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Author: William A. Stein
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7\section{Background}
8The proposed project reflects the interplay of abstract theory
9with explicit machine computation, as illustrated by the following
10quote of Bryan Birch~\cite{birch:bsd}:
11\begin{quote}
12I want to describe some computations undertaken by myself and
13Swin-nerton-Dyer on EDSAC by which we have calculated the
14zeta-functions of certain elliptic curves. As a result of these
15computations we have found an analogue for an elliptic curve of
16the Tamagawa number of an algebraic group; and conjectures (due to
17ourselves, due to Tate, and due to others) have proliferated.
18\end{quote}
19
20
21The PI is primarily interested in abelian varieties attached to
22modular forms via Shimura's construction \cite{shimura:factors},
23which we now recall. Let~$f=\sum a_n q^n$ be a weight~$2$ newform
24on $\Gamma_1(N)$. Then~$f$ corresponds to a differential on the
25modular curve $X_1(N)$, which is a curve whose affine points
26over~$\C$ correspond to isomorphism classes of pairs $(E,P)$,
27where~$E$ is an elliptic curve and $P\in E$ is a point of
28order~$N$.  We view the Hecke algebra
29$\T=\Z[T_1,T_2,T_3,\ldots]$
30 as a subring of the endomorphism ring of the Jacobian $J_1(N)$
31of $X_1(N)$. Let $I_f$ be the kernel of the homomorphism $\T\to 32\Z[a_1,a_2,a_3]$ that sends $T_n$ to $a_n$, and attach to~$f$ the
33quotient $$A_f=J_1(N)/I_f J_1(N).$$ Then $A_f$ is a simple abelian
34variety over~$\Q$ of dimension equal to the degree of the field
35$\Q(a_1,a_2,a_3,\ldots)$ generated by the coefficients of~$f$.  We
36also sometimes consider a similar construction with $J_1(N)$
37replaced by the Jacobian $J_0(N)$ of the modular curve $X_0(N)$
38that parametrizes isomorphism classes of pairs $(E,C)$, where $C$
39is a cyclic subgroup of $E$ of order~$N$.
40
41\begin{definition}[Modular abelian variety]
42A {\em newform abelian variety} is an abelian variety over~$\Q$ of
43the form $A_f$.  An abelian variety over a number field is a {\em
44modular abelian varieties} if it is a quotient of $J_1(N)$ for
45some~$N$.
46\end{definition}
47Over $\Q$, newform abelian varieties are simple and every modular
48abelian variety is isogenous to a product of copies of newform
49abelian varieties.  Newform abelian varieties are typically not
50absolutely simple.
51
52Newform abelian varieties $A_f$ are important. For example,the
53celebrated modularity theorem of C.~Breuil, B.~Conrad, F.~Diamond,
55asserts that every elliptic curve over~$\Q$ is isogenous to some
56$A_f$.  Also, J-P.~Serre conjectures that, up to twist, every
57two-dimensional odd irreducible mod~$p$ Galois representation
58appears in the torsion points on some $A_f$.
59
60Much of this research proposal is inspired by the following
61special case of the Birch and Swinnerton-Dyer conjecture:
62\begin{conjecture}[BSD Conjecture (special case)]\label{conj:bsd}
63Let $A$ be a modular abelian variety over~$\Q$.
64\begin{enumerate}%
65\item $L(A,1)=0$ if and
66only if $A(\Q)$ is infinite.%
67\item If $L(A,1)\neq 0$, then
68$69\frac{L(A,1)}{\Omega_{A}} =% 70\frac{\prod c_p \cdot \#\Sha(A)}% 71{\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}}, 72$
73where the objects and notation in this formula are discussed
74below.
75\end{enumerate}
76\end{conjecture}
77Here $L(A,s)$ is the $L$-series attached to $A$, which is entire
78because~$A$ is modular, so $L(A,1)$ makes sense.  The real volume
79$\Omega_{A}$ is the measure of $A(\R)$ with respect to a basis of
80differentials for the N\'eron model of $A$. For each prime $p\mid 81N$, the integer $c_p=\#\Phi_{A,p}(\F_p)$ is the {\em Tamagawa
82number} of~$A$ at~$p$, where $\Phi_{A,p}$ denotes the component
83group of the N\'eron model of~$A$ at $p$.  The dual of $A$ is
84denoted $A^{\vee}$, and in the conjecture $A(\Q)_{\tor}$ and
85$A^{\vee}(\Q)_{\tor}$ are the torsion subgroups. The {\em
86Shafarevich-Tate group} of $A$ is
87$88 \Sha(A) = \Ker\left(\H^1(\Q,A) \to \bigoplus_{p\leq 89 \infty} \H^1(\Q_p,A)\right), 90$
91which is a group that measures the failure of a local-to-global
92principle.  When $L(A,1)\neq 0$, Kato proved in \cite{kato:secret}
93that $\Sha(A)$ and $A(\Q)$ are finite, so $\#\Sha(A)$ makes sense
94and one implication of part 1 of the conjecture is known.
