\documentclass[11pt]{article}1\newcommand{\thisdocument}{Project Description}2\include{macros}3%\textheight=0.8\textheight45\begin{document}67\section{Introduction}8My research reflects the essential interplay of abstract theory9with explicit machine computation, which is illustrated by the10following quote of Bryan Birch~\cite{birch:bsd} about computations11that led to a central conjecture in number theory:12\begin{quote}13I want to describe some computations undertaken by myself and14Swinnerton-Dyer on EDSAC by which we have calculated the15zeta-functions of certain elliptic curves. As a result of these16computations we have found an analogue for an elliptic curve of17the Tamagawa number of an algebraic group; and conjectures (due to18ourselves, due to Tate, and due to others) have proliferated.19\end{quote}202122I am primarily interested in abelian varieties attached to modular23forms via Shimura's construction \cite{shimura:factors}.24Let~$f=\sum a_n q^n$ be a weight~$2$ newform on $\Gamma_1(N)$. We25may view~$f$ as a differential on the modular curve $X_1(N)$,26which is a curve whose affine points over~$\C$ correspond to27isomorphism classes of pairs $(E,P)$, where~$E$ is an elliptic28curve and $P\in E$ is a point of order~$N$. We view the Hecke29algebra30\[\T=\Z[T_1,T_2,T_3,\ldots]\]31as a subring of the endomorphism ring of the Jacobian $J_1(N)$32of $X_1(N)$. Let $I_f$ be the annihilator of~$f$ in $\T$, and33attach to~$f$ the quotient $$A_f=J_1(N)/I_f J_1(N).$$ Then $A_f$34is an abelian variety over~$\Q$ of dimension equal to the degree35of the field $\Q(a_1,a_2,a_3,\ldots)$ generated by the36coefficients of~$f$.3738The abelian varieties $A_f$ attached to newforms are important.39For example,the celebrated modularity theorem of C.~Breuil,40B.~Conrad, F.~Diamond, Taylor, and Wiles41\cite{breuil-conrad-diamond-taylor} asserts that every elliptic42curve over~$\Q$ is isogenous to some $A_f$. Also, Serre43conjectures that up to twist every two-dimensional odd irreducible44Galois representation appears in the torsion points on some $A_f$.4546My investigations into modular abelian varieties are inspired by47the following special case of the Birch and Swinnerton-Dyer48conjecture (see \cite{tate:bsd, lang:nt3}):49\begin{conjecture}[BSD Conjecture]50\[51\frac{L(A_f,1)}{\Omega_{A_f}} =%52\frac{\prod c_p \cdot \#\Sha(A_f/\Q)}%53{\#A_f(\Q)\cdot \#A_f^{\vee}(\Q)}.54\]55\end{conjecture}56Here $L(A_f,s)$ is the canonical $L$-series attached to $A_f$, the57real volume $\Omega_{A_f}$ is the measure of $A_f(\R)$ with58respect to a basis of differentials for the N\'eron model of59$A_f$, the $c_p$ are the Tamagawa numbers of $A_f$, the dual of60$A_f$ is denoted $A_f^{\vee}$, and61\[62\Sha(A_f/\Q) = \ker\left(\H^1(\Q,A_f) \to \bigoplus_{p\leq63\infty} \H^1(\Q_p,A_f)\right)64\] is the65Shafarevich-Tate group of $A_f$. When $A_f(\Q)$ is infinite, the66right hand side should be interpreted as~$0$, so, in particular,67the conjecture asserts that $L(A_f,1)=0$ if and only if $A_f(\Q)$68is infinite. Birch and Swinnerton-Dyer also conjectured that the69order of vanishing of $L(A_f,s)$ at $s=1$ equals the rank of70$A_f(\Q)$.717273\section{Computing with modular abelian varieties}7475The PI proposes to continue developing algorithms and making76available tools for computing with modular forms, modular abelian77varieties, and motives attached to modular forms. This includes78finishing a major new {\sc Magma} \cite{magma} package for79computing directly with modular abelian varieties over number80fields, extending the Modular Forms Database \cite{mfd}, and81searching for algorithms for computing the quantities appearing in82the Birch and Swinnerton-Dyer for modular abelian varieties and83the Bloch-Kato conjecture for modular motives.84858687\subsection{The Modular Forms Database}%88The Modular Forms Database \cite{mfd} is a freely-available89collection of data about objects attached to cuspidal modular90forms, that is a much used resource for number theorists. It is91analogous to Sloane's tables of integer sequences, and extends92Cremona's tables to dimension bigger than one and weight bigger93than two (\cite{cremona-tables} contain more precise data about94elliptic curves than \cite{mfd}).9596The PI proposes to greatly expand the database. The major97challenge is that data about modular abelian varieties of large98dimension takes a huge amount of space to store. For example, the99database currently occupies 40GB disk space. He proposes to find100better method for storing information about modular abelian so101that the database can grow larger, and to investigate methods used102by astronomers or the human genome project to see how they cope103with a torrent of data while making it available to their104colleagues.105106The PI implemented the current database using PostgreSQL coupled107with a Python web interface. To speed access and improve108efficiency, the PI is considering rewriting key portions of the109database using MySQL and PHP. Also, the database currently runs110on a three-year-old 933Mhz Pentium III, so the PI is requesting111more modern hardware.112113\subsubsection{M{\small AGMA} package for modular abelian varieties}%114{\sc Magma} is a nonprofit computer algebra system developed115primarily at the University of Sydney, which is supported mostly116by grant money from organizations such as the US National Security117Agency. {\sc Magma} is considered by many to be the most118comprehensive tool for research in number theory, finite group119theory, and cryptography, and it is widely distributed. The PI120has already written over 400 pages (26000 lines) of modular forms121code and extensive documentation that is distributed with {\sc122Magma}, and intends to ``publish'' future work in {\sc Magma}.123124An abelian variety $A$ over a number field is {\em modular} if it125is a quotient of $J_1(N)$ for some $N$. Modular abelian varieties126were studied intensively by Ken Ribet, Barry Mazur, and others127during recent decades, and studying them is popular because128results about them often yield surprising insight into number129theoretic questions. Computation with modular abelian varieties130is popular because most of them are easily described by giving a131level and the first few coefficients of a modular form, and the132$L$-functions of modular abelian varieties are particularly well133understood. This is in sharp contrast to the case of general134abelian varieties which, in general, can only be described by135unwieldy systems of polynomial equations, and whose $L$-functions136are very mysterious.137138The PI recently designed and partially implemented a general139package for computing with modular abelian varieties over number140fields. Several crucial algorithms still need to be developed or141refined. When available, this package will likely be useful for142people working with modular abelian varieties. The following143major problems arose in work on this package, and they must be144resolved in order to have a completely satisfactory system for145computing with modular abelian varieties:146\begin{itemize}147\item {\em Given a modular abelian variety $A$, efficiently148compute the endomorphism ring $\End(A)$ as a ring of matrices149acting $\H_1(A,\Z)$.} The PI has found a modular symbols solution150that draws on work of Ribet \cite{ribet:twistsendoalg} and Shimura151\cite{shimura:factors}, but it is too slow to be really useful in152practice. In \cite{merel:1585}, Merel uses Herbrand matrices and153Manin symbols to give efficient algorithms for computing with154Hecke operators. The PI intends to carry over Merel's method to155give an efficient algorithm to compute $\End(A)$.%156\item {\em Given $\End(A)\otimes\Q$, compute an isogeny157decomposition of $A$ as a product of simple abelian varieties.