\documentclass[11pt]{article}1\newcommand{\thisdocument}{Project Description}2\include{macros}34\begin{document}56\section{Background}7The proposed project reflects the interplay of abstract theory8with explicit machine computation, as illustrated by the following9quote of Bryan Birch~\cite{birch:bsd}:10\begin{quote}11I want to describe some computations undertaken by myself and12Swin-nerton-Dyer on EDSAC by which we have calculated the13zeta-functions of certain elliptic curves. As a result of these14computations we have found an analogue for an elliptic curve of15the Tamagawa number of an algebraic group; and conjectures (due to16ourselves, due to Tate, and due to others) have proliferated.17\end{quote}181920The PI is primarily interested in abelian varieties attached to21modular forms via Shimura's construction \cite{shimura:factors},22which we now recall. Let~$f=\sum a_n q^n$ be a weight~$2$ newform23on $\Gamma_1(N)$. Then~$f$ corresponds to a differential on the24modular curve $X_1(N)$, which is a curve whose affine points25over~$\C$ correspond to isomorphism classes of pairs $(E,P)$,26where~$E$ is an elliptic curve and $P\in E$ is a point of27order~$N$. We view the Hecke algebra28\[\T=\Z[T_1,T_2,T_3,\ldots]\]29as a subring of the endomorphism ring of the Jacobian $J_1(N)$30of $X_1(N)$. Let $I_f$ be the kernel of the homomorphism $\T\to31\Z[a_1,a_2,a_3]$ that sends $T_n$ to $a_n$, and attach to~$f$ the32quotient $$A_f=J_1(N)/I_f J_1(N).$$ Then $A_f$ is a simple abelian33variety over~$\Q$ of dimension equal to the degree of the field34$\Q(a_1,a_2,a_3,\ldots)$ generated by the coefficients of~$f$. We35also sometimes consider a similar construction with $J_1(N)$36replaced by the Jacobian $J_0(N)$ of the modular curve $X_0(N)$37that parametrizes isomorphism classes of pairs $(E,C)$, where $C$38is a cyclic subgroup of $E$ of order~$N$.3940\begin{definition}[Modular abelian variety]41A {\em newform abelian variety} is an abelian variety over~$\Q$ of42the form $A_f$. An abelian variety over a number field is a {\em43modular abelian varieties} if it is a quotient of $J_1(N)$ for44some~$N$.45\end{definition}46Over $\Q$, newform abelian varieties are simple and every modular47abelian variety is isogenous to a product of copies of newform48abelian varieties. Newform abelian varieties are typically not49absolutely simple.5051Newform abelian varieties $A_f$ are important. For example,the52celebrated modularity theorem of C.~Breuil, B.~Conrad, F.~Diamond,53R.~Taylor, and A.~Wiles \cite{breuil-conrad-diamond-taylor}54asserts that every elliptic curve over~$\Q$ is isogenous to some55$A_f$. Also, J-P.~Serre conjectures that, up to twist, every56two-dimensional odd irreducible mod~$p$ Galois representation57appears in the torsion points on some $A_f$.5859Much of this research proposal is inspired by the following60special case of the Birch and Swinnerton-Dyer conjecture:61\begin{conjecture}[BSD Conjecture (special case)]\label{conj:bsd}62Let $A$ be a modular abelian variety over~$\Q$.63\begin{enumerate}%64\item $L(A,1)=0$ if and65only if $A(\Q)$ is infinite.%66\item If $L(A,1)\neq 0$, then67\[68\frac{L(A,1)}{\Omega_{A}} =%69\frac{\prod c_p \cdot \#\Sha(A)}%70{\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}},71\]72where the objects and notation in this formula are discussed73below.74\end{enumerate}75\end{conjecture}76Here $L(A,s)$ is the $L$-series attached to $A$, which is entire77because~$A$ is modular, so $L(A,1)$ makes sense. The real volume78$\Omega_{A}$ is the measure of $A(\R)$ with respect to a basis of79differentials for the N\'eron model of $A$. For each prime $p\mid80N$, the integer $c_p=\#\Phi_{A,p}(\F_p)$ is the {\em Tamagawa81number} of~$A$ at~$p$, where $\Phi_{A,p}$ denotes the component82group of the N\'eron model of~$A$ at $p$. The dual of $A$ is83denoted $A^{\vee}$, and in the conjecture $A(\Q)_{\tor}$ and84$A^{\vee}(\Q)_{\tor}$ are the torsion subgroups. The {\em85Shafarevich-Tate group} of $A$ is86\[87\Sha(A) = \Ker\left(\H^1(\Q,A) \to \bigoplus_{p\leq88\infty} \H^1(\Q_p,A)\right),89\]90which is a group that measures the failure of a local-to-global91principle. When $L(A,1)\neq 0$, Kato proved