95
96\begin{remark}
97The general Birch and Swinnerton-Dyer conjecture (see
98\cite{tate:bsd, lang:nt3}) is a conjecture about any abelian
99variety $A$ over a global field~$K$. It asserts that the order of
100vanishing of $L(A,s)$ at $s=1$ equals the free rank of $A(K)$, and
101gives a formula for the leading coefficient of the Taylor
102expansion of $L(A,s)$ about $s=1$.
103\end{remark}
104
105The rest of this proposal is divided into two parts.  The first is
106about computing with modular forms and abelian varieties, and
107making the results of these computations available to the
108mathematical community.
109 The second is about visibility of Mordell-Weil and Shafarevich-Tate groups, the
110ultimate goal being to obtain relationships between parts 1 and 2
111of Conjecture~\ref{conj:bsd}.
112
113\section{Computing with modular forms}
114
115The PI proposes to continue developing algorithms and making
116available tools for computing with modular forms, modular abelian
117varieties, and motives attached to modular forms. This includes
118finishing a major new {\sc Magma} \cite{magma} package for
119computing directly with modular abelian varieties over number
120fields, extending the Modular Forms Database \cite{mfd}, and
121searching for algorithms for computing the quantities appearing in
122Conjecture~\ref{conj:bsd} and in the Bloch-Kato conjecture for
123modular motives.
124
125
126
127\subsection{The Modular Forms Database}%
128The Modular Forms Database \cite{mfd} is a freely-available
129collection of data about objects attached to cuspidal modular
130forms. It is analogous to Sloane's tables of integer sequences,
131and extends Cremona's tables \cite{cremona:onlinetables} to
132dimension bigger than one and weight bigger than two. Cremona's
133tables contain more refined data about elliptic curves than
134\cite{mfd}, but the PI intends to work with Cremona to make the
135modular forms database a superset of \cite{cremona:onlinetables}.
136
137The database is used world-wide by prominent number theorists,
138including Noam Elkies, Matthias Flach, Dorian Goldfeld, Benedict
139Gross, Ken Ono, and Don Zagier.
140
141The PI proposes to greatly expand the database.  A major challenge
142is that data about modular abelian varieties of large dimension
143takes a huge amount of space to store.  For example, the database
144currently occupies 40GB disk space.   He proposes to find and
145implement a better method for storing information about modular
146abelian varieties so that the database will be more useful.  He
147has found a method whereby a certain eigenvector is computed by
148the database server, which may (or may not!) enable storing
149coefficients of modular forms far more efficiently; however, he
150has not yet tried to implement it and study its properties.
151
152The PI proposes to improve the usability of the database.  It is
153currently implemented using a PostgreSQL database coupled with a
154Python web interface. To speed access and improve efficiency, he
155is considering rewriting key portions of the database using MySQL
156and PHP.  He hopes to rewrite key portions of the database in
157response to user feedback that he has received.   The database
158currently runs on a three-year-old 933Mhz Pentium III, which has
159unduly limited disk space and no offsite backup, so the PI is
160requesting a powerful modern computer with a large hard drive
161array and external hard drives for offsite backups.
162
163\subsubsection{M{\small AGMA} package for modular abelian varieties}\label{sec:magma}%
164The PI's software is published as part of the non-commercial {\sc
165Magma} computer algebra system. The core of {\sc Magma} is
166developed by a group of academics at the University of Sydney, who
167are supported mostly by grant money. {\sc Magma} is considered by
168many to be the most comprehensive tool for research in number
169theory, finite group theory, and cryptography, and is widely
170distributed.  The PI has already written over 400 pages (26000
171lines) of modular forms code and extensive documentation that is
172distributed with {\sc Magma}, and intends to publish'' future
173work in {\sc Magma}.