}158This is a standard and difficult problem in general, but it might159be possible to combine work of Allan Steel on his160``characteristic~$0$ Meataxe'' with special features of modular161abelian varieties to solve it in practice. It is absolutely {\em162essential} to solve this problem in order to explicitly enumerate163all modular abelian varieties over number fields of given level.164Such an enumeration would be a major step towards the ultimate165possible generalization of Cremona's tables \cite{cremona:tables}166to modular abelian varieties. %167\item {\em Given two modular abelian varieties over a number field168$K$, decide whether there is an isomorphism between them.} When169the endomorphism ring of each abelian variety is known and both170are simple, it is possible to reduce this problem to mostly171well-studied norm equations. This problem is analogous to the172problem of testing isomorphism for modules over a fixed ring,173which has been solved with much effort for many classes of rings.174One application of isomorphism testing is that it could be used to175prove that an abelian variety is not principally polarized.%176\end{itemize}177178We finish be describing recent work with $J_1(p)$ that was179partially inspired by PI's modular forms software. This180conjecture generalizes a conjecture of Ogg, which asserted that181$J_0(p)(\Q)_{\tor}$ is cyclic of order the numerator of182$(p-1)/12$, a fact that Mazur proved in \cite{mazur:eisenstein}.183\begin{conjecture}[Stein]\label{conj:tor}184Let $p$ be a prime. The torsion subgroup of $J_1(p)(\Q)$ is the185group generated by the cusps on $X_1(p)$ that lie over $\infty\in186X_0(p)$.187\end{conjecture}188Significant numerical evidence for this conjecture is given in189\cite{j1p}, and cuspidal subgroups of $J_1(p)$ are considered in190detail in \cite{kubert-lang}, where, e.g., they compute orders of191such groups in terms of Bernoulli numbers.192193Mazur's proof of Ogg's conjecture for $J_0(p)$ is deep, though the194proof for the odd part of $J_0(p)(\Q)_{\tor}$ is much easier. The195PI intends explore whether or not it is possible to build on196Mazur's method and prove results towards197Conjecture~\ref{conj:tor}. The PI also intends to develop his198computational methods for computing torsion subgroups in order to199answer, at least conjecturally, the following question.200\begin{question}201If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the202natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective? If203so, is the product of the orders of all $A_f(\Q)_{\tor}$ over all204classes of newforms $f$ equal to $\# J_1(p)(\Q)_{\tor}$?205\end{question}206The PI conjectured that the analogous questions for $J_0(p)$207should have ``yes'' answers, and in \cite{emerton:optimal}208M.~Emerton proved this conjecture.209210In \cite{j1p}, it is proved that $J_1(p)$ has trivial component211group (component groups are closely related to the Tamagawa212numbers $c_p$, which appear in the Birch and Swinnerton-Dyer213conjecture).214\begin{question}215If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the216component group of $A_f$ trivial?217\end{question}218Even assuming the full Birch and Swinnerton-Dyer conjecture,219extensive computations by the PI have not produced a conjectural220answer to this question. However, he and Bjorn Poonen formulated221a strategy to answer this question in some interesting cases by222using an explicit decomposition of $\End({A_f}_{\Qbar})$ to obtain223a curve whose Jacobian is $A_f$.224225\section{Visibility of Shafarevich-Tate and Mordell-Weil Groups}226\subsection{Computational problems}227Before describing work the PI proposes to do on228visibility and its uses in obtaining evidence for the Birch and229Swinnerton-Dyer conjecture, we recall the precise definition of230visibility over~$\Q$ (the definition makes sense over any global231field, but we restrict to $\Q$ below for simplicity).232233\begin{definition}[Visibility of Shafarevich-Tate Groups]234Suppose that $\iota:A\hra J$ is an inclusion of abelian varieties235over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$ with respect236to $\iota$ is the kernel of the induced map $\H^1(\Q,A)\to237\H^1(\Q,J)$. The {\em visible subgroup} of $\Sha(A/\Q)$ is the238intersection of $\Sha(A/\Q)$ with the visible subgroup of239$\H^1(\Q,A)$; equivalently, it is the kernel of the induced map240$\Sha(A/\Q)\to \Sha(J/\Q)$.241\end{definition}242243Before discussing theoretical questions about visibility, we244describe computational evidence for the Birch and Swinnerton-Dyer245conjecture for modular abelian varieties (and motives) that we246obtained using a theorem inspired by the definition of visibility.247In \cite{agashe-stein:visibility}, the PI and Agashe prove a248theorem that makes it possible to use abelian varieties of large249rank to explicitly construct subgroups of Shafarevich-Tate groups250of abelian varieties. The main theorem is that if $A$ and $B$251are abelian subvarieties of an abelian variety $J$, and252$B[p]\subset A$, then, under certain hypothesis, there is an253injection from $B(\Q)/p B(\Q)$ into the visible subgroup of254$\Sha(A/\Q)$. The paper concludes with the first ever example of255an abelian variety $A_f$ attached to a newform, of large dimension256($20$), whose Shafarevich-Tate group is provably nontrivial.257258In \cite{agashe-stein:bsd}, the PI and Agashe describe algorithms259that we found and I implemented in {\sc Magma} for computing with260modular abelian varieties. We then compute an odd divisor and261even multiple of the BSD conjectural order of $\Sha(A/\Q)$ for the262over ten thousand quotients $A=A_f$ of $J_0(N)$ with $L(A,1)\neq2630$. For over a hundred of these, our lower bound on the264conjectural order of $\Sha(A/\Q)$ is is divisible by an odd prime;265for a quarter of these we prove, using the main theorem of266\cite{agashe-stein:visibility}, that if $n$ is the conjectural267divisor of $\#\Sha(A/\Q)$, then there are at least~$n$ elements of268$\Sha(A/\Q)$ that are visible in $J_0(N)$. \emph{The PI intends269to investigate the remaining 75\% of the examples with $n>1$ by270considering the image of $A$ in $J_0(NM)$ for small integer $M$.}271Information about which primes $\ell$ to choose can be extracted272from Ribet's level raising theorem (see \cite{ribet:raising}). As273a test, the PI tried the first example not already done, which is274an $18$ dimension abelian variety of level $551$, whose275Shafarevich-Tate group conjecturally contains elements of order276$3$; these are not visible in $J_0(551)$, but he shows in277\cite{stein:bsdmagma} that they are visible in $J_(551\cdot 2)$.278279One objective of the PI's past NSF supported research was to280generalize visibility to the context of modular motives.281Fortunately, Neil Dummigan, Mark Watkins, and the PI did282significant work in this direction in283\cite{dummigan-stein-watkins:motives}. There we prove a theorem284that can sometimes be used to deduce the existence of visible285Shafarevich-Tate groups in motives attached to modular forms,286assuming a conjecture of Beilinson about ranks of Chow groups.287However, we give several pages of tables that suggest that288Shafarevich-Tate groups of modular motives of level~$N$ are very289rarely visible in the higher weight motivic analogue of $J_0(N)$,290much more rarely than for weight~$2$. Just as above, the question291remains to decide whether one expects these groups to be visible292in $J_0(N M)$ for some integer~$M$. It would be relatively293straightforward for the PI to do computations in this direction,294but none have been done yet. The PI intends to do such295computations.296297Before moving on to theoretical questions about visibility, we298pause to emphasize that the above computational investigations299into the Birch and Swinnerton-Dyer conjecture motivated the PI and300others to develop new algorithms for computing with modular301abelian varieties. For