve over a number field~$K$
644and all of the $p$ torsion on $E$ is defined over $K$. Properties
645of the Weil pairing imply that the field $\Q(\zeta_p)$ of $p$th
646roots of unity is contained in $K$. {\bf Question.} {\em Is there
647an elliptic curve defined over $\Q(\zeta_p)$ all of whose
648$p$-torsion is also defined over $\Q(\zeta_p)$?}   By combining
649the significant theory developed in \cite{merel:cyclo} with a
650nontrivial modular symbols computation, we show that the question
651has a no'' answer for all $p<1000$, except $p=2,3,5,13$. (A
652student of Merel showed that $13$ also has a no'' answer.)
653
654
655\subsubsection{Modularity of icosahedral Galois representations} In \cite{buzzard-stein:artin},
656K.~Buzzard and I prove $8$ new cases of the Artin conjecture about
657modularity of icosahedral Galois representations, only $3$ of
658which are covered by the subsequent landmark work of Taylor which
659gave infinitely many new examples. Buzzard and I push through an
660explicit application of \cite{buzzard-taylor} by combining various
661theorems with significant modular symbols computations over the
662finite 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
667theoretical and computational point of view questions about
668$p$-adic approximation of infinite slope modular eigenforms by
669modular 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
671eigenform has infinite slope precisely when $a_p=0$.  When~$f$ is
672an eigenform having infinite slope, Naomi Jochnowitz asked if for
673every~$n$ 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,
675but prove that if~$f$ is a twist of a finite slope eigenform,
676then~$f$ can be approximated. We also investigate computationally
677which forms can be approximated and how the weight of~$g$ grows as
678a function of~$n$. These computations lead to intriguing
679unanswered questions.
680
681\section{Project Proposal}
682
683\subsection{Visibility of Shafarevich-Tate groups at higher level}
684
685The following conjecture is the central open problem in visibility
686theory.
687\begin{conjecture}[Stein]\label{conj:allvis}
688Let~$A$ be a modular abelian variety.%
689\begin{enumerate}%
690\item Then there is an integer~$N$ and a morphism $f:A\to J_1(N)$
691such 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
693terms of properties of~$A$.  (Part of the conjecture is to give a
694reasonable interpretation of natural.)
695\end{enumerate}
696\end{conjecture}
697
698If true, Conjecture~\ref{conj:allvis} would imply finiteness of
699the Shafarevich-Tate group of~$A$, which would massively
700strengthen many current results towards the Birch and
701Swinnerton-Dyer conjecture.  In \cite{agashe-stein:visibility}, I
702proved that each element of the Shafarevich-Tate group of~$A$ is
703visible in some modular abelian variety~$B$, but in this
704construction~$B$ depends on the element. As a first step toward
705Conjecture~\ref{conj:allvis}, I hope to use my result to prove
706that if $\Sha(A)$ is finite then part 1 of the conjecture is true.
707The 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
709is essential to know {\em exactly} how these abelian varieties fit
710together.  I strongly believe resolving this difficulty is within
711reach and will lead to new ideas. (See
712Conjecture~\ref{conj:mwallmodular} below for a similar situation.)
713
714The first part of Conjecture~\ref{conj:allvis} for a single
715element of $\Sha(A)$ is analogous to the easy-to-prove assertion
716that each ideal class in the ring of integers of a number field
717becomes principal in a suitable extension field, where the
718extension depends on the ideal class. The second part of the
719conjecture is reminiscent of the existence of the Hilbert class
720field of a number field, and deeper investigation into it may
721prove crucial to understanding Shafarevich-Tate groups.
722
723I intend to revisit the computations of \cite{agashe-stein:bsd}
724and see how far visibility at higher level goes toward
725constructing the odd part of $\Sha(A/\Q)$ in the remaining $131$
726cases not already covered.
727
728Suppose $A=A_f$ with $f\in S_2(\Gamma_1(N))$ and $L(A_f,1)\neq 0$.
729The following discussion illustrates one way in which ideas from
730visibility have vague unexplored implications for the BSD
731conjecture, namely for the assertion that if $p\mid \#\Sha(A)$
732then $p$ divides the conjectural order of $\Sha(A)$.   This is
733only one of many similar ideas.