174
175As mentioned above, an abelian variety $A$ over a number field~$K$
176is {\em modular} if it is a quotient of $J_1(N)$ for some $N$.
177Modular abelian varieties were studied intensively by Ken Ribet,
178Barry Mazur, and others during recent decades, and studying them
179is popular because results about them often yield surprising
180insight into number theoretic questions.  Computation with modular
181abelian varieties is attractive because they are much easier to
182describe than arbitrary abelian varieties, and their $L$-functions
183are reasonably well understood when~$K$ is an abelian extension
184of~$\Q$.
185
186The PI recently designed and partially implemented a general
187system for computing with modular abelian varieties over number
188fields.  He hopes to develop and refine several crucial components
189of the system.  For example, three major problems arose, and the
190PI intends to resolve them in order to have a completely
191satisfactory system for computing with modular abelian varieties.
192\begin{enumerate}
193\item {\em Given a modular abelian variety $A$, efficiently
194compute the endomorphism ring $\End(A)$ as a ring of matrices
195acting on $\H_1(A,\Z)$.} The PI has found a modular symbols
196solution that draws on work of Ribet \cite{ribet:twistsendoalg}
197and Shimura \cite{shimura:factors}, but it is too slow to be
198really useful in practice.  In \cite{merel:1585}, Merel uses
199Herbrand matrices and Manin symbols to give efficient algorithms
200for computing with Hecke operators. The PI intends to carry over
201Merel's method to
202give an efficient algorithm to compute $\End(A)$.%
203\item {\em Given $\End(A)\otimes\Q$, compute an isogeny
204decomposition of $A$ as a product of simple abelian varieties.}
205This is a standard and difficult problem in general, but it might
206be possible to combine work of Allan Steel on his
207characteristic~$0$ Meataxe'' with special features of modular
208abelian varieties to solve it in practice.  It is absolutely {\em
209essential} to solve this problem in order to explicitly enumerate
210all modular abelian varieties over $\Qbar$ of given level~$N$.
211Such an enumeration would be a major step towards the ultimate
212possible generalization of Cremona's tables
213\cite{cremona:onlinetables} to modular abelian varieties.
214Computation of a decomposition is also crucial to other
215algorithms, e.g., computing
216complements and duals of abelian subvarieties.%
217\item {\em Given two modular abelian varieties over a number field
218$K$, decide whether there is an isomorphism between them.} When
219the endomorphism ring of each abelian variety is known and both
220are simple, it is possible to reduce this problem to the solution
221of a norm equation, which has been studied extensively in many
222cases. This problem is analogous to the problem of testing
223isomorphism for modules over a fixed ring, which has been solved
224with much effort for many classes of rings. One application is
225to proving that specific abelian varieties can not be principally polarized.%
226\end{enumerate}
227
228\subsection{An Example: The Arithmetic of $J_1(p)$}
229We finish be describing recent work of the PI on the modular
230Jacobian $J_1(p)$, where~$p$ is a prime, that was partly inspired
231by computation. The following conjecture generalizes a conjecture
232of Ogg, which asserts that $J_0(p)(\Q)_{\tor}$ is cyclic of order
233the numerator of $(p-1)/12$, a fact that Mazur proved in
234\cite{mazur:eisenstein}.
235\begin{conjecture}[Stein]\label{conj:tor}
236Let $p$ be a prime. The torsion subgroup of $J_1(p)(\Q)$ is the
237group generated by the cusps on $X_1(p)$ that lie over $\infty\in 238X_0(p)$.
239\end{conjecture}
240The PI gives significant numerical evidence for this conjecture in
242$J_1(p)$ are considered in detail in \cite{kubert-lang}, where,
243e.g.,  they compute orders of such groups in terms of Bernoulli
244numbers.
245
246Mazur's proof of Ogg's conjecture for $J_0(p)$ is deep, though the
247proof for the odd part of $J_0(p)(\Q)_{\tor}$ is much easier.  The
248PI intends to explore whether or not it is possible to build on
249Mazur's method and prove results towards
250Conjecture~\ref{conj:tor}.  The PI also intends to develop his
251computational methods for computing torsion subgroups in order to
252answer, at least conjecturally, the following question.
253\begin{question}
254If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
255natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective? Is
256the product of the orders of all $A_f(\Q)_{\tor}$ over all classes
257of newforms $f$ equal to $\# J_1(p)(\Q)_{\tor}$?