example, in \cite{stein:compgroup}, Conrad302and the PI use Grothendieck's monodromy pairing to give a complete303proof of correctness of an algorithm the PI found for computing304the order of the component group of certain purely toric abelian305varieties. This algorithm makes it practical to compute component306groups of quotients of $A_f$ of $J_0(N)$ at primes~$p$ that307exactly divide $N$; without such an algorithm it would probably be308difficult to get anywhere in computational investigations into the309Birch and Swinnerton-Dyer conjecture for abelian varieties;310indeed, the only other paper in this direction is \cite{evidence},311which restricts to the case of dimension~$2$ Jacobians.312313314\subsection{Theoretical problems}315Suppose $A_f$ is a quotient of $J_1(N)$ attached to a newform and316let $A=A_f^{\vee}\subset J_1(N)$ be its dual. One expects that317most of $\Sha(A)$ is {\em not} visible in $J_1(N)$. The following318conjecture then arises.319\begin{conjecture}\label{conj:allvis}320For each $x\in \Sha(A/\Q)$, there is an integer $M$ and a morphism321$f:A\to J_1(NM)$ of finite degree coprime to the order of~$x$ such322that the image of~$x$ in $\Sha(f(A)/\Q)$ is visible in $J_1(NM)$.323\end{conjecture}324In \cite{agashe-stein:visibility}, the PI proved that if $x\in325\H^1(\Q,A)$ then there is an abelian variety $B$ and an inclusion326$\iota:A\to B$ such that $x$ is visible in $B$; moreover, $B$ is a327quotient of $J_1(NM)$ for some $M$. The PI hopes to prove328Conjecture~\ref{conj:allvis} by understanding much more precisely329how $A$, $B$, and $J_1(NM)$ are related.330331A more analytical approach to Conjecture~\ref{conj:allvis} is to332assume the rank statement of the Birch and Swinnerton-Dyer333conjecture and relate when elements of $\Sha(A/\Q)$ becoming334visible at level $NM$ to when there is a congruence between $f$335and a newform $g$ of level $NM$ with $L(g,1)=0$. Such an approach336leads one to wish to formulate a refinement of Ribet's level337raising theorem that includes a statement about the behavior of338the value at $1$ of the $L$-function attached to the form at339higher level. The PI intends to do further computations in order340to give a good conjectural refinement of Ribet's theorem.341342343The Gross-Zagier theorem asserts that points on elliptic curves of344rank $1$ come from Heegner points, and that points on curves of345rank bigger than one do not. Over fifteen years later, it still346seems mysterious to give an interpretation of points on elliptic347curves of rank higher than~$1$. The PI introduced the following348definition, in hopes of eventually creating a framework for giving349a conjectural explanation.350351\begin{definition}[Visibility of Mordell-Weil Groups]352Suppose that $\pi : J\to A$ is a surjective morphism of abelian353varieties with connected kernel~$C$. Let $\delta : A(\Q)\to354\H^1(\Q,C)$ be the connecting homomorphism of Galois cohomology.355An element $x\in A(\Q)$ is \emph{$n$-visible in $\H^1$} (with356respect to $\pi$) if $\delta(x)$ has order divisible by~$n$, and357$x$ is \emph{$n$-visible in $\Sha$} if moreover $\delta(x)\in358\Sha(C/\Q)$.359\end{definition}360361The following theorem is not difficult to prove by combining362Kato's powerful results towards the Birch and Swinnerton-Dyer363conjecture (see \cite{kato:secret,rubin:kato}) with a nonvanishing364theorem of Rohrlich \cite{rohrlich:cyclo}.365\begin{theorem}[Stein]\label{thm:allmwvis}366If~$A$ is a modular abelian variety and $x\in A(\Q)$ has infinite367order, then for every integer~$n$ there is a covering $J\to A$368with connected kernel such that $x$ is $n$-visible in $\H^1$.369\end{theorem}370%The key idea of the proof is that if $p$ is any prime and371%$\Q_\infty$ is the cyclotomic $\Z_p$-extension of $\Q$, then by a372%nonvanishing theorem of Rohrlich and Kato's theorem , the group373%$A(\Q_\infty)$ is finitely generated. From this we deduce that374%there is an (abelian) extension $K$ of $\Q$ such that $n$ divides375%the order of the image of $x$ in $A(\Q)/\Tr_{K/\Q}(A(K))$. Trace376%defines a morphism from the restriction of scalars377%$\pi:J=\Res_{K/\Q}(A_K)$ to $A$ with connected kernel. Then $x$378%is $n$-visible with respect to $\pi$.379380\begin{conjecture}[Stein]381If~$A$ be a modular abelian variety and $x\in A(\Q)$ has infinite382order, then for every integer~$n$, there is an abelian variety $B$383and a surjective morphism $\pi:B\to A$ with connected kernel such384that $x$ is $n$-visible in $\Sha$ with respect to~$\pi$.385\end{conjecture}386387The PI proved partial results towards this conjecture in388\cite{stein:nonsquaresha}. Suppose $E$ is an elliptic curve over389$\Q$ with conductor $N$, and let $f$ be the newform attached to390$E$. Fix a prime~$p\nmid N \prod c_p$ and such that the mod~$p$391Galois representation attached to $E$ is surjective. Suppose392$\chi:(\Z/\ell\Z)\to\mu_p$ is a Dirichlet character with393$\ell\nmid N$ such that394\[395L(E,\chi,1)\neq 0 \qquad\text{and}\qquad396a_{\ell}(E) \not\con \ell+1 \pmod{p},397\]398and let $K$ be the corresponding abelian extension. In399\cite{stein:nonsquaresha}, the PI uses Kato's theorem and400restriction of scalars to construct abelian varieties $A$ and $J$401and an exact sequence402\[4030 \to E(\Q)/ p E(\Q) \to \Sha(A/\Q) \to \Sha(J/\Q) \to \Sha(E/\Q)404\to 0.405\]406Here $A$ is an abelian variety of rank $0$ with $L(A,1)\neq 0$.407Thus all of $E(\Q)$ is $p$-visible in $\Sha$ and one can interpret408$E(\Q)/ p E(\Q)$ as a visible subgroup of a rank~$0$ abelian409variety~$A$.410411There are two problems with this picture. First, the PI does not412know a proof that such a $\chi$ exists (he has verified existence413of $\chi$ in thousands of examples). Second, even if the existence414of $\chi$ were known it seems unlikely that visibility of415Mordell-Weil groups in an abstract restriction of scalars abelian416variety is likely to yield a satisfactory interpretation of417$E(\Q)$ when $E$ has large rank. To answer these problems, the PI418intends to study the system of all possible abelian varieties $B$419in which $E(\Q)$ induces visible elements of Shafarevich-Tate420groups, and moreover to try to keep track of how these abelian421varieties $B$ are related to $J_1(NM)$ for various $M$. This422will involve a combination of computer computation of examples423followed hopefully by the development of a concise notation for424keeping track of all relevant data, and eventually perhaps new425ideas that clarify our understanding of elliptic curves of rank426bigger than one.427428\bibliographystyle{amsalpha}429\bibliography{biblio}430\end{document}431432433434%%%%%%%%%%%435436437438\begin{conjecture}[Stein]\label{conj:nonvanishchi}439Assume~$p$ is an odd prime and the mod~$p$ Galois representation440attached to $E$ is surjective. Then there are infinitely many441Dirichlet characters $\chi$ as above that satisfies the following442hypothesis:443\[444L(E,\chi,1)\neq 0 \qquad\text{and}\qquad445a_{\ell}(E) \not\con \ell+1 \pmod{p}.446\]447\end{conjecture}448I have verified this conjecture numerically in thousands of449examples, and I hope to prove something about it by assuming it is450false and constructing many relations between modular symbols.451Analytic methods involving averaging special values of452$L$-functions seem incapable of handling twists of high degree.453454\begin{definition}[Modularity of Mordell-Weil]\label{defn:modmw}455If $A$ is a modular abelian variety, and $n$ is an integer, we say456that the Mordell-Weil group of $A$ is {\em $n$-modular of level457$M$} if there is a quotient $\pi:J_1(M)\to A'$, with connected458kernel, such that $A'$ is isogenous to $A$ and%459$\pi(J_1(M))\subset n A'(\Q)$. We say that the Mordell-Weil group460of $A$ is {\em modular} if it is $n$-modular for every461integer~$n$.462\end{definition}463464I think the following conjecture is