in \cite{kato:secret}92that $\Sha(A)$ and $A(\Q)$ are finite, so $\#\Sha(A)$ makes sense93and one implication of part 1 of the conjecture is known.9495\begin{remark}96The general Birch and Swinnerton-Dyer conjecture (see97\cite{tate:bsd, lang:nt3}) is a conjecture about any abelian98variety $A$ over a global field~$K$. It asserts that the order of99vanishing of $L(A,s)$ at $s=1$ equals the free rank of $A(K)$, and100gives a formula for the leading coefficient of the Taylor101expansion of $L(A,s)$ about $s=1$.102\end{remark}103104The rest of this proposal is divided into two parts. The first is105about computing with modular forms and abelian varieties, and106making the results of these computations available to the107mathematical community.108The second is about visibility of Mordell-Weil and Shafarevich-Tate groups, the109ultimate goal being to obtain relationships between parts 1 and 2110of Conjecture~\ref{conj:bsd}.111112\section{Computing with modular forms}113114The PI proposes to continue developing algorithms and making115available tools for computing with modular forms, modular abelian116varieties, and motives attached to modular forms. This includes117finishing a major new {\sc Magma} \cite{magma} package for118computing directly with modular abelian varieties over number119fields, extending the Modular Forms Database \cite{mfd}, and120searching for algorithms for computing the quantities appearing in121Conjecture~\ref{conj:bsd} and in the Bloch-Kato conjecture for122modular motives.123124125126\subsection{The Modular Forms Database}%127The Modular Forms Database \cite{mfd} is a freely-available128collection of data about objects attached to cuspidal modular129forms. It is analogous to Sloane's tables of integer sequences,130and extends Cremona's tables \cite{cremona:onlinetables} to131dimension bigger than one and weight bigger than two. Cremona's132tables contain more refined data about elliptic curves than133\cite{mfd}, but the PI intends to work with Cremona to make the134modular forms database a superset of \cite{cremona:onlinetables}.135136The database is used world-wide by prominent number theorists,137including Noam Elkies, Matthias Flach, Dorian Goldfeld, Benedict138Gross, Ken Ono, and Don Zagier.139140The PI proposes to greatly expand the database. A major challenge141is that data about modular abelian varieties of large dimension142takes a huge amount of space to store. For example, the database143currently occupies 40GB disk space. He proposes to find and144implement a better method for storing information about modular145abelian varieties so that the database will be more useful. He146has found a method whereby a certain eigenvector is computed by147the database server, which may (or may not!) enable storing148coefficients of modular forms far more efficiently; however, he149has not yet tried to implement it and study its properties.150151The PI proposes to improve the usability of the database. It is152currently implemented using a PostgreSQL database coupled with a153Python web interface. To speed access and improve efficiency, he154is considering rewriting key portions of the database using MySQL155and PHP. He hopes to rewrite key portions of the database in156response to user feedback that he has received. The database157currently runs on a three-year-old 933Mhz Pentium III, which has158unduly limited disk space and no offsite backup, so the PI is159requesting a powerful modern computer with a large hard drive160array and external hard drives for offsite backups.161162\subsubsection{M{\small AGMA} package for modular abelian varieties}\label{sec:magma}%163The PI's software is published as part of the non-commercial {\sc164Magma} computer algebra system. The core of {\sc Magma} is165developed by a group of academics at the University of Sydney, who166are supported mostly by grant money. {\sc Magma} is considered by167many to be the most comprehensive tool for research in number168theory, finite group theory, and cryptography, and is widely169distributed. The PI has already written over 400 pages (26000170lines) of modular forms code and extensive documentation that is171distributed