258\end{question}
259The PI conjectured that the analogous questions for $J_0(p)$
260should have yes'' answers, and in \cite{emerton:optimal}
261M.~Emerton proved this conjecture.  There he also proved that the
262natural map from the component group of $J_0(p)$ to that of $A_f$
264component group of $J_1(p)$ is trivial, which suggests the
265following question.
266\begin{question}\label{ques:comp}
267If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the
268component group of $A_f$ trivial?
269\end{question}
270Even assuming the full BSD conjecture (and a conjecture about a
271Manin constant), the PI has not yet produced enough data to give a
272conjectural answer to this question.   He has many examples in
273which the conjecture predicts that either $\Sha(A_f)$ is
274nontrivial or the component group of $A_f$ is nontrivial.  He and
275B.~Poonen formulated, and hope to carry out, a strategy to decide
276which of these two is nontrivial by using an explicit description
277of $\End({A_f/\Qbar})$  to obtain a curve whose Jacobian is $A_f$.
278Note that computing $\End(A_f/\Qbar)$ in general is the second
279problem in Section~\ref{sec:magma} above.
280
281\section{Visibility}
282The underlying motivation for this part of the proposal is to
283prove implications between the two parts of
284Conjecture~\ref{conj:bsd}, in examples and eventually in some
286the BSD conjecture for an abelian variety~$B$ to information about
287the second part of the conjecture for a related abelian
288variety~$A$.  Visibility provides a conceptual framework in which
289to organize our ideas.
290
291\subsection{Computational problems}\label{sec:compprob}
292Barry Mazur introduced visibility in order to unify various
293constructions of Shafarevich-Tate groups.
294\begin{definition}[Visibility of Shafarevich-Tate Groups]
295Suppose that $$\iota:A\hra J$$ is an inclusion of abelian
296varieties over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$
297with respect to $J$ is
298$299\Vis_J\H^1(\Q,A) := \Ker(\H^1(\Q,A)\to \H^1(\Q,J)). 300$
301The {\em visible subgroup} of $\Sha(A)$ in~$J$ is the intersection
302of $\Sha(A)$ with $\Vis_J\H^1(\Q,A)$; equivalently,
303$304 \Vis_J\Sha(A) := \Ker(\Sha(A)\to \Sha(J)). 305$
306\end{definition}
307The terminology visible'' arises from the fact that if
308$x\in\Sha(A)$ is visible in~$J$, then a principal homogenous
309space~$X$ corresponding to~$x$ can be realized as a subvariety
310of~$J$.
311
312Before discussing theoretical questions about visibility, we
313describe computational evidence for the Birch and Swinnerton-Dyer
314conjecture for modular abelian varieties (and motives) that the PI
315and A.~Agashe obtained using theorems inspired by the definition
316of visibility. In \cite{agashe-stein:visibility}, the PI and
317Agashe prove a theorem that makes it possible to use abelian
318varieties of positive rank to explicitly construct subgroups of
319Shafarevich-Tate groups of other abelian varieties.   The main
320theorem is that if $A$ and $B$ are abelian subvarieties of an
321abelian variety $J$, and $B[p]\subset A$, then, under certain
322hypothesis, there is an injection
323$B(\Q)/p B(\Q) \hra \Vis_J\Sha(A).$
324The paper concludes with the first ever example of an abelian
325variety $A_f$ attached to a newform, of large dimension ($20$),
326whose Shafarevich-Tate group has order that is provably divisible
327by an odd prime ($5$).
328
329The PI has used the result described above to give evidence for
330the BSD conjecture for many $A\subset J_0(N)$, where $A$ is
331attached to a newform of level~$N\leq 2333$.  The PI proposes to
332give similar evidence using visibility in $J_0(NM)$ for small~$M$.
333More precisely, \cite{agashe-stein:bsd} describes the computation
334of an odd divisor of the BSD conjectural order of $\Sha(A)$ for
335over ten thousand $A\subset J_0(N)$ with $L(A,1)\neq 0$ (these are
336{\em all} simple $A$ with $N\leq 2333$ and $L(A,1)\neq 0$). For
337over a hundred of these, the divisor of the conjectural order of
338$\Sha(A)$ is divisible by an odd prime; for a quarter of these the
339PI and Agashe prove that if~$n$ is the conjectural divisor of the
340order of $\Sha(A)$, then there are at least~$n$ elements of
341$\Sha(A)$ that are visible in $J_0(N)$.