within reach.465\begin{conjecture}[Stein]\label{conj:mwallmodular}466The Mordell-Weil group $A(\Q)$ of every modular abelian variety is467itself modular, in the sense of Definition~\ref{defn:modmw}.468\end{conjecture}469This is closely related to Theorem~\ref{thm:allmwvis} since the470restriction of scalars of a modular abelian variety is again471modular. However it is unclear exactly how $A$, $\Res_{K/\Q}(A)$,472and $J_1(M)$ all fit together, and it is essential to understand473{\em exactly} how they fit together in order to verify the474conjecture. Also, for a given $n$, it would be interesting to475decide if the Mordell-Weil group is $n$-visible of level $M$ for476some naturally defined~$M$.477478479480481482483\section{OLD STUFF}484%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%485486487488489[summary/intro/the point]490491492\begin{definition}[Visibility]493Suppose that $\iota:A\hra J$ is an inclusion of abelian varieties494over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$ with respect495to $\iota$ is the kernel of the induced map $\H^1(\Q,A)\to496\H^1(\Q,J)$. The visible subgroup of $\Sha(A/\Q)$ is the497intersection of $\Sha(A/\Q)$ with the visible subgroup of498$\H^1(\Q,A)$, or equivalently, the kernel of the induced map499$\Sha(A/\Q)\to \Sha(J/\Q)$.500\end{definition}501502The paper \cite{agashe-stein:visibility} extends work of Cremona503and Mazur \cite{mazur:visord3, cremona-mazur} to lay the504foundations for studying visibility of Shafarevich-Tate groups of505abelian varieties. In it, I use a restriction of scalars506construction to prove that if $A$ is an abelian variety and $x\in507\Sha(A/\Q)$ then there is an inclusion $A\hra B$ such that $x$ is508visible in $B$. We then prove that if $A$ and $B$ are abelian509subvarieties of an abelian variety $J$, and $B[p]\subset A$, then,510under certain hypothesis, there is an injection from $B(\Q)/p511B(\Q)$ into the visible subgroup of $\Sha(A/\Q)$. We apply this512theorem to prove that $25$ divides the order of the visible513subgroup of the Shafarevich-Tate group of an abelian variety of514dimension $20$ and level $389$. We also give the first explicit515example of an element of the Shafarevich-Tate group of an elliptic516curve that only becomes visible at higher level.517518\subsubsection{Computational evidence for the BSD conjecture}519In \cite{agashe-stein:bsd}, Agashe and I520describe a number of algorithms that we found and I implemented in521{\sc Magma} for computing with modular abelian varieties. We then522compute a divisor and multiple of the BSD conjectural order of523$\Sha(A/\Q)$ for the 10360 (optimal) abelian variety quotients524$A=A_f$ of $J_0(N)$ with $L(A,1)\neq 0$ and $f$ of level $N\leq5252333$. For $168$ of these $A$ our divisor is divisible by an odd526prime, and for $37$ of these $168$, we prove using visibility that527if $n$ is the prime--to-$2$ part of the conjectural order of528$\Sha(A/\Q)$, then there are $n$ elements of $\Sha(A/\Q)$ that are529visible in $J_0(N)$. The challenge remains to use other530techniques, e.g., visibility at higher level, to show that there531are $n$ elements in $\Sha(A/\Q)$ in the remaining $131$ cases. For532example, in \cite{stein:bsdmagma} we do this for the first533example, which has level $551$ and whose $\Sha$ becomes visible at534level $2\cdot 551$.535536\subsubsection{Visibility for modular motives} One objective of my537past NSF supported research was to generalize visibility theory to538the context of modular motives. Fortunately, Neil Dummigan, Mark539Watkins, and I succeeded in carrying out such a program in540\cite{dummigan-stein-watkins:motives}. There we prove a general541theorem that can be used in many cases to deduce the existence of542visible Shafarevich-Tate groups in motives attached to modular543forms, assuming a conjecture of Beilinson about ranks of Chow544groups. We give several pages of tables that suggest that545Shafarevich-Tate groups of modular motives of level~$N$ are very546rarely visible in the higher weight motivic analogue of $J_0(N)$,547much more rarely than for weight~$2$.548549550\subsubsection{Nonsquare Shafarevich-Tate groups} In the paper551\cite{stein:nonsquaresha} I give a surprising application of552visibility to understanding the possibilities for the orders of553Shafarevich-Tate groups. Before \cite{stein:nonsquaresha} no554examples of Shafarevich-Tate groups of order $p\cdot n^2$ were555known for any odd prime $p$, and the literature even suggested556such examples do not exist ([swinnerton-dyer]). In557\cite{poonen-stoll}, Poonen and Stoll give the first examples of558Shafarevich-Tate groups of order $2\cdot n^2$, which inspired me559to look for examples of order $3\cdot n^2$. Using an adaptation560of the ideas from visibility along with a deep theorem of Kato on561the Birch and Swinnerton-Dyer conjecture, I was able to construct,562under suitable hypothesis, an abelian variety $A$ with563$\#\Sha(A/\Q)=p\cdot n^2$; I then used a computer to verify the564hypothesis for all odd primes $p<25000$. The $p$-part of565$\Sha(A/\Q)$ is visible in an abelian variety that is isogenous to566$A\times E$, where $E$ is an elliptic curve. To remove the567hypothesis on~$p$ would require proving a nonvanishing result568about twists of~$L$ functions (see Section~\ref{conj:nonvanishchi}569below).570571\subsection{The Arithmetic of modular abelian varieties}572573\subsubsection{$J_1(p)$ has connected fibers} In the paper \cite{conrad-edixhoven-stein:j1p},574Conrad, S.~Edixhoven, and I prove a remarkable uniformity result575for the component group of the N\'eron model of $J_1(p)$: {\em It576has order~$1$ for all primes $p$!} We do this by determining the577closed fiber at~$p$ of a model for $X_1(p)$, then do intersection578theory computations to find a regular model for $X_1(p)$ over579$\Z$. In this paper, I use theorems of Mazur, Kato, and a580computation to determine the primes~$p$ such that $J_1(p)$ has581rank~$0$. I also do significant computations of many of the582invariants appearing in the BSD conjecture for each of the simple583factors of $J_1(p)$. The results of these numerical computations584combined with the main theoretical result of the paper on585component groups suggest several questions, which I intend to586address (see Section~\ref{sec:j1pques}).587588\subsubsection{Component groups of purely toric abelian varieties}589In \cite{stein:compgroup}, Conrad and I use Grothendieck's590monodromy pairing to give a complete proof of correctness of an591algorithm I found for computing the order of the component group592of certain purely toric abelian varieties. I found this algorithm593after reading a letter from Ribet to Mestre, which addressed594certain numerical relations for elliptic curves in terms of595Mestre's method of graphs. In \cite{kohel-stein:ants4}, D.~Kohel596and I explain how to calculate Tamagawa numbers (the $c_p$ in the597BSD conjecture, which are orders of component groups) for purely598toric modular abelian varieties using this algorithm. In general,599we only obtain the Tamagawa number up to a bounded power of $2$, a600shortcoming I intend to remedy with further work. We also do not601determine the structure of the underlying component groups, which602is something I hope to do.603604\subsubsection{The BSD conjecture for Jacobians of genus two605curves} The paper \cite{empirical} is about the BSD conjecture for606$32$ modular Jacobians of genus~$2$ curves. I learned of an early607version of \cite{empirical} before it was published, and was608shocked by the table of orders of Shafarevich-Tate groups that it609contained. I used the equation-free algorithms I developed in610\cite{stein:phd} to do the computations in a new way, and found611that the most striking example in the paper, a Shafarevich-Tate612group of order $49$, was incorrect. I was made a coauthor and613wrote a section of the paper describing my methods.614615\subsection{Other research}616\subsubsection{Application of Kolyvagin's Euler system}%617In \cite{stein:index} I give an innovative application of618Kolyvagin's Euler system to an old question of E.