with {\sc Magma}, and intends to ``publish'' future172work in {\sc Magma}.173174As mentioned above, an abelian variety $A$ over a number field~$K$175is {\em modular} if it is a quotient of $J_1(N)$ for some $N$.176Modular abelian varieties were studied intensively by Ken Ribet,177Barry Mazur, and others during recent decades, and studying them178is popular because results about them often yield surprising179insight into number theoretic questions. Computation with modular180abelian varieties is attractive because they are much easier to181describe than arbitrary abelian varieties, and their $L$-functions182are reasonably well understood when~$K$ is an abelian extension183of~$\Q$.184185The PI recently designed and partially implemented a general186system for computing with modular abelian varieties over number187fields. He hopes to develop and refine several crucial components188of the system. For example, three major problems arose, and the189PI intends to resolve them in order to have a completely190satisfactory system for computing with modular abelian varieties.191\begin{enumerate}192\item {\em Given a modular abelian variety $A$, efficiently193compute the endomorphism ring $\End(A)$ as a ring of matrices194acting on $\H_1(A,\Z)$.} The PI has found a modular symbols195solution that draws on work of Ribet \cite{ribet:twistsendoalg}196and Shimura \cite{shimura:factors}, but it is too slow to be197really useful in practice. In \cite{merel:1585}, Merel uses198Herbrand matrices and Manin symbols to give efficient algorithms199for computing with Hecke operators. The PI intends to carry over200Merel's method to201give an efficient algorithm to compute $\End(A)$.%202\item {\em Given $\End(A)\otimes\Q$, compute an isogeny203decomposition of $A$ as a product of simple abelian varieties.}204This is a standard and difficult problem in general, but it might205be possible to combine work of Allan Steel on his206``characteristic~$0$ Meataxe'' with special features of modular207abelian varieties to solve it in practice. It is absolutely {\em208essential} to solve this problem in order to explicitly enumerate209all modular abelian varieties over $\Qbar$ of given level~$N$.210Such an enumeration would be a major step towards the ultimate211possible generalization of Cremona's tables212\cite{cremona:onlinetables} to modular abelian varieties.213Computation of a decomposition is also crucial to other214algorithms, e.g., computing215complements and duals of abelian subvarieties.%216\item {\em Given two modular abelian varieties over a number field217$K$, decide whether there is an isomorphism between them.} When218the endomorphism ring of each abelian variety is known and both219are simple, it is possible to reduce this problem to the solution220of a norm equation, which has been studied extensively in many221cases. This problem is analogous to the problem of testing222isomorphism for modules over a fixed ring, which has been solved223with much effort for many classes of rings. One application is224to proving that specific abelian varieties can not be principally polarized.%225\end{enumerate}226227\subsection{An Example: The Arithmetic of $J_1(p)$}228We finish be describing recent work of the PI on the modular229Jacobian $J_1(p)$, where~$p$ is a prime, that was partly inspired230by computation. The following conjecture generalizes a conjecture231of Ogg, which asserts that $J_0(p)(\Q)_{\tor}$ is cyclic of order232the numerator of $(p-1)/12$, a fact that Mazur proved in233\cite{mazur:eisenstein}.234\begin{conjecture}[Stein]\label{conj:tor}235Let $p$ be a prime. The torsion subgroup of $J_1(p)(\Q)$ is the236group generated by the cusps on $X_1(p)$ that lie over $\infty\in237X_0(p)$.238\end{conjecture}239The PI gives significant numerical evidence for this conjecture in240\cite{conrad-edixhoven-stein:j1p}, and cuspidal subgroups of241$J_1(p)$ are considered in detail in \cite{kubert-lang}, where,242e.g., they compute orders of such groups in terms of Bernoulli243numbers.244245Mazur's proof of Ogg's conjecture for $J_0(p)$ is deep, though the246proof for the odd part of $J_0(p)(\Q)_{\tor}$ is much easier. The247PI intends to