342
343The PI intends to investigate the remaining 75\% of the $A$ with
344$n>1$ by considering the image of $A$ in $J_0(NM)$ for small
345integers~$M$. Information about which~$M$ to choose can be
346extracted from Ribet's level raising theorem (see
347\cite{ribet:raising}).  As a test, the PI recently tried the first
348example with conjectural odd $\Sha(A)$ that is not visible in
349$J_0(N)$ (this is an $18$ dimensional abelian variety $A$ of level
350$551$ such that $9\mid \#\Sha(A)$). He showed in
351\cite{stein:bsdmagma} that there are elements of order~$3$ in
352$\Sha(A)$ that are visible in $J_0(551\cdot 2)$.   Since the
353dimension of $J_0(NM)$ grows very quickly, a huge amount of
354computer memory will be required to investigate visibility at
355higher level.  Fortunately, the PI recently received a grant from
356Sun Microsystems for a \$67,000 computer that contains 22GB of 357contiguously addressable RAM (the processors are relatively slow 358and the hard drive is small, making this computer less suitable as 359a platform for the modular forms database, which requires a large 360hard drive but not so much RAM). 361 362Some of these ideas generalize to the context of Grothendieck 363motives, which A.~Scholl attached to newforms of weight greater 364than two. N.~Dummigan, M.~Watkins, and the PI did work in this 365direction in \cite{dummigan-stein-watkins:motives}. There we prove 366a theorem that can sometimes be used to deduce the existence of 367visible Shafarevich-Tate groups in motives attached to modular 368forms, assuming a conjecture of Beilinson about ranks of Chow 369groups. However, we give several pages of tables that suggest that 370Shafarevich-Tate groups of modular motives of level~$N$are rarely 371visible in the higher-weight motivic analogue of$J_0(N)$, much 372more rarely than for weight~$2$. Just as above, the question 373remains to decide whether one expects these groups to be visible 374in the analogue of$J_0(N M)$for some integer~$M$. It would be 375relatively straightforward for the PI to do computations in this 376direction, and he intends to do so. 377 378Before moving on to theoretical questions about visibility, we 379pause to emphasize that the above computational investigations 380into the Birch and Swinnerton-Dyer conjecture motivated the PI and 381others to develop new algorithms for computing with modular 382abelian varieties. For example, in \cite{conrad-stein:compgroup}, 383B.~Conrad and the PI use Grothendieck's monodromy pairing to give 384an algorithm for computing orders of component groups of certain 385purely toric abelian varieties. This algorithm makes it practical 386to compute component groups of quotients$A_f$of$J_0(N)$at 387primes~$p$that exactly divide$N$. Without such an algorithm it 388would probably be difficult to get very far in computational 389investigations into the Birch and Swinnerton-Dyer conjecture for 390abelian varieties; indeed, the only other paper in this direction 391is \cite{empirical}, which restricts to the case of Jacobians of 392genus~$2$curves. 393 394 395\subsection{Theoretical problems} 396\subsubsection{Visibility at higher level} 397Suppose$A_f$is a quotient of$J_1(N)$attached to a newform and 398let$A=A_f^{\vee}\subset J_1(N)$be its dual. One expects that 399most of$\Sha(A)$is {\em not} visible in$J_1(N)$. The following 400conjecture then arises. 401\begin{conjecture}[Stein]\label{conj:allvis} For each$x\in \Sha(A)$, there is an integer 402$M$and a morphism$f:A\to J_1(NM)$, of finite degree and coprime 403to the order of~$x$, such that the image of$x$in$\Sha(f(A))$is 404visible in$J_1(NM)$. 405\end{conjecture} 406In \cite{agashe-stein:visibility}, the PI proved that if$x\in