~Artin, S.~Lang,619and Tate (see \cite{lang-tate}).620621Let $X$ be a curve over $\Q$ (say) of622genus~$g$. The {\em index} of $X$ is the greatest common divisor623of the degrees of the extensions of $\Q$ in which $X$ has a624rational point. Then the canonical divisor has degree $2g-2$, so625the index of $X$ divides $2g-2$. When $g=1$ this is no condition626at all. {\bf Question:} {\em For every integer $n$, is there a627genus one curve with index exactly~$n$?}628629In \cite{lang-tate}, Lang and Tate prove that for each $n$ there630is a genus one curve~$X$ over some number field~$K$ (which depends631on $n$) such that~$X$ has index~$n$. In \cite{stein:index}, I632prove that if~$K$ is a fixed number field, then for any~$n$ not633divisible by~$8$ there is a genus one curve~$X$ over~$K$ of634index~$n$. The proof involves reinterpreting genus one curves and635the notion of index in terms of Galois cohomology, then finding636nontrivial Galois cohomology classes with the requisite properties637in the Euler system of Heegner points on $X_0(17)$.638639\subsubsection{Elliptic curves with full torsion} In \cite{merel-stein}, L.~Merel640and I investigate a natural question about fields of definition641that is connected with points on modular curves. Let $p$ be a642prime. Suppose $E$ is an elliptic curve over a number field~$K$643and all of the $p$ torsion on $E$ is defined over $K$. Properties644of the Weil pairing imply that the field $\Q(\zeta_p)$ of $p$th645roots of unity is contained in $K$. {\bf Question.} {\em Is there646an elliptic curve defined over $\Q(\zeta_p)$ all of whose647$p$-torsion is also defined over $\Q(\zeta_p)$?} By combining648the significant theory developed in \cite{merel:cyclo} with a649nontrivial modular symbols computation, we show that the question650has a ``no'' answer for all $p<1000$, except $p=2,3,5,13$. (A651student of Merel showed that $13$ also has a ``no'' answer.)652653654\subsubsection{Modularity of icosahedral Galois representations} In \cite{buzzard-stein:artin},655K.~Buzzard and I prove $8$ new cases of the Artin conjecture about656modularity of icosahedral Galois representations, only $3$ of657which are covered by the subsequent landmark work of Taylor which658gave infinitely many new examples. Buzzard and I push through an659explicit application of \cite{buzzard-taylor} by combining various660theorems with significant modular symbols computations over the661finite field of order~$5$.662663664\subsubsection{Approximating $p$-adic modular forms} In665\cite{coleman-stein:padicapprox}, R.~Coleman and I consider from a666theoretical and computational point of view questions about667$p$-adic approximation of infinite slope modular eigenforms by668modular eigenforms of finite slope. The slope of an eigenform669$f=\sum a_n q^n$ is the $p$-adic valuation of $a_p$, so an670eigenform has infinite slope precisely when $a_p=0$. When~$f$ is671an eigenform having infinite slope, Naomi Jochnowitz asked if for672every~$n$ there is an eigenform~$g$ of finite slope such that673$f\equiv g\pmod{p^n}$. We show that the answer in general is no,674but prove that if~$f$ is a twist of a finite slope eigenform,675then~$f$ can be approximated. We also investigate computationally676which forms can be approximated and how the weight of~$g$ grows as677a function of~$n$. These computations lead to intriguing678unanswered questions.679680\section{Project Proposal}681682\subsection{Visibility of Shafarevich-Tate groups at higher level}683684The following conjecture is the central open problem in visibility685theory.686\begin{conjecture}[Stein]\label{conj:allvis}687Let~$A$ be a modular abelian variety.%688\begin{enumerate}%689\item Then there is an integer~$N$ and a morphism $f:A\to J_1(N)$690such that every element of $\Sha(f(A))$ is visible in $J_1(N)$.%691\item The level $N$ should be determined in some natural way in692terms of properties of~$A$. (Part of the conjecture is to give a693reasonable interpretation of natural.)694\end{enumerate}695\end{conjecture}696697If true, Conjecture~\ref{conj:allvis} would imply finiteness of698the Shafarevich-Tate group of~$A$, which would massively699strengthen many current results towards the Birch and700Swinnerton-Dyer conjecture. In \cite{agashe-stein:visibility}, I701proved that each element of the Shafarevich-Tate group of~$A$ is702visible in some modular abelian variety~$B$, but in this703construction~$B$ depends on the element. As a first step toward704Conjecture~\ref{conj:allvis}, I hope to use my result to prove705that if $\Sha(A)$ is finite then part 1 of the conjecture is true.706The main obstruction is that it is unclear how $A$, $B$ and707$J_1(N)$ all fit together, and in order to prove the conjecture it708is essential to know {\em exactly} how these abelian varieties fit709together. I strongly believe resolving this difficulty is within710reach and will lead to new ideas. (See711Conjecture~\ref{conj:mwallmodular} below for a similar situation.)712713The first part of Conjecture~\ref{conj:allvis} for a single714element of $\Sha(A)$ is analogous to the easy-to-prove assertion715that each ideal class in the ring of integers of a number field716becomes principal in a suitable extension field, where the717extension depends on the ideal class. The second part of the718conjecture is reminiscent of the existence of the Hilbert class719field of a number field, and deeper investigation into it may720prove crucial to understanding Shafarevich-Tate groups.721722I intend to revisit the computations of \cite{agashe-stein:bsd}723and see how far visibility at higher level goes toward724constructing the odd part of $\Sha(A/\Q)$ in the remaining $131$725cases not already covered.726727Suppose $A=A_f$ with $f\in S_2(\Gamma_1(N))$ and $L(A_f,1)\neq 0$.728The following discussion illustrates one way in which ideas from729visibility have vague unexplored implications for the BSD730conjecture, namely for the assertion that if $p\mid \#\Sha(A)$731then $p$ divides the conjectural order of $\Sha(A)$. This is732only one of many similar ideas.733734Suppose $x\in \Sha(A/\Q)[p]$ is an element of prime order $p$ that735is visible in $J_1(NM)$ for some $M$. Then in most cases there736should be a factor $A_g$ of $J_1(NM)$ that has positive737Mordell-Weil rank such that $x$ is in the image of $A_g(\Q)$ under738some map. Usually this should imply that~$g$ that is congruent739to~$f$ modulo a prime of characteristic~$p$; then by Kato's740theorem \cite{kato:secret, rubin:kato} we must have $L(g,1)=0$,741since otherwise $A_g$ would have rank $0$. Because congruences742between eigenforms usually induce congruences between special743values of $L$ functions, this will often imply that744\[745L(A_f,1)/\Omega_{A_f} \con L(A_g,1)/\omega = 0 \pmod{p}.746\]747748%%%% I just commented this out because isn't it trivially true by switching parity in functional equation749%This discussion also motivates the following conjecture, which may750%be viewed as an analytic shadow of visibility:751%\begin{conjecture}[Stein]752%Suppose $f$ is a newform and $p$ is a prime such that753%$L(A_f,1)/\Omega_{A_f}\con 0\pmod{p}$. Then there exists a754%newform $g$ that is congruent to $f$ modulo~$p$ such that755%$L(g,1)=0$.756%\end{conjecture}757%Note that the conjecture is trivially true in case $L(f,1)=0$,758%since we just take $g=f$.759760761762\subsection{Visibility of Mordell-Weil Groups of abelian varieties}763Turning Mazur's visibility idea on its head, I introduced the764notion of visibility of Mordell-Weil groups.765766\begin{definition}[Visibility of Mordell-Weil]767Suppose that $\pi : J\to A$ is a surjective morphism of