734
735Suppose $x\in \Sha(A/\Q)[p]$ is an element of prime order $p$ that
736is visible in $J_1(NM)$ for some $M$. Then in most cases there
737should be a factor $A_g$ of $J_1(NM)$ that has positive
738Mordell-Weil rank such that $x$ is in the image of $A_g(\Q)$ under
739some map. Usually this should imply that~$g$ that is congruent
740to~$f$ modulo a prime of characteristic~$p$; then by Kato's
741theorem \cite{kato:secret, rubin:kato} we must have $L(g,1)=0$,
742since otherwise $A_g$ would have rank $0$.  Because congruences
743between eigenforms usually induce congruences between special
744values 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}
764Turning Mazur's visibility idea on its head, I introduced the
765notion of visibility of Mordell-Weil groups.
766
767\begin{definition}[Visibility of Mordell-Weil]
768Suppose that $\pi : J\to A$ is a surjective morphism of abelian
769varieties with connected kernel~$C$.  Let $\delta : A(\Q)\to 770\H^1(\Q,C)$ be the connecting homomorphism.  An element $x\in 771A(\Q)$ is \emph{$n$-visible} with respect to $\pi$ if $\delta(x)$
772has 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}
777If~$A$ is a modular abelian variety and $x\in A(\Q)$, then for
778every integer~$n$ there is a covering $J\to A$ with connected
779kernel such that $x$ is $n$-visible in $\H^1(\Q,J)$.
780\end{theorem}
781The 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
783nonvanishing theorem of Rohrlich \cite{rohrlich:cyclo} and Kato's
784theorem \cite{kato:secret,rubin:kato}, the group $A(\Q_\infty)$ is
785finitely generated. From this we deduce that there is an (abelian)
786extension $K$ of $\Q$ such that $n$ divides the order of the image
787of $x$ in $A(\Q)/\Tr_{K/\Q}(A(K))$. Trace defines a morphism from
788the restriction of scalars $\pi:J=\Res_{K/\Q}(A_K)$ to $A$ with
789connected kernel.  Then $x$ is $n$-visible with respect to $\pi$.
790
791\begin{conjecture}[Stein]
792Let $A$ 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
798My attempts so far to prove this conjecture led to the paper
799\cite{stein:nonsquaresha}, the connection being as follows.
800Suppose $E$ is an elliptic curve over $\Q$ with $E(\Q)=\Z{}x$, and
801let $f$ 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
803several carefully chosen hypothesis, and let $K$ be the
804corresponding abelian extension. By chasing the appropriate
805diagrams and using results about \'etale cohomology and  N\'eron
806models, I show that if $J=\Res_{K/\Q}(E)$ then $x$ is $p$-visible
807in $\Sha$ with respect to $J\to E$.  This means that
808$809 E(\Q)/pE(\Q)\isom \Z/p\Z\subset \Sha(\ker(J\to E)), 810$
811which is where the nonsquare part of $\Sha$ comes from.
812\begin{conjecture}[Stein]\label{conj:nonvanishchi}
813Assume~$p$ is an odd prime and the mod~$p$ Galois representation
814attached to $E$ is surjective.  Then there are infinitely many
815Dirichlet characters $\chi$ as above that satisfies the following
816hypothesis:
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}
822I have verified this conjecture numerically in thousands of
823examples, and I hope to prove something about it by assuming it is
824false and constructing many relations between modular symbols.
825Analytic 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}
829If $A$ is a modular abelian variety, and $n$ is an integer, we say
830that 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
832kernel, such that $A'$ is isogenous to $A$ and%
833$\pi(J_1(M))\subset n A'(\Q)$.  We say that the Mordell-Weil group
834of $A$ is {\em modular} if it is $n$-modular for every
835integer~$n$.
836\end{definition}
837
838I think the following conjecture is within reach.
839\begin{conjecture}[Stein]\label{conj:mwallmodular}
840The Mordell-Weil group $A(\Q)$ of every modular abelian variety is
841itself modular, in the sense of Definition~\ref{defn:modmw}.
842\end{conjecture}
843This is closely related to Theorem~\ref{thm:allmwvis} since the
844restriction of scalars of a modular abelian variety is again
845modular.  However it is unclear exactly how $A$, $\Res_{K/\Q}(A)$,
846and $J_1(M)$ all fit together, and it is essential to understand
847{\em exactly} how they fit together in order to verify the
848conjecture.   Also, for a given $n$, it would be interesting to
849decide if the Mordell-Weil group is $n$-visible of level $M$ for
850some naturally defined~$M$.
851
852
853
854\subsection{Computing with modular abelian varieties}
855Bryan Birch once commented to me in reference to computation that
856It is always a good idea to try to prove true theorems.''  To
857this end, the author proposes to continue developing algorithms
858and making available tools for computing with modular forms,
859modular abelian varieties, and motives attached to modular forms.
