\documentclass[11pt]{article} \newcommand{\thisdocument}{Project Description} \include{macros} %\textheight=0.8\textheight \begin{document} \section{Introduction} My research reflects the essential interplay of abstract theory with explicit machine computation, which is illustrated by the following quote of Bryan Birch~\cite{birch:bsd} about computations that led to a central conjecture in number theory: \begin{quote} I want to describe some computations undertaken by myself and Swinnerton-Dyer on EDSAC by which we have calculated the zeta-functions of certain elliptic curves. As a result of these computations we have found an analogue for an elliptic curve of the Tamagawa number of an algebraic group; and conjectures (due to ourselves, due to Tate, and due to others) have proliferated. \end{quote} I am primarily interested in abelian varieties attached to modular forms via Shimura's construction \cite{shimura:factors}. Let~$f=\sum a_n q^n$ be a weight~$2$ newform on $\Gamma_1(N)$. We may view~$f$ as a differential on the modular curve $X_1(N)$, which is a curve whose affine points over~$\C$ correspond to isomorphism classes of pairs $(E,P)$, where~$E$ is an elliptic curve and $P\in E$ is a point of order~$N$. We view the Hecke algebra \[\T=\Z[T_1,T_2,T_3,\ldots]\] as a subring of the endomorphism ring of the Jacobian $J_1(N)$ of $X_1(N)$. Let $I_f$ be the annihilator of~$f$ in $\T$, and attach to~$f$ the quotient $$A_f=J_1(N)/I_f J_1(N).$$ Then $A_f$ is an abelian variety over~$\Q$ of dimension equal to the degree of the field $\Q(a_1,a_2,a_3,\ldots)$ generated by the coefficients of~$f$. The abelian varieties $A_f$ attached to newforms are important. For example,the celebrated modularity theorem of C.~Breuil, B.~Conrad, F.~Diamond, Taylor, and Wiles \cite{breuil-conrad-diamond-taylor} asserts that every elliptic curve over~$\Q$ is isogenous to some $A_f$. Also, Serre conjectures that up to twist every two-dimensional odd irreducible Galois representation appears in the torsion points on some $A_f$. My investigations into modular abelian varieties are inspired by the following special case of the Birch and Swinnerton-Dyer conjecture (see \cite{tate:bsd, lang:nt3}): \begin{conjecture}[BSD Conjecture] \[ \frac{L(A_f,1)}{\Omega_{A_f}} =% \frac{\prod c_p \cdot \#\Sha(A_f/\Q)}% {\#A_f(\Q)\cdot \#A_f^{\vee}(\Q)}. \] \end{conjecture} Here $L(A_f,s)$ is the canonical $L$-series attached to $A_f$, the real volume $\Omega_{A_f}$ is the measure of $A_f(\R)$ with respect to a basis of differentials for the N\'eron model of $A_f$, the $c_p$ are the Tamagawa numbers of $A_f$, the dual of $A_f$ is denoted $A_f^{\vee}$, and \[ \Sha(A_f/\Q) = \ker\left(\H^1(\Q,A_f) \to \bigoplus_{p\leq \infty} \H^1(\Q_p,A_f)\right) \] is the Shafarevich-Tate group of $A_f$. When $A_f(\Q)$ is infinite, the right hand side should be interpreted as~$0$, so, in particular, the conjecture asserts that $L(A_f,1)=0$ if and only if $A_f(\Q)$ is infinite. Birch and Swinnerton-Dyer also conjectured that the order of vanishing of $L(A_f,s)$ at $s=1$ equals the rank of $A_f(\Q)$. \section{Computing with modular abelian varieties} The PI proposes to continue developing algorithms and making available tools for computing with modular forms, modular abelian varieties, and motives attached to modular forms. This includes finishing a major new {\sc Magma} \cite{magma} package for computing directly with modular abelian varieties over number fields, extending the Modular Forms Database \cite{mfd}, and searching for algorithms for computing the quantities appearing in the Birch and Swinnerton-Dyer for modular abelian varieties and the Bloch-Kato conjecture for modular motives. \subsection{The Modular Forms Database}% The Modular Forms Database \cite{mfd} is a freely-available collection of data about objects attached to cuspidal modular forms, that is a much used resource for number theorists. It is analogous to Sloane's tables of integer sequences, and extends Cremona's tables to dimension bigger than one and weight bigger than two (\cite{cremona-tables} contain more precise data about elliptic curves than \cite{mfd}). The PI proposes to greatly expand the database. The major challenge is that data about modular abelian varieties of large dimension takes a huge amount of space to store. For example, the database currently occupies 40GB disk space. He proposes to find better