explore whether or not it is possible to build on248Mazur's method and prove results towards249Conjecture~\ref{conj:tor}. The PI also intends to develop his250computational methods for computing torsion subgroups in order to251answer, at least conjecturally, the following question.252\begin{question}253If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the254natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective? Is255the product of the orders of all $A_f(\Q)_{\tor}$ over all classes256of newforms $f$ equal to $\# J_1(p)(\Q)_{\tor}$?257\end{question}258The PI conjectured that the analogous questions for $J_0(p)$259should have ``yes'' answers, and in \cite{emerton:optimal}260M.~Emerton proved this conjecture. There he also proved that the261natural map from the component group of $J_0(p)$ to that of $A_f$262is surjective. By \cite{conrad-edixhoven-stein:j1p}, the263component group of $J_1(p)$ is trivial, which suggests the264following question.265\begin{question}\label{ques:comp}266If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the267component group of $A_f$ trivial?268\end{question}269Even assuming the full BSD conjecture (and a conjecture about a270Manin constant), the PI has not yet produced enough data to give a271conjectural answer to this question. He has many examples in272which the conjecture predicts that either $\Sha(A_f)$ is273nontrivial or the component group of $A_f$ is nontrivial. He and274B.~Poonen formulated, and hope to carry out, a strategy to decide275which of these two is nontrivial by using an explicit description276of $\End({A_f/\Qbar})$ to obtain a curve whose Jacobian is $A_f$.277Note that computing $\End(A_f/\Qbar)$ in general is the second278problem in Section~\ref{sec:magma} above.279280\section{Visibility}281The underlying motivation for this part of the proposal is to282prove implications between the two parts of283Conjecture~\ref{conj:bsd}, in examples and eventually in some284generality. That is, we link information about the first part of285the BSD conjecture for an abelian variety~$B$ to information about286the second part of the conjecture for a related abelian287variety~$A$. Visibility provides a conceptual framework in which288to organize our ideas.289290\subsection{Computational problems}\label{sec:compprob}291Barry Mazur introduced visibility in order to unify various292constructions of Shafarevich-Tate groups.293\begin{definition}[Visibility of Shafarevich-Tate Groups]294Suppose that $$\iota:A\hra J$$ is an inclusion of abelian295varieties over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$296with respect to $J$ is297\[298\Vis_J\H^1(\Q,A) := \Ker(\H^1(\Q,A)\to \H^1(\Q,J)).299\]300The {\em visible subgroup} of $\Sha(A)$ in~$J$ is the intersection301of $\Sha(A)$ with $\Vis_J\H^1(\Q,A)$; equivalently,302\[303\Vis_J\Sha(A) := \Ker(\Sha(A)\to \Sha(J)).304\]305\end{definition}306The terminology ``visible'' arises from the fact that if307$x\in\Sha(A)$ is visible in~$J$, then a principal homogenous308space~$X$ corresponding to~$x$ can be realized as a subvariety309of~$J$.310311Before discussing theoretical questions about visibility, we312describe computational evidence for the Birch and Swinnerton-Dyer313conjecture for modular abelian varieties (and motives) that the PI314and A.~Agashe obtained using theorems inspired by the definition315of visibility. In \cite{agashe-stein:visibility}, the PI and316Agashe prove a theorem that makes it possible to use abelian317varieties of positive rank to explicitly construct subgroups of318Shafarevich-Tate groups of other abelian varieties. The main319theorem is that if $A$ and $B$ are abelian subvarieties of an320abelian variety $J$, and $B[p]\subset A$, then, under certain321hypothesis, there is an injection322\[ B(\Q)/p B(\Q) \hra \Vis_J\Sha(A).