abelian768varieties with connected kernel~$C$. Let $\delta : A(\Q)\to769\H^1(\Q,C)$ be the connecting homomorphism. An element $x\in770A(\Q)$ is \emph{$n$-visible} with respect to $\pi$ if $\delta(x)$771has order divisible by~$n$, and $x$ is \emph{$n$-visible in772$\Sha$} if moreover $\delta(x)\in \Sha(C/\Q)$.773\end{definition}774775\begin{theorem}\label{thm:allmwvis}776If~$A$ is a modular abelian variety and $x\in A(\Q)$, then for777every integer~$n$ there is a covering $J\to A$ with connected778kernel such that $x$ is $n$-visible in $\H^1(\Q,J)$.779\end{theorem}780The key idea of the proof is that if $p$ is any prime and781$\Q_\infty$ is the cyclotomic $\Z_p$-extension of $\Q$, then by a782nonvanishing theorem of Rohrlich \cite{rohrlich:cyclo} and Kato's783theorem \cite{kato:secret,rubin:kato}, the group $A(\Q_\infty)$ is784finitely generated. From this we deduce that there is an (abelian)785extension $K$ of $\Q$ such that $n$ divides the order of the image786of $x$ in $A(\Q)/\Tr_{K/\Q}(A(K))$. Trace defines a morphism from787the restriction of scalars $\pi:J=\Res_{K/\Q}(A_K)$ to $A$ with788connected kernel. Then $x$ is $n$-visible with respect to $\pi$.789790\begin{conjecture}[Stein]791Let $A$ be a modular abelian variety, let $x\in A(\Q)$, and let792$n$ be a positive integer. Then there is a surjective morphism793$\pi:J\to A$ with connected kernel such that $x$ is $n$-visible in794$\Sha$ with respect to~$\pi$.795\end{conjecture}796797My attempts so far to prove this conjecture led to the paper798\cite{stein:nonsquaresha}, the connection being as follows.799Suppose $E$ is an elliptic curve over $\Q$ with $E(\Q)=\Z{}x$, and800let $f$ be the newform attached to $f$. Fix a prime~$p$. Suppose801$\chi:(\Z/\ell\Z)\to\mu_p$ is a Dirichlet character that satisfies802several carefully chosen hypothesis, and let $K$ be the803corresponding abelian extension. By chasing the appropriate804diagrams and using results about \'etale cohomology and N\'eron805models, I show that if $J=\Res_{K/\Q}(E)$ then $x$ is $p$-visible806in $\Sha$ with respect to $J\to E$. This means that807\[808E(\Q)/pE(\Q)\isom \Z/p\Z\subset \Sha(\ker(J\to E)),809\]810which is where the nonsquare part of $\Sha$ comes from.811\begin{conjecture}[Stein]\label{conj:nonvanishchi}812Assume~$p$ is an odd prime and the mod~$p$ Galois representation813attached to $E$ is surjective. Then there are infinitely many814Dirichlet characters $\chi$ as above that satisfies the following815hypothesis:816\[817L(E,\chi,1)\neq 0 \qquad\text{and}\qquad818a_{\ell}(E) \not\con \ell+1 \pmod{p}.819\]820\end{conjecture}821I have verified this conjecture numerically in thousands of822examples, and I hope to prove something about it by assuming it is823false and constructing many relations between modular symbols.824Analytic methods involving averaging special values of825$L$-functions seem incapable of handling twists of high degree.826827\begin{definition}[Modularity of Mordell-Weil]\label{defn:modmw}828If $A$ is a modular abelian variety, and $n$ is an integer, we say829that the Mordell-Weil group of $A$ is {\em $n$-modular of level830$M$} if there is a quotient $\pi:J_1(M)\to A'$, with connected831kernel, such that $A'$ is isogenous to $A$ and%832$\pi(J_1(M))\subset n A'(\Q)$. We say that the Mordell-Weil group833of $A$ is {\em modular} if it is $n$-modular for every834integer~$n$.835\end{definition}836837I think the following conjecture is within reach.838\begin{conjecture}[Stein]\label{conj:mwallmodular}839The Mordell-Weil group $A(\Q)$ of every modular abelian variety is840itself modular, in the sense of Definition~\ref{defn:modmw}.841\end{conjecture}842This is closely related to Theorem~\ref{thm:allmwvis} since the843restriction of scalars of a modular abelian variety is again844modular. However it is unclear exactly how $A$, $\Res_{K/\Q}(A)$,845and $J_1(M)$ all fit together, and it is essential to understand846{\em exactly} how they fit together in order to verify the847conjecture. Also, for a given $n$, it would be interesting to848decide if the Mordell-Weil group is $n$-visible of level $M$ for849some naturally defined~$M$.850851852853\subsection{Computing with modular abelian varieties}854Bryan Birch once commented to me in reference to computation that855``It is always a good idea to try to prove true theorems.'' To856this end, the author proposes to continue developing algorithms857and making available tools for computing with modular forms,858modular abelian varieties, and motives attached to modular forms.859This includes finishing a major new {\sc Magma} \cite{magma}860package for computing directly with modular abelian varieties over861number fields, extending the Modular Forms Database \cite{mfd},862and searching for algorithms for computing the quantities863appearing in the Birch and Swinnerton-Dyer for modular abelian864varieties and the Bloch-Kato conjecture for modular motives. The865results of this work should give an explicit picture of modular866abelian varieties that could never be obtained from general867theory.868869870\subsubsection{The Modular forms database} The modular forms871database \cite{mfd} contains a large collection of information872about objects attached to cuspidal eigenforms. Though greatly873appreciated by the many mathematicians who use it, the database874currently only scratches the surface of what it should contain.875876The database is stored using the database system PostgreSQL, and I877wrote the web user interface in Python. During Summer 2003 the878Harvard undergraduate Dimitar Jetchev did extensive work on the879database, and this pointed out significant deficiencies in how it880is currently implemented. It is more difficult than it should be881to modify the web interface to the database, the data is not882compressed well, and there is no way to submit new data to the883database using the web page. I intend to completely rewrite the884database using MySQL and PHP, and investigate better algorithms885for storing $q$-expansions of modular forms much more efficiently.886Currently the limit on the database is not the difficulty of887computing modular forms, but the space and time used in storing888them. This could be partially remedied by moving the database to889a more modern computer (it currently runs on a three year old890Pentium III), something I am requesting in this grant.891892\subsubsection{Example database queries that have not yet been done}893\begin{itemize}894\item Suppose $d=2,3,4,5$, say. Using the algorithm described in895\cite{agashe-stein:bsd}, compute a multiple of the order of the896torsion subgroup of $A_f(\Q)$ for each $d$-dimensional $A_f$ in897the database. What is the maximum number that occurs? After what898level do no new numbers appear? For small $d$ such a computation899may suggest a conjectural generalization to modular abelian900varieties of Mazur's theorem on torsion points on elliptic curves.%901902\item Make a conjectural list of all number fields of degree~$d$903(for $d=2,3,4,5$, say) that arise as the field generated by the904eigenvalues of a newform in the database. Coleman has conjectured905that for each~$d$ only finitely many number fields of degree~$d$906appear. When do new $d$ seem to stop appearing?%907\end{itemize}908909\subsubsection{M{\small AGMA} package for modular abelian varieties} I910wrote the modular forms and modular symbols packages that are part911of the {\sc Magma} computer algebra system \cite{magma}. I spent912June 2003 in Sydney, Australia and did exciting work on designing913and implementing a very general package for computing with modular914abelian varieties over number fields. Much work is left to be915done to finish this package, and several crucial algorithms still916need to be developed or refined. When available this package917will likely be greatly appreciated by anybody working with modular918abelian varieties.919920The following are problems that arose in work on this package,921which must be resolved in order to have a satisfactory system for922computing with modular abelian varieties. I need to solve all of923these problems.924\begin{itemize}925\item {\bf Endomorphism ring over $\Qbar$:} {\em Giving a modular926abelian variety $A$, explicitly (and efficiently) compute the927endomorphism ring $\End(A)$ as a ring of matrices acting928$\H_1(A,\Z)$.