method for storing information about modular abelian so that the database can grow larger, and to investigate methods used by astronomers or the human genome project to see how they cope with a torrent of data while making it available to their colleagues. The PI implemented the current database using PostgreSQL coupled with a Python web interface. To speed access and improve efficiency, the PI is considering rewriting key portions of the database using MySQL and PHP. Also, the database currently runs on a three-year-old 933Mhz Pentium III, so the PI is requesting more modern hardware. \subsubsection{M{\small AGMA} package for modular abelian varieties}% {\sc Magma} is a nonprofit computer algebra system developed primarily at the University of Sydney, which is supported mostly by grant money from organizations such as the US National Security Agency. {\sc Magma} is considered by many to be the most comprehensive tool for research in number theory, finite group theory, and cryptography, and it is widely distributed. The PI has already written over 400 pages (26000 lines) of modular forms code and extensive documentation that is distributed with {\sc Magma}, and intends to ``publish'' future work in {\sc Magma}. An abelian variety $A$ over a number field is {\em modular} if it is a quotient of $J_1(N)$ for some $N$. Modular abelian varieties were studied intensively by Ken Ribet, Barry Mazur, and others during recent decades, and studying them is popular because results about them often yield surprising insight into number theoretic questions. Computation with modular abelian varieties is popular because most of them are easily described by giving a level and the first few coefficients of a modular form, and the $L$-functions of modular abelian varieties are particularly well understood. This is in sharp contrast to the case of general abelian varieties which, in general, can only be described by unwieldy systems of polynomial equations, and whose $L$-functions are very mysterious. The PI recently designed and partially implemented a general package for computing with modular abelian varieties over number fields. Several crucial algorithms still need to be developed or refined. When available, this package will likely be useful for people working with modular abelian varieties. The following major problems arose in work on this package, and they must be resolved in order to have a completely satisfactory system for computing with modular abelian varieties: \begin{itemize} \item {\em Given a modular abelian variety $A$, efficiently compute the endomorphism ring $\End(A)$ as a ring of matrices acting $\H_1(A,\Z)$.} The PI has found a modular symbols solution that draws on work of Ribet \cite{ribet:twistsendoalg} and Shimura \cite{shimura:factors}, but it is too slow to be really useful in practice. In \cite{merel:1585}, Merel uses Herbrand matrices and Manin symbols to give efficient algorithms for computing with Hecke operators. The PI intends to carry over Merel's method to give an efficient algorithm to compute $\End(A)$.% \item {\em Given $\End(A)\otimes\Q$, compute an isogeny decomposition of $A$ as a product of simple abelian varieties.} This is a standard and difficult problem in general, but it might be possible to combine work of Allan Steel on his ``characteristic~$0$ Meataxe'' with special features of modular abelian varieties to solve it in practice. It is absolutely {\em essential} to solve this problem in order to explicitly enumerate all modular abelian varieties over number fields of given level. Such an enumeration would be a major step towards the ultimate possible generalization of Cremona's tables \cite{cremona:tables} to modular abelian varieties. % \item {\em Given two modular abelian varieties over a number field $K$, decide whether there is an isomorphism between them.} When the endomorphism ring of each abelian variety is known and both are simple, it is possible to reduce this problem to mostly well-studied norm equations. This problem is analogous to the problem of testing isomorphism for modules over a fixed ring, which has been solved with much effort for many classes of rings. One application of isomorphism testing is that it could be used to prove that an abelian variety is not principally polarized.