407\H^1(\Q,A)$, then there is an abelian variety~$B$and an inclusion 408$\iota:A\to B$such that~$x$is visible in~$B$; moreover,~$B$is a 409quotient of$J_1(NM)$for some~$M$. This theorem is the main 410reason why the PI makes Conjecture~\ref{conj:allvis}. The PI hopes 411to prove Conjecture~\ref{conj:allvis} by understanding the precise 412relationship between$A$,$B$, and$J_1(NM)$. First he will 413investigate explicitly the example with$N=551$described in 414Section~\ref{sec:compprob} above. 415 416A more analytical, and possibly deeper, approach to 417Conjecture~\ref{conj:allvis} is to assume the rank statement of 418the Birch and Swinnerton-Dyer conjecture and relate when elements 419of$\Sha(A)$becoming visible at level$NM$to when there is a 420congruence between$f$and a newform$g$of level$NM$with 421$L(g,1)=0$. Such an approach leads one to try to formulate a 422refinement of Ribet's level raising theorem that includes a 423statement about the behavior of the value at$1$of the 424$L$-function attached to the form at higher level. The PI intends 425to do further computations in the hopes of finding a satisfactory 426conjectural refinement of Ribet's theorem, which he then hopes to 427subsequently prove. 428 429The PI also proposes to investigate whether there is an~$M$that 430is minimal with respect to some property, such that every element 431of$\Sha(A)$is simultaneously visible in$J_1(NM)$. This is well 432worth looking into, since the payoffs could be huge---the 433existence of such an~$M$would imply finiteness of$\Sha(A)$, 434since$\Vis_J(\Sha(A))$is always finite. Finiteness of$\Sha(A)$435is a mysterious open problem when$L(A,1)=0$and$A$is not a 436quotient of$J_0(N)$with$\ord_{s=1}L(A,s)=\dim A$. 437 438\subsubsection{Visibility of Mordell-Weil groups} 439The Gross-Zagier theorem asserts that points on elliptic curves of 440rank$1$come from Heegner points, and that points on curves of 441rank bigger than one do not. It seems difficult to describe 442where points on elliptic curves of rank bigger than~$1$come 443from''. The PI introduced the following definition, in hopes of 444eventually creating a framework for giving a conjectural 445explanation. 446 447\begin{definition}[Visibility of Mordell-Weil Groups] 448Suppose that$\pi : J\to A$is a surjective morphism of abelian 449varieties with connected kernel. The {\em visible quotient of 450$A(\Q)$} with respect to~$J$(and$\pi$) is 451$452 \Vis^J(A(\Q)) := \Coker(J(\Q)\to A(\Q)). 453$ 454\end{definition} 455 456Visibility of Mordell-Weil groups is closely connected to 457visibility of Shafarevich-Tate groups. If$C$is the kernel 458of~$\pi$and$\delta : A(\Q)\to \H^1(\Q,C)$is the connecting 459homomorphism of Galois cohomology, then$\delta$induces an 460isomorphism 461$462 \tilde{\delta}: \Vis^J(A(\Q)) \isom \Vis_J(\H^1(\Q,C)). 463$ 464Note that this implies$\Vis^J(A(\Q))$is finite. Let 465$466 \Vis^J_{\scriptsize\Sha}(A(\Q)) := \tilde{\delta}^{-1}(\Vis_J(\Sha(C))). 467$ 468 469 470Though we have introduced nothing fundamentally new, this 471different point of view suggested questions that seemed unnatural 472before, which inspired the following theorem and conjecture (the 473proof of the theorem relies on \cite{kato:secret,rubin:kato} and 474\cite{rohrlich:cyclo}): 475\begin{theorem}[Stein]\label{thm:allmwvis} 476Let$A$be an elliptic curve. If$x\in A(\Q)$has order~$n$(set 477$n=0$if$x$has infinite order), then for every divisor$d$of 478$n$, there is surjective morphism$J\to A$, with connected kernel, 479such that the image of~$x$in$\Vis^J(A(\Q))$has order~$d$. 480\end{theorem} 481 482\begin{conjecture}[Stein] 483Suppose~$A$is a modular abelian variety and$x\in A(\Q)$has 484order~$n$. For every divisor~$d$of$\,n$there is a surjective 485morphism$J\to A$, with connected kernel, such that the image of 486$\,x$in$\Vis^J(A(\Q))$lies in$\Vis_{\scriptsize \Sha}^J(A(\Q))$and 487has order~$d$. 488\end{conjecture} 489 490We now describe partial results about this conjecture that the PI 491proved in \cite{stein:nonsquaresha}. Suppose$E$is an elliptic 492curve over$\Q$with conductor$N$, and let$f$be the newform 493attached to~$E$. Fix a prime~$p\nmid 2 N \prod c_p$such that the 494Galois representation$\Gal(\Qbar)\to \Aut(E[p])$is surjective. 