} I have a modular symbols solution that draw on929work of Ribet \cite{ribet:twistsendoalg} and Shimura930\cite{shimura:factors} but is too slow to be really useful in931practice; however, similar Manin symbols methods must932exist and be very efficient.%933\item {\bf Decomposition:} {\em Given the endomorphism ring of an934abelian variety $A$, compute its decomposition as a product of935simple abelian varieties.} This is a standard and difficult936problem in general, but it might be possible to combine work of937Allan Steel on a ``characteristic~$0$ Meataxe'' with special938features of modular abelian varieties to solve it in practice. It939is absolutely {\em essential} to solve this problem in order to940explicitly enumerate all modular abelian varieties over number941fields of given level.%942\item {\bf Isomorphism testing:} {\em Given943two modular abelian varieties over a number field $K$, represented944as explicit quotients of Jacobians $J_1(N)$, decide whether there945is an isomorphism between them.} I have solved this problem when946the two modular abelian varieties are simple. There are analogues947of this problem in other categories, which I intend to investigate.%948\end{itemize}949950\subsection{Some conjectures that were inspired by my computations}951952\subsubsection{Questions about $J_1(p)$}\label{sec:j1pques} The following953conjecture generalizes a famous conjecture of Ogg that954$J_0(p)(\Q)_{\tor}$ is cyclic of order the numerator of955$(p-1)/12$, a fact that Mazur proved in \cite{mazur:eisenstein}.956\begin{conjecture}[Stein]957The torsion subgroup of $J_1(p)(\Q)$ is exactly the group958generated by the cusps on $X_1(p)$ that lie over $\infty\in959X_0(p)$. This is a group of order960\[961\frac{p}{2^{p-3}} \cdot\prod_{\eps\neq 1} B_{2,\eps}962\]963where the product is over the nontrivial even Dirichlet964characters~$\varepsilon$ of conductor dividing~$p$, and965$B_{2,\eps}$ is a generalized Bernoulli number.966\end{conjecture}967Mazur's complete proof of the analogue of this statement for968$J_0(p)$ is quite deep, though the proof for the prime-to-$2$ part969of $J_0(p)(\Q)_{\tor}$ is much easier. I hope to mimic Mazur's970method and prove the conjecture above for the prime-to-$2$ part of971$J_1(p)(\Q)_{\tor}$.972973More generally, I would like to investigate the torsion in974quotients of $J_1(p)$.975\begin{question}976If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the977natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective? If978so, is the product of the orders of all $A_f(\Q)_{\tor}$ over all979classes of newforms $f$ equal to $\# J_1(p)(\Q)_{\tor}$?980\end{question}981I conjectured that the analogous questions for $J_0(p)$ should have982``yes'' answers, and in \cite{emerton:optimal} M.~Emerton983subsequently proved this conjecture. It is still not clear if984one should make this conjecture for $J_1(p)$.985986\subsubsection{Congruences between modular forms of prime level}987Ken Ribet enticed me into studying modular forms as a graduate988student by asking me the following question: ``Is there a prime989$p$ so that $p$ is ramified in the Hecke algebra $\T$ attached to990$S_2(\Gamma_0(p))$?'' I answered his question by showing that991$p=389$ is the only prime less than $50000$ that ramifies in the992associated Hecke algebra of level~$p$. The question remains993whether $p=389$ is the only such example, and this seems extremely994difficult to say anything useful about. However a related995question exhibits a shockingly clear pattern, and this related996question is the question Ribet was really interested in for his997application to images of Galois representations998\cite{ribet:torsion}.9991000Let $p$ be a prime and $k$ a positive even integer. Let $\T$ be1001the Hecke algebra attached to $S_k(\Gamma_0(p))$ and let $d(k,p)$1002be the valuation at $p$ of the index of $\T$ in its normalization.1003The following conjecture is backed up by significant numerical1004evidence, and was discovered by staring at tables and looking for1005a pattern.1006\begin{conjecture}[Stein, F.~Calegari]1007Suppose $p\geq k-1$. Then1008$$1009d(k,p) = \left\lfloor\frac{p}{12}\right\rfloor\cdot1010\binom{m}{2} + a(p,m),1011$$1012where the function $a$ only depends on $p$ modulo $12$ as1013follows:1014$$1015a(p,m) =1016\begin{cases}10170 & \text{if $p\equiv 1\pmod{12}$,}\\10183\cdot\displaystyle\binom{\lceil \frac{m}{3}\rceil}{2} & \text{if $p\equiv 5\pmod{12}$,}\\10192\cdot\displaystyle\binom{\lceil \frac{m}{2}\rceil}{2} & \text{if $p\equiv 7\pmod{12}$,}\\1020a(5,m)+a(7,m) & \text{if $p\equiv 11\pmod{12}$.}1021\end{cases}1022$$1023\end{conjecture}10241025The situation of interest to Ribet is $k=2$, in which case the1026conjecture simply asserts that $\T\otimes\Z_p$ is normal, i.e.,1027{\em there or no congruences in characteristic $p$ between1028non-Galois conjugate newforms in $S_2(\Gamma_0(p))$.} Calegari1029has given a conjectural interpretation of some of the congruences1030that the conjecture asserts must exist, which I intend to study1031further.10321033\section{Summary}1034This research proposal depicts an intricate network of ongoing1035investigations into the arithmetic of modular abelian varieties,1036which unite a theoretical and computational point of view. The1037basic foundations of visibility theory are nearly complete, but1038solutions to the questions about visibility outlined in this1039proposal demand a new level of precision in our understanding of1040the web of modular abelian varieties. I am determined to advance1041our understanding in this direction.10421043My work has produced results and tools that are of use to other1044mathematicians who are exploring the world of modular forms. By1045supporting my research, you will assure the sustained development1046of this technology.104710481049105010511052Let $p$ be a prime and $k$ a positive even integer. Let $\T$ be1053the Hecke algebra attached to the space $S_k(\Gamma_0(p))$ of cusp1054forms for $\Gamma_0(p)$ and let $d(k,p)$ be the valuation at $p$1055of the index of $\T$ in its normalization. The following1056conjecture is backed up by significant numerical evidence, and was1057discovered by staring at tables computed using the PIs {\sc Magma}1058code and looking for a pattern.1059\begin{conjecture}[Stein, F.~Calegari]1060Suppose $p\geq k-1$. Then1061$$1062d(k,p) = \left\lfloor\frac{p}{12}\right\rfloor\cdot1063\binom{m}{2} + a(p,m),1064$$1065where the function $a$ only depends on $p$ modulo $12$ as1066follows:1067$$1068a(p,m) =1069\begin{cases}10700 & \text{if $p\equiv 1\pmod{12}$,}\\10713\cdot\displaystyle\binom{\lceil \frac{m}{3}\rceil}{2} & \text{if $p\equiv 5\pmod{12}$,}\\10722\cdot\displaystyle\binom{\lceil \frac{m}{2}\rceil}{2} & \text{if $p\equiv 7\pmod{12}$,}\\1073a(5,m)+a(7,m) & \text{if $p\equiv 11\pmod{12}$.}1074\end{cases}1075$$1076\end{conjecture}1077The conjecture is of interest to Ribet in the case $k=2$, because1078it is a hypotheses to the main argument of \cite{ribet:torsion}.1079For a long time it was unclear what to conjecture when $k=2$;1080finally, investigation into what happens at higher weight1081suggested the above conjectural formula, which specializes in1082weight $2$ to the assertion that $\T\otimes \Z_p$ is normal. The1083PI has no idea how to prove this conjecture when $k=2$, but1084intends to at least find similar conjectures when $\Gamma_0(p)$ is1085replaced by $\Gamma_1(p)$ and when~$p$ is replaced by a composite1086number.1087108810891090\newpage10911092\newcommand{\etalchar}[1]{$^{#1}$}1093\providecommand{\bysame}{\leavevmode\hbox1094to3em{\hrulefill}\thinspace}1095\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }1096% \MRhref is called by the amsart/book/proc definition of \MR.1097\providecommand{\MRhref}[2]{%1098\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}1099} \providecommand{\href}[2]{#2}1100\begin{thebibliography}{BCDT01}11011102\bibitem[AS]{agashe-stein:bsd}1103A.