% \end{itemize} We finish be describing recent work with $J_1(p)$ that was partially inspired by PI's modular forms software. This conjecture generalizes a conjecture of Ogg, which asserted that $J_0(p)(\Q)_{\tor}$ is cyclic of order the numerator of $(p-1)/12$, a fact that Mazur proved in \cite{mazur:eisenstein}. \begin{conjecture}[Stein]\label{conj:tor} Let $p$ be a prime. The torsion subgroup of $J_1(p)(\Q)$ is the group generated by the cusps on $X_1(p)$ that lie over $\infty\in X_0(p)$. \end{conjecture} Significant numerical evidence for this conjecture is given in \cite{j1p}, and cuspidal subgroups of $J_1(p)$ are considered in detail in \cite{kubert-lang}, where, e.g., they compute orders of such groups in terms of Bernoulli numbers. Mazur's proof of Ogg's conjecture for $J_0(p)$ is deep, though the proof for the odd part of $J_0(p)(\Q)_{\tor}$ is much easier. The PI intends explore whether or not it is possible to build on Mazur's method and prove results towards Conjecture~\ref{conj:tor}. The PI also intends to develop his computational methods for computing torsion subgroups in order to answer, at least conjecturally, the following question. \begin{question} If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective? If so, is the product of the orders of all $A_f(\Q)_{\tor}$ over all classes of newforms $f$ equal to $\# J_1(p)(\Q)_{\tor}$? \end{question} The PI conjectured that the analogous questions for $J_0(p)$ should have ``yes'' answers, and in \cite{emerton:optimal} M.~Emerton proved this conjecture. In \cite{j1p}, it is proved that $J_1(p)$ has trivial component group (component groups are closely related to the Tamagawa numbers $c_p$, which appear in the Birch and Swinnerton-Dyer conjecture). \begin{question} If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the component group of $A_f$ trivial? \end{question} Even assuming the full Birch and Swinnerton-Dyer conjecture, extensive computations by the PI have not produced a conjectural answer to this question. However, he and Bjorn Poonen formulated a strategy to answer this question in some interesting cases by using an explicit decomposition of $\End({A_f}_{\Qbar})$ to obtain a curve whose Jacobian is $A_f$. \section{Visibility of Shafarevich-Tate and Mordell-Weil Groups} \subsection{Computational problems} Before describing work the PI proposes to do on visibility and its uses in obtaining evidence for the Birch and Swinnerton-Dyer conjecture, we recall the precise definition of visibility over~$\Q$ (the definition makes sense over any global field, but we restrict to $\Q$ below for simplicity). \begin{definition}[Visibility of Shafarevich-Tate Groups] Suppose that $\iota:A\hra J$ is an inclusion of abelian varieties over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$ with respect to $\iota$ is the kernel of the induced map $\H^1(\Q,A)\to \H^1(\Q,J)$. The {\em visible subgroup} of $\Sha(A/\Q)$ is the intersection of $\Sha(A/\Q)$ with the visible subgroup of $\H^1(\Q,A)$; equivalently, it is the kernel of the induced map $\Sha(A/\Q)\to \Sha(J/\Q)$. \end{definition} Before discussing theoretical questions about visibility, we describe computational evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties (and motives) that we obtained using a theorem inspired by the definition of visibility. In \cite{agashe-stein:visibility}, the PI and Agashe prove a theorem that makes it possible to use abelian varieties of large rank to explicitly construct subgroups of Shafarevich-Tate groups of abelian varieties. The main theorem is that if $A$ and $B$ are abelian subvarieties of an abelian variety $J$, and $B[p]\subset A$, then, under certain hypothesis, there is an injection from $B(\Q)/p B(\Q)$ into the visible subgroup of $\Sha(A/\Q)$. The paper concludes with the first ever example of an abelian variety $A_f$ attached to a newform, of large dimension ($20$), whose Shafarevich-Tate group is provably nontrivial. In \cite{agashe-stein:bsd}, the PI and Agashe describe algorithms that we found and I implemented in {\sc Magma} for computing with modular abelian varieties. We then compute an odd divisor and even multiple of the BSD conjectural order of $\Sha(A/\Q)$ for the over ten thousand quotients $A=A_f$ of $J_0(N)$ with $L(A,1)\neq 0$. For over a hundred of these, our lower bound on the conjectural order of $\Sha(A/\Q)$ is is divisible by an odd prime; for a quarter of these we prove, using the main theorem of \cite{agashe-stein:visibility}, that if $n$ is the conjectural divisor of $\#\Sha(A/\Q)$, then there are at least~$n$ elements of $\Sha(A/\Q)$ that are visible in $J_0(N)$. \emph{The PI intends to investigate the remaining 75\% of the examples with $n>1$ by considering the image of $A$ in $J_0(NM)$ for small integer $M$.