\]323The paper concludes with the first ever example of an abelian324variety $A_f$ attached to a newform, of large dimension ($20$),325whose Shafarevich-Tate group has order that is provably divisible326by an odd prime ($5$).327328The PI has used the result described above to give evidence for329the BSD conjecture for many $A\subset J_0(N)$, where $A$ is330attached to a newform of level~$N\leq 2333$. The PI proposes to331give similar evidence using visibility in $J_0(NM)$ for small~$M$.332More precisely, \cite{agashe-stein:bsd} describes the computation333of an odd divisor of the BSD conjectural order of $\Sha(A)$ for334over ten thousand $A\subset J_0(N)$ with $L(A,1)\neq 0$ (these are335{\em all} simple $A$ with $N\leq 2333$ and $L(A,1)\neq 0$). For336over a hundred of these, the divisor of the conjectural order of337$\Sha(A)$ is divisible by an odd prime; for a quarter of these the338PI and Agashe prove that if~$n$ is the conjectural divisor of the339order of $\Sha(A)$, then there are at least~$n$ elements of340$\Sha(A)$ that are visible in $J_0(N)$.341342The PI intends to investigate the remaining 75\% of the $A$ with343$n>1$ by considering the image of $A$ in $J_0(NM)$ for small344integers~$M$. Information about which~$M$ to choose can be345extracted from Ribet's level raising theorem (see346\cite{ribet:raising}). As a test, the PI recently tried the first347example with conjectural odd $\Sha(A)$ that is not visible in348$J_0(N)$ (this is an $18$ dimensional abelian variety $A$ of level349$551$ such that $9\mid \#\Sha(A)$). He showed in350\cite{stein:bsdmagma} that there are elements of order~$3$ in351$\Sha(A)$ that are visible in $J_0(551\cdot 2)$. Since the352dimension of $J_0(NM)$ grows very quickly, a huge amount of353computer memory will be required to investigate visibility at354higher level. Fortunately, the PI recently received a grant from355Sun Microsystems for a \$67,000 computer that contains 22GB of356contiguously addressable RAM (the processors are relatively slow357and the hard drive is small, making this computer less suitable as358a platform for the modular forms database, which requires a large359hard drive but not so much RAM).360361Some of these ideas generalize to the context of Grothendieck362motives, which A.~Scholl attached to newforms of weight greater363than two. N.~Dummigan, M.~Watkins, and the PI did work in this364direction in \cite{dummigan-stein-watkins:motives}. There we prove365a theorem that can sometimes be used to deduce the existence of366visible Shafarevich-Tate groups in motives attached to modular367forms, assuming a conjecture of Beilinson about ranks of Chow368groups. However, we give several pages of tables that suggest that369Shafarevich-Tate groups of modular motives of level~$N$ are rarely370visible in the higher-weight motivic analogue of $J_0(N)$, much371more rarely than for weight~$2$. Just as above, the question372remains to decide whether one expects these groups to be visible373in the analogue of $J_0(N M)$ for some integer~$M$. It would be374relatively straightforward for the PI to do computations in this375direction, and he intends to do so.376377Before moving on to theoretical questions about visibility, we378pause to emphasize that the above computational investigations379into the Birch and Swinnerton-Dyer conjecture motivated the PI and380others to develop new algorithms for computing with modular381abelian varieties. For example, in \cite{conrad-stein:compgroup},382B.~Conrad and the PI use Grothendieck's monodromy pairing to give383an algorithm for computing orders of component groups of certain384purely toric abelian varieties. This algorithm makes it practical385to compute component groups of quotients $A_f$ of $J_0(N)$ at386primes~$p$ that exactly divide $N$. Without such an algorithm it387would probably be difficult to get very far in computational388investigations into the Birch and Swinnerton-Dyer conjecture for389abelian varieties; indeed, the only other paper in this direction390is \cite{empirical}, which restricts to the case of Jacobians of391genus~$2$ curves.392393394\subsection{Theoretical problems}395\subsubsection{Visibility at higher level}396Suppose $A_f$ is a quotient of $J_1(N)$ attached to a newform and397let $A=A_f^{\vee}\subset J_1(N)$ be its dual. One expects that398most of $\Sha(A)$ is {\em not} visible in $J_1(N)$. The following399conjecture then arises.400\begin{conjecture}[Stein]\label{conj:allvis} For each $x\in \Sha(A)$, there is an integer401$M$ and a morphism $f:A\to J_1(NM)$, of