860This includes finishing a major new {\sc Magma} \cite{magma}
861package for computing directly with modular abelian varieties over
862number fields, extending the Modular Forms Database \cite{mfd},
863and searching for algorithms for computing the quantities
864appearing in the Birch and Swinnerton-Dyer for modular abelian
865varieties and the Bloch-Kato conjecture for modular motives.  The
866results of this work should give an explicit picture of modular
867abelian varieties that could never be obtained from general
868theory.
869
870
871\subsubsection{The Modular forms database} The modular forms
872database \cite{mfd} contains a large collection of information
873about objects attached to cuspidal eigenforms.  Though greatly
874appreciated by the many mathematicians who use it, the database
875currently only scratches the surface of what it should contain.
876
877The database is stored using the database system PostgreSQL, and I
878wrote the web user interface in Python.  During Summer 2003 the
879Harvard undergraduate Dimitar Jetchev did extensive work on the
880database, and this pointed out significant deficiencies in how it
881is currently implemented.   It is more difficult than it should be
882to modify the web interface to the database, the data is not
883compressed well, and there is no way to submit new data to the
884database using the web page.  I intend to completely rewrite the
885database using MySQL and PHP, and investigate better algorithms
886for storing $q$-expansions of modular forms much more efficiently.
887Currently the limit on the database is not the difficulty of
888computing modular forms, but the space and time used in storing
889them.  This could be partially remedied by moving the database to
890a more modern computer (it currently runs on a three year old
891Pentium 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
897torsion subgroup of $A_f(\Q)$ for each $d$-dimensional $A_f$ in
898the database.  What is the maximum number that occurs?  After what
899level do no new numbers appear?  For small $d$ such a computation
900may suggest a conjectural generalization to modular abelian
901varieties 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
905eigenvalues of a newform in the database. Coleman has conjectured
906that for each~$d$ only finitely many number fields of degree~$d$
907appear. When do new $d$ seem to stop appearing?%
908\end{itemize}
909
910\subsubsection{M{\small AGMA} package for modular abelian varieties} I
911wrote the modular forms and modular symbols packages that are part
912of the {\sc Magma} computer algebra system \cite{magma}.  I spent
913June 2003 in Sydney, Australia and did exciting work on designing
914and implementing a very general package for computing with modular
915abelian varieties over number fields.  Much work is left to be
916done to finish this package, and several crucial algorithms still
917need to be developed or refined.   When available this package
918will likely be greatly appreciated by anybody working with modular
919abelian varieties.
920
921The following are problems that arose in work on this package,
922which must be resolved in order to have a satisfactory system for
923computing with modular abelian varieties.  I need to solve all of
924these problems.
925\begin{itemize}
926\item {\bf Endomorphism ring over $\Qbar$:} {\em Giving a modular
927abelian variety $A$, explicitly (and efficiently) compute the
928endomorphism ring $\End(A)$ as a ring of matrices acting
929$\H_1(A,\Z)$.}   I have a modular symbols solution that draw on
930work of Ribet \cite{ribet:twistsendoalg} and Shimura
931\cite{shimura:factors} but is too slow to be really useful in
932practice; however, similar Manin symbols methods must
933exist and be very efficient.%
934\item {\bf Decomposition:} {\em Given the endomorphism ring of an
935abelian variety $A$, compute its decomposition as a product of
936simple abelian varieties.}  This is a standard and difficult
937problem in general, but it might be possible to combine work of
938Allan Steel on a characteristic~$0$ Meataxe'' with special
939features of modular abelian varieties to solve it in practice.  It
940is absolutely {\em essential} to solve this problem in order to
941explicitly enumerate all modular abelian varieties over number
942fields of given level.%
943 \item {\bf Isomorphism testing:} {\em Given
944two modular abelian varieties over a number field $K$, represented
945as explicit quotients of Jacobians $J_1(N)$, decide whether there
946is an isomorphism between them.}  I have solved this problem when
947the two modular abelian varieties are simple.  There are analogues
948of 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
954conjecture 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]
958The torsion subgroup of $J_1(p)(\Q)$ is exactly the group
959generated by the cusps on $X_1(p)$ that lie over $\infty\in 960X_0(p)$.  This is a group of order
961$962\frac{p}{2^{p-3}} \cdot\prod_{\eps\neq 1} B_{2,\eps} 963$
964where the product is over the nontrivial even Dirichlet
965characters~$\varepsilon$ of conductor dividing~$p$, and
966$B_{2,\eps}$ is a generalized Bernoulli number.