} Information about which primes $\ell$ to choose can be extracted from Ribet's level raising theorem (see \cite{ribet:raising}). As a test, the PI tried the first example not already done, which is an $18$ dimension abelian variety of level $551$, whose Shafarevich-Tate group conjecturally contains elements of order $3$; these are not visible in $J_0(551)$, but he shows in \cite{stein:bsdmagma} that they are visible in $J_(551\cdot 2)$. One objective of the PI's past NSF supported research was to generalize visibility to the context of modular motives. Fortunately, Neil Dummigan, Mark Watkins, and the PI did significant work in this direction in \cite{dummigan-stein-watkins:motives}. There we prove a theorem that can sometimes be used to deduce the existence of visible Shafarevich-Tate groups in motives attached to modular forms, assuming a conjecture of Beilinson about ranks of Chow groups. However, we give several pages of tables that suggest that Shafarevich-Tate groups of modular motives of level~$N$ are very rarely visible in the higher weight motivic analogue of $J_0(N)$, much more rarely than for weight~$2$. Just as above, the question remains to decide whether one expects these groups to be visible in $J_0(N M)$ for some integer~$M$. It would be relatively straightforward for the PI to do computations in this direction, but none have been done yet. The PI intends to do such computations. Before moving on to theoretical questions about visibility, we pause to emphasize that the above computational investigations into the Birch and Swinnerton-Dyer conjecture motivated the PI and others to develop new algorithms for computing with modular abelian varieties. For example, in \cite{stein:compgroup}, Conrad and the PI use Grothendieck's monodromy pairing to give a complete proof of correctness of an algorithm the PI found for computing the order of the component group of certain purely toric abelian varieties. This algorithm makes it practical to compute component groups of quotients of $A_f$ of $J_0(N)$ at primes~$p$ that exactly divide $N$; without such an algorithm it would probably be difficult to get anywhere in computational investigations into the Birch and Swinnerton-Dyer conjecture for abelian varieties; indeed, the only other paper in this direction is \cite{evidence}, which restricts to the case of dimension~$2$ Jacobians. \subsection{Theoretical problems} Suppose $A_f$ is a quotient of $J_1(N)$ attached to a newform and let $A=A_f^{\vee}\subset J_1(N)$ be its dual. One expects that most of $\Sha(A)$ is {\em not} visible in $J_1(N)$. The following conjecture then arises. \begin{conjecture}\label{conj:allvis} For each $x\in \Sha(A/\Q)$, there is an integer $M$ and a morphism $f:A\to J_1(NM)$ of finite degree coprime to the order of~$x$ such that the image of~$x$ in $\Sha(f(A)/\Q)$ is visible in $J_1(NM)$. \end{conjecture} In \cite{agashe-stein:visibility}, the PI proved that if $x\in \H^1(\Q,A)$ then there is an abelian variety $B$ and an inclusion $\iota:A\to B$ such that $x$ is visible in $B$; moreover, $B$ is a quotient of $J_1(NM)$ for some $M$. The PI hopes to prove Conjecture~\ref{conj:allvis} by understanding much more precisely how $A$, $B$, and $J_1(NM)$ are related. A more analytical approach to Conjecture~\ref{conj:allvis} is to assume the rank statement of the Birch and Swinnerton-Dyer conjecture and relate when elements of $\Sha(A/\Q)$ becoming visible at level $NM$ to when there is a congruence between $f$ and a newform $g$ of level $NM$ with $L(g,1)=0$. Such an approach leads one to wish to formulate a refinement of Ribet's level raising theorem that includes a statement about the behavior of the value at $1$ of the $L$-function attached to the form at higher level. The PI intends to do further computations in order to give a good conjectural refinement of Ribet's theorem. The Gross-Zagier theorem asserts that points on elliptic curves of rank $1$ come from Heegner points, and that points on curves of rank bigger than one do not. Over fifteen years later, it still seems mysterious to give an interpretation