495\begin{conjecture}[Stein]\label{conj:nonvanishtwist} 496There is a prime~$\ell\nmid N$and a surjective Dirichlet 497character$\chi:(\Z/\ell\Z)^*\to\mu_p$such that 498$499 L(E,\chi,1)\neq 0 \qquad\text{and}\qquad 500 a_{\ell}(E) \not\con \ell+1 \pmod{p}. 501$ 502\end{conjecture} 503According to Sarnak and Kowalski, this conjecture does not seem 504amenable to standard analytic averaging arguments. The PI has 505verified this conjecture for the elliptic curve of rank~$1$and 506conductor~$37$and all$p\leq 25000$. In almost all cases, the 507smallest$\ell\nmid N$such that$a_{\ell}(E) \not \con
508\ell+1\pmod{p}$and$\ell\con 1\pmod{p}$satisfies the conjecture. 509 510The PI proved the following theorem in \cite{stein:nonsquaresha}. 511\begin{theorem}[Stein]\label{thm:exact} 512Let$E$be an elliptic curve over~$\Q$and suppose~$p$and$\chi$513are as in Conjecture~\ref{conj:nonvanishtwist} above. Then there 514is an exact sequence$0\to A\to J\to E\to 0$that induces an exact 515sequence 516$5170 \to E(\Q)/ p E(\Q) \to \Sha(A) \to \Sha(J) \to \Sha(E) \to 0. 518$ 519In particular, 520$521 E(\Q)/p E(\Q) \isom \Vis_{\scriptsize\Sha}^J(E(\Q)) \isom \Vis_J(\Sha(A)). 522$ 523\end{theorem} 524%When the hypothesis of the theorem are satisfied, the conclusion 525%explains$E(\Q)\otimes\F_p$in terms of the Shafarevich-Tate group 526%of an abelian variety with analytic rank~$0$. It thus links parts 527%1 and 2 of Conjecture~\ref{conj:bsd}. The PI proposes to explore 528%the significance of this further. 529 530%There are two problems with this picture. First, the PI does not 531%know a proof of Conjecture~\ref{conj:nonvanishtwist}, and is 532%having difficulty finding one. Second, even if 533%Conjecture~\ref{conj:nonvanishtwist} were known, it is unclear 534%what implications the isomorphism 535%$536%E(\Q)/p E(\Q) \isom \Vis_J(\Sha(A)) 537%$ 538%has for Conjecture~\ref{conj:bsd}. 539% 540 541We finish this research proposal by explaining how 542Theorem~\ref{thm:exact} may lead to a link between the two parts 543of the BSD Conjecture (Conjecture~\ref{conj:bsd}). Suppose$E$is 544an elliptic curve over~$\Q$and$L(E,1)=0$. Then part 1 of 545Conjecture~\ref{conj:bsd} asserts that$E(\Q)$is infinite. Under 546our hypothesis that$L(E,1)=0$, a standard argument shows that 547$$\frac{L(A,1)}{\Omega_A} \con 0\pmod{p},$$ where$A$is as in 548Theorem~\ref{thm:exact}. If part 2 of Conjecture~\ref{conj:bsd} 549were true, there would be an element$x\in \Sha(A)$of order~$p$550(the proof of Theorem~\ref{thm:exact} rules out the possibility 551that~$p$divides a Tamagawa number). If, in addition,~$x$were 552visible in$J$, then$E(\Q)$would be infinite, since$E(\Q)$has 553no elements of order~$p$. Part 2 of Conjecture~\ref{conj:bsd} does 554not assert that$x$is visible in~$J$, so one can only hope that a 555close examination of an eventual proof of part 2 of 556Conjecture~\ref{conj:bsd} would yield some insight into whether or 557not$x$is visible. Alternatively, one could try to replace the 558isomorphism$E(\Q)/p E(\Q) \isom \Vis_J(\Sha(A)) $by an 559isomorphism 560$561\mbox{\rm Sel}^{(p)}(E)\isom \Sha(A)[I] 562$ 563where$I$is an appropriate ideal in the ring$\Z[\mu_p]$of 564endomorphism of~$A$. Then an appropriate refinement of part 2 of 565Conjecture~\ref{conj:bsd} might imply that$\Sha(A)[I]$contains 566an element of order~$p$, which would imply that either$E(\Q)$is 567infinite or$\Sha(E/\Q)[p]\$ is nonzero.
568
569One can also work orthogonally to the above approach by
570investigating similar situations coming from level raising, where
571isomorphisms like the ones above may arise.  The PI intends to
572investigate this cluster of ideas from various directions in hopes
573of finding a new perspective on where points on elliptic curves of
574rank bigger than one come from it.
575
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