967\end{conjecture}
968Mazur'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
970of $J_0(p)(\Q)_{\tor}$ is much easier.  I hope to mimic Mazur's
971method and prove the conjecture above for the prime-to-$2$ part of
972$J_1(p)(\Q)_{\tor}$.
973
974More generally, I would like to investigate the torsion in
975quotients of $J_1(p)$.
976\begin{question}
977If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
978natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective?  If
979so, is the product of the orders of all $A_f(\Q)_{\tor}$ over all
980classes 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
983yes'' answers, and in \cite{emerton:optimal} M.~Emerton
984subsequently proved this conjecture.   It is still not clear if
985one should make this conjecture for $J_1(p)$.
986
987\subsubsection{Congruences between modular forms of prime level}
988Ken Ribet enticed me into studying modular forms as a graduate
989student 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
993associated Hecke algebra of level~$p$.  The question remains
994whether $p=389$ is the only such example, and this seems extremely
995difficult to say anything useful about.   However a related
996question exhibits a shockingly clear pattern, and this related
997question is the question Ribet was really interested in for his
998application to images of Galois representations
999\cite{ribet:torsion}.
1000
1001Let $p$ be a prime and $k$ a positive even integer.  Let $\T$ be
1002the Hecke algebra attached to $S_k(\Gamma_0(p))$ and let $d(k,p)$
1003be the valuation at $p$ of the index of $\T$ in its normalization.
1004The following conjecture is backed up by significant numerical
1005evidence, and was discovered by staring at tables and looking for
1006a pattern.
1007\begin{conjecture}[Stein, F.~Calegari]
1008Suppose $p\geq k-1$.  Then
1009  $$1010d(k,p) = \left\lfloor\frac{p}{12}\right\rfloor\cdot 1011 \binom{m}{2} + a(p,m), 1012$$
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
1026The situation of interest to Ribet is $k=2$, in which case the
1027conjecture simply asserts that $\T\otimes\Z_p$ is normal, i.e.,
1028{\em there or no congruences in characteristic $p$ between
1029non-Galois conjugate newforms in $S_2(\Gamma_0(p))$.}  Calegari
1030has given a conjectural interpretation of some of the congruences
1031that the conjecture asserts must exist, which I intend to study
1032further.
1033
1034\section{Summary}
1035This research proposal depicts an intricate network of ongoing
1036investigations into the arithmetic of modular abelian varieties,
1037which unite a theoretical and computational point of view.  The
1038basic foundations of visibility theory are nearly complete, but
1039solutions to the questions about visibility outlined in  this
1040proposal demand a new level of precision in our understanding of
1041the web of modular abelian varieties. I am determined to advance
1042our understanding in this direction.
1043
1044My work has produced results and tools that are of use to other
1045mathematicians who are exploring the world of modular forms.   By
1046supporting my research, you will assure the sustained development
1047of this technology.
1048
1049
1050
1051
1052
1053Let $p$ be a prime and $k$ a positive even integer.  Let $\T$ be
1054the Hecke algebra attached to the space $S_k(\Gamma_0(p))$ of cusp
1055forms for $\Gamma_0(p)$ and let $d(k,p)$ be the valuation at $p$
1056of the index of $\T$ in its normalization. The following
1057conjecture is backed up by significant numerical evidence, and was
1058discovered by staring at tables computed using the PIs {\sc Magma}
1059code and looking for a pattern.
1060\begin{conjecture}[Stein, F.~Calegari]
1061Suppose $p\geq k-1$.  Then
1062  $$1063d(k,p) = \left\lfloor\frac{p}{12}\right\rfloor\cdot 1064 \binom{m}{2} + a(p,m), 1065$$
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}
1078The conjecture is of interest to Ribet in the case $k=2$, because
1079it is a hypotheses to the main argument of \cite{ribet:torsion}.
1080For a long time it was unclear what to conjecture when $k=2$;
1081finally, investigation into what happens at higher weight
1082suggested the above conjectural formula, which specializes in
1083weight $2$ to the assertion that $\T\otimes \Z_p$ is normal. The
1084PI has no idea how to prove this conjecture when $k=2$, but
1085intends to at least find similar conjectures when $\Gamma_0(p)$ is
1086replaced by $\Gamma_1(p)$ and when~$p$ is replaced by a composite
1087number.