of points on elliptic curves of rank higher than~$1$. The PI introduced the following definition, in hopes of eventually creating a framework for giving a conjectural explanation. \begin{definition}[Visibility of Mordell-Weil Groups] Suppose that $\pi : J\to A$ is a surjective morphism of abelian varieties with connected kernel~$C$. Let $\delta : A(\Q)\to \H^1(\Q,C)$ be the connecting homomorphism of Galois cohomology. An element $x\in A(\Q)$ is \emph{$n$-visible in $\H^1$} (with respect to $\pi$) if $\delta(x)$ has order divisible by~$n$, and $x$ is \emph{$n$-visible in $\Sha$} if moreover $\delta(x)\in \Sha(C/\Q)$. \end{definition} The following theorem is not difficult to prove by combining Kato's powerful results towards the Birch and Swinnerton-Dyer conjecture (see \cite{kato:secret,rubin:kato}) with a nonvanishing theorem of Rohrlich \cite{rohrlich:cyclo}. \begin{theorem}[Stein]\label{thm:allmwvis} If~$A$ is a modular abelian variety and $x\in A(\Q)$ has infinite order, then for every integer~$n$ there is a covering $J\to A$ with connected kernel such that $x$ is $n$-visible in $\H^1$. \end{theorem} %The key idea of the proof is that if $p$ is any prime and %$\Q_\infty$ is the cyclotomic $\Z_p$-extension of $\Q$, then by a %nonvanishing theorem of Rohrlich and Kato's theorem , the group %$A(\Q_\infty)$ is finitely generated. From this we deduce that %there is an (abelian) extension $K$ of $\Q$ such that $n$ divides %the order of the image of $x$ in $A(\Q)/\Tr_{K/\Q}(A(K))$. Trace %defines a morphism from the restriction of scalars %$\pi:J=\Res_{K/\Q}(A_K)$ to $A$ with connected kernel. Then $x$ %is $n$-visible with respect to $\pi$. \begin{conjecture}[Stein] If~$A$ be a modular abelian variety and $x\in A(\Q)$ has infinite order, then for every integer~$n$, there is an abelian variety $B$ and a surjective morphism $\pi:B\to A$ with connected kernel such that $x$ is $n$-visible in $\Sha$ with respect to~$\pi$. \end{conjecture} The PI proved partial results towards this conjecture in \cite{stein:nonsquaresha}. Suppose $E$ is an elliptic curve over $\Q$ with conductor $N$, and let $f$ be the newform attached to $E$. Fix a prime~$p\nmid N \prod c_p$ and such that the mod~$p$ Galois representation attached to $E$ is surjective. Suppose $\chi:(\Z/\ell\Z)\to\mu_p$ is a Dirichlet character with $\ell\nmid N$ such that \[ L(E,\chi,1)\neq 0 \qquad\text{and}\qquad a_{\ell}(E) \not\con \ell+1 \pmod{p}, \] and let $K$ be the corresponding abelian extension. In \cite{stein:nonsquaresha}, the PI uses Kato's theorem and restriction of scalars to construct abelian varieties $A$ and $J$ and an exact sequence \[ 0 \to E(\Q)/ p E(\Q) \to \Sha(A/\Q) \to \Sha(J/\Q) \to \Sha(E/\Q) \to 0. \] Here $A$ is an abelian variety of rank $0$ with $L(A,1)\neq 0$. Thus all of $E(\Q)$ is $p$-visible in $\Sha$ and one can interpret $E(\Q)/ p E(\Q)$ as a visible subgroup of a rank~$0$ abelian variety~$A$. There are two problems with this picture. First, the PI does not know a proof that such a $\chi$ exists (he has verified existence of $\chi$ in thousands of examples). Second, even if the existence of $\chi$ were known it seems unlikely that visibility of Mordell-Weil groups in an abstract restriction of scalars abelian variety is likely to yield a satisfactory interpretation of $E(\Q)$ when $E$ has large rank. To answer these problems, the PI intends to study the system of all possible abelian varieties $B$ in which $E(\Q)$ induces visible elements of Shafarevich-Tate groups, and moreover to try to keep track of how these abelian varieties $B$ are related to $J_1(NM)$ for various $M$. This will involve a combination of computer computation of examples followed hopefully by the development of a concise notation for keeping track of all relevant data, and eventually perhaps new ideas that clarify our understanding of elliptic curves of rank bigger than one. \bibliographystyle{amsalpha} \bibliography{biblio} \end{document} %%%%%%%%%%% \begin{conjecture}[Stein]\label{conj:nonvanishchi} Assume~$p$ is an odd prime and the mod~$p$ Galois representation attached to $E$ is surjective. Then there are infinitely many Dirichlet characters $\chi$ as above that satisfies the following hypothesis: \[ L(E,\chi,1)\neq 0 \qquad\text{and}\qquad a_{\ell}(E) \not\con \ell+1 \pmod{p}. \] \end{conjecture} I have verified this conjecture numerically in thousands of