finite degree and coprime402to the order of~$x$, such that the image of $x$ in $\Sha(f(A))$ is403visible in $J_1(NM)$.404\end{conjecture}405In \cite{agashe-stein:visibility}, the PI proved that if $x\in406\H^1(\Q,A)$, then there is an abelian variety~$B$ and an inclusion407$\iota:A\to B$ such that~$x$ is visible in~$B$; moreover,~$B$ is a408quotient of $J_1(NM)$ for some~$M$. This theorem is the main409reason why the PI makes Conjecture~\ref{conj:allvis}. The PI hopes410to prove Conjecture~\ref{conj:allvis} by understanding the precise411relationship between $A$, $B$, and $J_1(NM)$. First he will412investigate explicitly the example with $N=551$ described in413Section~\ref{sec:compprob} above.414415A more analytical, and possibly deeper, approach to416Conjecture~\ref{conj:allvis} is to assume the rank statement of417the Birch and Swinnerton-Dyer conjecture and relate when elements418of $\Sha(A)$ becoming visible at level $NM$ to when there is a419congruence between $f$ and a newform $g$ of level $NM$ with420$L(g,1)=0$. Such an approach leads one to try to formulate a421refinement of Ribet's level raising theorem that includes a422statement about the behavior of the value at $1$ of the423$L$-function attached to the form at higher level. The PI intends424to do further computations in the hopes of finding a satisfactory425conjectural refinement of Ribet's theorem, which he then hopes to426subsequently prove.427428The PI also proposes to investigate whether there is an~$M$ that429is minimal with respect to some property, such that every element430of $\Sha(A)$ is simultaneously visible in $J_1(NM)$. This is well431worth looking into, since the payoffs could be huge---the432existence of such an~$M$ would imply finiteness of $\Sha(A)$,433since $\Vis_J(\Sha(A))$ is always finite. Finiteness of $\Sha(A)$434is a mysterious open problem when $L(A,1)=0$ and $A$ is not a435quotient of $J_0(N)$ with $\ord_{s=1}L(A,s)=\dim A$.436437\subsubsection{Visibility of Mordell-Weil groups}438The Gross-Zagier theorem asserts that points on elliptic curves of439rank $1$ come from Heegner points, and that points on curves of440rank bigger than one do not. It seems difficult to describe441where points on elliptic curves of rank bigger than~$1$ ``come442from''. The PI introduced the following definition, in hopes of443eventually creating a framework for giving a conjectural444explanation.445446\begin{definition}[Visibility of Mordell-Weil Groups]447Suppose that $\pi : J\to A$ is a surjective morphism of abelian448varieties with connected kernel. The {\em visible quotient of449$A(\Q)$} with respect to~$J$ (and $\pi$) is450\[451\Vis^J(A(\Q)) := \Coker(J(\Q)\to A(\Q)).452\]453\end{definition}454455Visibility of Mordell-Weil groups is closely connected to456visibility of Shafarevich-Tate groups. If $C$ is the kernel457of~$\pi$ and $\delta : A(\Q)\to \H^1(\Q,C)$ is the connecting458homomorphism of Galois cohomology, then $\delta$ induces an459isomorphism460\[461\tilde{\delta}: \Vis^J(A(\Q)) \isom \Vis_J(\H^1(\Q,C)).462\]463Note that this implies $\Vis^J(A(\Q))$ is finite. Let464\[465\Vis^J_{\scriptsize\Sha}(A(\Q)) := \tilde{\delta}^{-1}(\Vis_J(\Sha(C))).466\]467468469Though we have introduced nothing fundamentally new, this470different point of view suggested questions that seemed unnatural471before, which inspired the following theorem and conjecture (the472proof of the theorem relies on \cite{kato:secret,rubin:kato} and473\cite{rohrlich:cyclo}):474\begin{theorem}[Stein]\label{thm:allmwvis}475Let $A$ be an elliptic curve. If $x\in A(\Q)$ has order~$n$ (set476$n=0$ if $x$ has infinite order), then for every divisor $d$ of477$n$, there is surjective morphism $J\to A$, with connected kernel,478such that the image of~$x$ in $\Vis^J(A(\Q))$ has order~$d$.479\end{theorem}480481\begin{conjecture}[Stein]482Suppose~$A$ is a modular abelian variety and $x\in A(\Q)$ has483order~$n$. For every divisor~$d$ of $\,n$ there is a surjective484morphism $J\to A$, with connected kernel, such that the image of485$\,x$ in $\Vis^J(A(\Q))$ lies in $\Vis_{\scriptsize \Sha}^J(A(\Q))$ and486has order~$d$.487\end{conjecture}488489We now describe partial results