~Agashe and W.\thinspace{}A. Stein, \emph{Visible {E}vidence for1104the {B}irch1105and {S}winnerton-{D}yer {C}onjecture for {M}odular {A}belian {V}arieties of1106{A}nalytic {R}ank~$0$}, To appear in Mathematics of Computation.11071108\bibitem[AS02]{agashe-stein:visibility}1109\bysame, \emph{Visibility of {S}hafarevich-{T}ate groups of1110abelian varieties},1111J. Number Theory \textbf{97} (2002), no.~1, 171--185.11121113\bibitem[BCDT01]{breuil-conrad-diamond-taylor}1114C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor, \emph{On the1115modularity of1116elliptic curves over {$\bold Q$}: wild 3-adic exercises}, J. Amer. Math. Soc.1117\textbf{14} (2001), no.~4, 843--939 (electronic).11181119\bibitem[BCP97]{magma}1120W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra1121system. {I}.1122{T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,1123235--265, Computational algebra and number theory (London, 1993).11241125\bibitem[Bir71]{birch:bsd}1126B.\thinspace{}J. Birch, \emph{Elliptic curves over1127\protect{${\mathbf{Q}}$:1128{A}} progress report}, 1969 Number Theory Institute (Proc. Sympos. Pure1129Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math.1130Soc., Providence, R.I., 1971, pp.~396--400.11311132\bibitem[BS02]{buzzard-stein:artin}1133K.~Buzzard and W.\thinspace{}A. Stein, \emph{A mod five approach1134to modularity1135of icosahedral {G}alois representations}, Pacific J. Math. \textbf{203}1136(2002), no.~2, 265--282.11371138\bibitem[BT99]{buzzard-taylor}1139K.~Buzzard and R.~Taylor, \emph{Companion forms and weight one1140forms}, Ann. of1141Math. (2) \textbf{149} (1999), no.~3, 905--919.11421143\bibitem[CES03]{conrad-edixhoven-stein:j1p}1144B.~Conrad, S.~Edixhoven, and W.\thinspace{}A. Stein,1145\emph{${J}_1(p)$ {H}as1146{C}onnected {F}ibers}, To appear in Documenta Mathematica (2003).11471148\bibitem[CM00]{cremona-mazur}1149J.\thinspace{}E. Cremona and B.~Mazur, \emph{Visualizing elements1150in the1151{S}hafarevich-{T}ate group}, Experiment. Math. \textbf{9} (2000), no.~1,115213--28.11531154\bibitem[Col03]{coleman-stein:padicapprox}1155R.~Coleman, \emph{Approximation of infinite-slope modular1156eigenforms by1157finite-slope eigenforms}, To appear in the Dwork Proceedings (2003).11581159\bibitem[CS02]{stein:compgroup}1160B.~Conrad and W.\thinspace{}A. Stein, \emph{Component {G}roups of1161{P}urely1162{T}oric {Q}uotients}, To appear in Math Research Letters (2002).11631164\bibitem[DWS]{dummigan-stein-watkins:motives}1165N.~Dummigan, M.~Watkins, and W.\thinspace{}A. Stein,1166\emph{{Constructing1167Elements in Shafarevich-Tate Groups of Modular Motives}}, To appear in the1168Swinnerton-Dyer proceedings.11691170\bibitem[Eme01]{emerton:optimal}1171M.~Emerton, \emph{Optimal {Q}uotients of {M}odular {J}acobians},1172preprint1173(2001).11741175\bibitem[FpS{\etalchar{+}}01]{empirical}1176E.\thinspace{}V. Flynn, F.~\protect{Lepr\'{e}vost},1177E.\thinspace{}F. Schaefer,1178W.\thinspace{}A. Stein, M.~Stoll, and J.\thinspace{}L. Wetherell,1179\emph{Empirical evidence for the {B}irch and {S}winnerton-{D}yer conjectures1180for modular {J}acobians of genus 2 curves}, Math. Comp. \textbf{70} (2001),1181no.~236, 1675--1697 (electronic).11821183\bibitem[Kat]{kato:secret}1184K.~Kato, \emph{$p$-adic {H}odge theory and values of zeta1185functions of modular1186forms}, Preprint, 244 pages.11871188\bibitem[KS00]{kohel-stein:ants4}1189D.\thinspace{}R. Kohel and W.\thinspace{}A. Stein, \emph{Component1190{G}roups of1191{Q}uotients of \protect{$J_0(N)$}}, Proceedings of the 4th International1192Symposium (ANTS-IV), Leiden, Netherlands, July 2--7, 2000 (Berlin), Springer,11932000.11941195\bibitem[Lan91]{lang:nt3}1196S.~Lang, \emph{Number theory. {I}{I}{I}}, Springer-Verlag, Berlin,11971991,1198Diophantine geometry.11991200\bibitem[LT58]{lang-tate}1201S.~Lang and J.~Tate, \emph{Principal homogeneous spaces over1202abelian1203varieties}, Amer. J. Math. \textbf{80} (1958), 659--684.12041205\bibitem[Maz77]{mazur:eisenstein}1206B.~Mazur, \emph{Modular curves and the \protect{Eisenstein}1207ideal}, Inst.1208Hautes \'Etudes Sci. Publ. Math. (1977), no.~47, 33--186 (1978).12091210\bibitem[Maz99]{mazur:visord3}1211\bysame, \emph{Visualizing elements of order three in the1212{S}hafarevich-{T}ate1213group}, Asian J. Math. \textbf{3} (1999), no.~1, 221--232, Sir Michael1214Atiyah: a great mathematician of the twentieth century.12151216\bibitem[Mer01]{merel:cyclo}1217L.~Merel, \emph{Sur la nature non-cyclotomique des points d'ordre1218fini des1219courbes elliptiques}, Duke Math. J. \textbf{110} (2001), no.~1, 81--119, With1220an appendix by E. Kowalski and P. Michel.12211222\bibitem[MS01]{merel-stein}1223L.~Merel and W.\thinspace{}A. Stein, \emph{The field generated by1224the points of1225small prime order on an elliptic curve}, Internat. Math. Res. Notices (2001),1226no.~20, 1075--1082.12271228\bibitem[PS99]{poonen-stoll}1229B.~Poonen and M.~Stoll, \emph{The {C}assels-{T}ate pairing on1230polarized abelian1231varieties}, Ann. of Math. (2) \textbf{150} (1999), no.~3, 1109--1149.123212331234\bibitem[Rib80]{ribet:twistsendoalg}1235K.\thinspace{}A. Ribet, \emph{Twists of modular forms and1236endomorphisms of1237abelian varieties}, Math. Ann. \textbf{253} (1980), no.~1, 43--62.123812391240\bibitem[Rib99]{ribet:torsion}1241\bysame, \emph{Torsion points on ${J}\sb 0({N})$ and {G}alois1242representations},1243Arithmetic theory of elliptic curves (Cetraro, 1997), Springer, Berlin, 1999,1244pp.~145--166.12451246\bibitem[Roh84]{rohrlich:cyclo}1247D.\thinspace{}E. Rohrlich, \emph{On {$L$}-functions of elliptic1248curves and1249cyclotomic towers}, Invent. Math. \textbf{75} (1984), no.~3, 409--423.12501251\bibitem[Rub98]{rubin:kato}1252K.~Rubin, \emph{Euler systems and modular elliptic curves}, Galois1253representations in arithmetic algebraic geometry (Durham, 1996), Cambridge1254Univ. Press, Cambridge, 1998, pp.~351--367.12551256\bibitem[Shi73]{shimura:factors}1257G.~Shimura, \emph{On the factors of the jacobian variety of a1258modular function1259field}, J. Math. Soc. Japan \textbf{25} (1973), no.~3, 523--544.12601261\bibitem[Ste]{stein:nonsquaresha}1262W.\thinspace{}A. Stein, \emph{Shafarevich-tate groups of nonsquare1263order},1264Proceedings of MCAV 2002, Progress of Mathematics (to appear).12651266\bibitem[Ste00]{stein:phd}1267\bysame, \emph{Explicit approaches to modular abelian varieties},1268Ph.D. thesis,1269University of California, Berkeley (2000).12701271\bibitem[Ste02]{stein:index}1272\bysame, \emph{There are genus one curves over {$\mathbf{Q}$} of1273every odd1274index}, J. Reine Angew. Math. \textbf{547} (2002), 139--147.12751276\bibitem[Ste03a]{mfd}1277\bysame, \emph{The {M}odular {F}orms {D}atabase}, \newline{\tt1278http://modular.fas.harvard.edu/Tables} (2003).12791280\bibitem[Ste03b]{stein:bsdmagma}1281\bysame, \emph{Studying the {B}irch and {S}winnerton-{D}yer1282{C}onjecture for1283{M}odular {A}belian {V}arieties {U}sing {\sc Magma}}, To appear in J.~Cannon, ed.,1284{\em Computational Experiments in Algebra and Geometry}, Springer-Verlag1285(2003).12861287\bibitem[Tat95]{tate:bsd}1288J.~Tate, \emph{On the conjectures of {B}irch and1289{S}winnerton-{D}yer and a1290geometric analog}, S\'eminaire Bourbaki, Vol.\ 9, Soc. Math. France, Paris,12911995, pp.~Exp.\ No.\ 306, 415--440.12921293\end{thebibliography}129412951296\end{document}1297129812991300\end{document}13011302