examples, and I hope to prove something about it by assuming it is false and constructing many relations between modular symbols. Analytic methods involving averaging special values of $L$-functions seem incapable of handling twists of high degree. \begin{definition}[Modularity of Mordell-Weil]\label{defn:modmw} If $A$ is a modular abelian variety, and $n$ is an integer, we say that the Mordell-Weil group of $A$ is {\em $n$-modular of level $M$} if there is a quotient $\pi:J_1(M)\to A'$, with connected kernel, such that $A'$ is isogenous to $A$ and% $\pi(J_1(M))\subset n A'(\Q)$. We say that the Mordell-Weil group of $A$ is {\em modular} if it is $n$-modular for every integer~$n$. \end{definition} I think the following conjecture is within reach. \begin{conjecture}[Stein]\label{conj:mwallmodular} The Mordell-Weil group $A(\Q)$ of every modular abelian variety is itself modular, in the sense of Definition~\ref{defn:modmw}. \end{conjecture} This is closely related to Theorem~\ref{thm:allmwvis} since the restriction of scalars of a modular abelian variety is again modular. However it is unclear exactly how $A$, $\Res_{K/\Q}(A)$, and $J_1(M)$ all fit together, and it is essential to understand {\em exactly} how they fit together in order to verify the conjecture. Also, for a given $n$, it would be interesting to decide if the Mordell-Weil group is $n$-visible of level $M$ for some naturally defined~$M$. \section{OLD STUFF} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [summary/intro/the point] \begin{definition}[Visibility] Suppose that $\iota:A\hra J$ is an inclusion of abelian varieties over $\Q$. The {\em visible subgroup} of $\H^1(\Q,A)$ with respect to $\iota$ is the kernel of the induced map $\H^1(\Q,A)\to \H^1(\Q,J)$. The visible subgroup of $\Sha(A/\Q)$ is the intersection of $\Sha(A/\Q)$ with the visible subgroup of $\H^1(\Q,A)$, or equivalently, the kernel of the induced map $\Sha(A/\Q)\to \Sha(J/\Q)$. \end{definition} The paper \cite{agashe-stein:visibility} extends work of Cremona and Mazur \cite{mazur:visord3, cremona-mazur} to lay the foundations for studying visibility of Shafarevich-Tate groups of abelian varieties. In it, I use a restriction of scalars construction to prove that if $A$ is an abelian variety and $x\in \Sha(A/\Q)$ then there is an inclusion $A\hra B$ such that $x$ is visible in $B$. We then prove that if $A$ and $B$ are abelian subvarieties of an abelian variety $J$, and $B[p]\subset A$, then, under certain hypothesis, there is an injection from $B(\Q)/p B(\Q)$ into the visible subgroup of $\Sha(A/\Q)$. We apply this theorem to prove that $25$ divides the order of the visible subgroup of the Shafarevich-Tate group of an abelian variety of dimension $20$ and level $389$. We also give the first explicit example of an element of the Shafarevich-Tate group of an elliptic curve that only becomes visible at higher level. \subsubsection{Computational evidence for the BSD conjecture} In \cite{agashe-stein:bsd}, Agashe and I describe a number of algorithms that we found and I implemented in {\sc Magma} for computing with modular abelian varieties. We then compute a divisor and multiple of the BSD conjectural order of $\Sha(A/\Q)$ for the 10360 (optimal) abelian variety quotients $A=A_f$ of $J_0(N)$ with $L(A,1)\neq 0$ and $f$ of level $N\leq 2333$. For $168$ of these $A$ our divisor is divisible by an odd prime, and for $37$ of these $168$, we prove using visibility that if $n$ is the prime--to-$2$ part of the conjectural order of $\Sha(A/\Q)$, then there are $n$ elements of $\Sha(A/\Q)$ that are visible in $J_0(N)$. The challenge remains to use other techniques, e.g., visibility at higher level, to show that there are $n$ elements in $\Sha(A/\Q)$ in the remaining $131$ cases. For example, in \cite{stein:bsdmagma} we do this for the first example, which has level $551$ and whose $\Sha$ becomes visible at level $2\cdot 551$. \subsubsection{Visibility for modular motives} One objective of my past NSF supported research was to generalize visibility theory to the context of modular motives. Fortunately, Neil Dummigan, Mark Watkins, and I succeeded in carrying out such a program in \cite{dummigan-stein-watkins:motives}. There we prove a general theorem that can be used in many cases to deduce the existence of visible Shafarevich-Tate groups in motives attached to modular forms, assuming a conjecture of