about this conjecture that the PI490proved in \cite{stein:nonsquaresha}. Suppose $E$ is an elliptic491curve over $\Q$ with conductor $N$, and let $f$ be the newform492attached to~$E$. Fix a prime~$p\nmid 2 N \prod c_p$ such that the493Galois representation $\Gal(\Qbar)\to \Aut(E[p])$ is surjective.494\begin{conjecture}[Stein]\label{conj:nonvanishtwist}495There is a prime~$\ell\nmid N$ and a surjective Dirichlet496character $\chi:(\Z/\ell\Z)^*\to\mu_p$ such that497\[498L(E,\chi,1)\neq 0 \qquad\text{and}\qquad499a_{\ell}(E) \not\con \ell+1 \pmod{p}.500\]501\end{conjecture}502According to Sarnak and Kowalski, this conjecture does not seem503amenable to standard analytic averaging arguments. The PI has504verified this conjecture for the elliptic curve of rank~$1$ and505conductor~$37$ and all $p\leq 25000$. In almost all cases, the506smallest $\ell\nmid N$ such that $a_{\ell}(E) \not \con507\ell+1\pmod{p}$ and $\ell\con 1\pmod{p}$ satisfies the conjecture.508509The PI proved the following theorem in \cite{stein:nonsquaresha}.510\begin{theorem}[Stein]\label{thm:exact}511Let $E$ be an elliptic curve over~$\Q$ and suppose~$p$ and $\chi$512are as in Conjecture~\ref{conj:nonvanishtwist} above. Then there513is an exact sequence $0\to A\to J\to E\to 0$ that induces an exact514sequence515\[5160 \to E(\Q)/ p E(\Q) \to \Sha(A) \to \Sha(J) \to \Sha(E) \to 0.517\]518In particular,519\[520E(\Q)/p E(\Q) \isom \Vis_{\scriptsize\Sha}^J(E(\Q)) \isom \Vis_J(\Sha(A)).521\]522\end{theorem}523%When the hypothesis of the theorem are satisfied, the conclusion524%explains $E(\Q)\otimes\F_p$ in terms of the Shafarevich-Tate group525%of an abelian variety with analytic rank~$0$. It thus links parts526%1 and 2 of Conjecture~\ref{conj:bsd}. The PI proposes to explore527%the significance of this further.528529%There are two problems with this picture. First, the PI does not530%know a proof of Conjecture~\ref{conj:nonvanishtwist}, and is531%having difficulty finding one. Second, even if532%Conjecture~\ref{conj:nonvanishtwist} were known, it is unclear533%what implications the isomorphism534%\[535%E(\Q)/p E(\Q) \isom \Vis_J(\Sha(A))536%\]537%has for Conjecture~\ref{conj:bsd}.538%539540We finish this research proposal by explaining how541Theorem~\ref{thm:exact} may lead to a link between the two parts542of the BSD Conjecture (Conjecture~\ref{conj:bsd}). Suppose $E$ is543an elliptic curve over~$\Q$ and $L(E,1)=0$. Then part 1 of544Conjecture~\ref{conj:bsd} asserts that $E(\Q)$ is infinite. Under545our hypothesis that $L(E,1)=0$, a standard argument shows that546$$\frac{L(A,1)}{\Omega_A} \con 0\pmod{p},$$ where $A$ is as in547Theorem~\ref{thm:exact}. If part 2 of Conjecture~\ref{conj:bsd}548were true, there would be an element $x\in \Sha(A)$ of order~$p$549(the proof of Theorem~\ref{thm:exact} rules out the possibility550that~$p$ divides a Tamagawa number). If, in addition,~$x$ were551visible in $J$, then $E(\Q)$ would be infinite, since $E(\Q)$ has552no elements of order~$p$. Part 2 of Conjecture~\ref{conj:bsd} does553not assert that $x$ is visible in~$J$, so one can only hope that a554close examination of an eventual proof of part 2 of555Conjecture~\ref{conj:bsd} would yield some insight into whether or556not $x$ is visible. Alternatively, one could try to replace the557isomorphism $E(\Q)/p E(\Q) \isom \Vis_J(\Sha(A)) $ by an558isomorphism559\[560\mbox{\rm Sel}^{(p)}(E)\isom \Sha(A)[I]561\]562where $I$ is an appropriate ideal in the ring $\Z[\mu_p]$ of563endomorphism of~$A$. Then an appropriate refinement of part 2 of564Conjecture~\ref{conj:bsd} might imply that $\Sha(A)[I]$ contains565an element of order~$p$, which would imply that either $E(\Q)$ is566infinite or $\Sha(E/\Q)[p]$ is nonzero.567568One can also work orthogonally to the above approach by569investigating similar situations coming from level raising, where570isomorphisms like the ones above may arise. The PI intends to571investigate this cluster of ideas from various directions in hopes572of finding a new perspective on where points on elliptic curves of573rank bigger than one come from it.574575\newpage576\bibliographystyle{amsalpha}577\bibliography{biblio}578\end{document}579580