1088
1089
1090
1091\newpage
1092
1093\newcommand{\etalchar}[1]{$^{#1}$}
1094\providecommand{\bysame}{\leavevmode\hbox
1095to3em{\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}
1104A.~Agashe and W.\thinspace{}A. Stein, \emph{Visible {E}vidence for
1105the {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
1111abelian varieties},
1112  J. Number Theory \textbf{97} (2002), no.~1, 171--185.
1113
1114\bibitem[BCDT01]{breuil-conrad-diamond-taylor}
1115C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor, \emph{On the
1116modularity 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}
1121W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra
1122system. {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}
1127B.\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}
1134K.~Buzzard and W.\thinspace{}A. Stein, \emph{A mod five approach
1135to modularity
1136  of icosahedral {G}alois representations}, Pacific J. Math. \textbf{203}
1137  (2002), no.~2, 265--282.
1138
1139\bibitem[BT99]{buzzard-taylor}
1140K.~Buzzard and R.~Taylor, \emph{Companion forms and weight one
1141forms}, Ann. of
1142  Math. (2) \textbf{149} (1999), no.~3, 905--919.
1143
1144\bibitem[CES03]{conrad-edixhoven-stein:j1p}
1145B.~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}
1150J.\thinspace{}E. Cremona and B.~Mazur, \emph{Visualizing elements
1151in the
1152  {S}hafarevich-{T}ate group}, Experiment. Math. \textbf{9} (2000), no.~1,
1153  13--28.
1154
1155\bibitem[Col03]{coleman-stein:padicapprox}
1156R.~Coleman, \emph{Approximation of infinite-slope modular
1157eigenforms by
1158  finite-slope eigenforms}, To appear in the Dwork Proceedings (2003).
1159
1160\bibitem[CS02]{stein:compgroup}
1161B.~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}
1166N.~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}
1172M.~Emerton, \emph{Optimal {Q}uotients of {M}odular {J}acobians},
1173preprint
1174  (2001).
1175
1176\bibitem[FpS{\etalchar{+}}01]{empirical}
1177E.\thinspace{}V. Flynn, F.~\protect{Lepr\'{e}vost},
1178E.\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}
1185K.~Kato, \emph{$p$-adic {H}odge theory and values of zeta
1186functions of modular
1187  forms}, Preprint, 244 pages.
1188
1189\bibitem[KS00]{kohel-stein:ants4}
1190D.\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
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1197S.~Lang, \emph{Number theory. {I}{I}{I}}, Springer-Verlag, Berlin,
11981991,
1199  Diophantine geometry.
1200
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1202S.~Lang and J.~Tate, \emph{Principal homogeneous spaces over
1203abelian
1204  varieties}, Amer. J. Math. \textbf{80} (1958), 659--684.
1205
1206\bibitem[Maz77]{mazur:eisenstein}
1207B.~Mazur, \emph{Modular curves and the \protect{Eisenstein}
1208ideal}, Inst.
1209  Hautes \'Etudes Sci. Publ. Math. (1977), no.~47, 33--186 (1978).
1210
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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}
1218L.~Merel, \emph{Sur la nature non-cyclotomique des points d'ordre
1219fini 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}
1224L.~Merel and W.\thinspace{}A. Stein, \emph{The field generated by
1225the 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}
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1231polarized abelian
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1233
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1237endomorphisms of
1238  abelian varieties}, Math. Ann. \textbf{253} (1980), no.~1, 43--62.
1239
1240
1241\bibitem[Rib99]{ribet:torsion}
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1243representations},
1244  Arithmetic theory of elliptic curves (Cetraro, 1997), Springer, Berlin, 1999,
1245  pp.~145--166.
1246
1247\bibitem[Roh84]{rohrlich:cyclo}
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1249curves and
1250  cyclotomic towers}, Invent. Math. \textbf{75} (1984), no.~3, 409--423.
1251
1252\bibitem[Rub98]{rubin:kato}
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1260  field}, J. Math. Soc. Japan \textbf{25} (1973), no.~3, 523--544.
1261
1262\bibitem[Ste]{stein:nonsquaresha}
1263W.\thinspace{}A. Stein, \emph{Shafarevich-tate groups of nonsquare
1264order},
1265  Proceedings of MCAV 2002, Progress of Mathematics (to appear).
1266
1267\bibitem[Ste00]{stein:phd}
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1269Ph.D. thesis,
1270  University of California, Berkeley (2000).
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1272\bibitem[Ste02]{stein:index}
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1275  index}, J. Reine Angew. Math. \textbf{547} (2002), 139--147.
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1277\bibitem[Ste03a]{mfd}
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1279  http://modular.fas.harvard.edu/Tables} (2003).
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1293
1294\end{thebibliography}
1295
1296
1297\end{document}
1298
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