Beilinson about ranks of Chow groups. We give several pages of tables that suggest that Shafarevich-Tate groups of modular motives of level~$N$ are very rarely visible in the higher weight motivic analogue of $J_0(N)$, much more rarely than for weight~$2$. \subsubsection{Nonsquare Shafarevich-Tate groups} In the paper \cite{stein:nonsquaresha} I give a surprising application of visibility to understanding the possibilities for the orders of Shafarevich-Tate groups. Before \cite{stein:nonsquaresha} no examples of Shafarevich-Tate groups of order $p\cdot n^2$ were known for any odd prime $p$, and the literature even suggested such examples do not exist ([swinnerton-dyer]). In \cite{poonen-stoll}, Poonen and Stoll give the first examples of Shafarevich-Tate groups of order $2\cdot n^2$, which inspired me to look for examples of order $3\cdot n^2$. Using an adaptation of the ideas from visibility along with a deep theorem of Kato on the Birch and Swinnerton-Dyer conjecture, I was able to construct, under suitable hypothesis, an abelian variety $A$ with $\#\Sha(A/\Q)=p\cdot n^2$; I then used a computer to verify the hypothesis for all odd primes $p<25000$. The $p$-part of $\Sha(A/\Q)$ is visible in an abelian variety that is isogenous to $A\times E$, where $E$ is an elliptic curve. To remove the hypothesis on~$p$ would require proving a nonvanishing result about twists of~$L$ functions (see Section~\ref{conj:nonvanishchi} below). \subsection{The Arithmetic of modular abelian varieties} \subsubsection{$J_1(p)$ has connected fibers} In the paper \cite{conrad-edixhoven-stein:j1p}, Conrad, S.~Edixhoven, and I prove a remarkable uniformity result for the component group of the N\'eron model of $J_1(p)$: {\em It has order~$1$ for all primes $p$!} We do this by determining the closed fiber at~$p$ of a model for $X_1(p)$, then do intersection theory computations to find a regular model for $X_1(p)$ over $\Z$. In this paper, I use theorems of Mazur, Kato, and a computation to determine the primes~$p$ such that $J_1(p)$ has rank~$0$. I also do significant computations of many of the invariants appearing in the BSD conjecture for each of the simple factors of $J_1(p)$. The results of these numerical computations combined with the main theoretical result of the paper on component groups suggest several questions, which I intend to address (see Section~\ref{sec:j1pques}). \subsubsection{Component groups of purely toric abelian varieties} In \cite{stein:compgroup}, Conrad and I use Grothendieck's monodromy pairing to give a complete proof of correctness of an algorithm I found for computing the order of the component group of certain purely toric abelian varieties. I found this algorithm after reading a letter from Ribet to Mestre, which addressed certain numerical relations for elliptic curves in terms of Mestre's method of graphs. In \cite{kohel-stein:ants4}, D.~Kohel and I explain how to calculate Tamagawa numbers (the $c_p$ in the BSD conjecture, which are orders of component groups) for purely toric modular abelian varieties using this algorithm. In general, we only obtain the Tamagawa number up to a bounded power of $2$, a shortcoming I intend to remedy with further work. We also do not determine the structure of the underlying component groups, which is something I hope to do. \subsubsection{The BSD conjecture for Jacobians of genus two curves} The paper \cite{empirical} is about the BSD conjecture for $32$ modular Jacobians of genus~$2$ curves. I learned of an early version of \cite{empirical} before it was published, and was shocked by the table of orders of Shafarevich-Tate groups that it contained. I used the equation-free algorithms I developed in \cite{stein:phd} to do the computations in a new way, and found that the most striking example in the paper, a Shafarevich-Tate group of order $49$, was incorrect. I was made a coauthor and wrote a section of the paper describing my methods. \subsection{Other research} \subsubsection{Application of Kolyvagin's Euler system}% In \cite{stein:index} I give an innovative application of Kolyvagin's Euler system to an old question of E.~Artin, S.~Lang, and Tate (see \cite{lang-tate}). Let $X$ be a curve over $\Q$ (say) of genus~$g$. The {\em index} of $X$ is the greatest common divisor of the degrees of the extensions of $\Q$ in which $X$ has a rational point. Then the canonical divisor has degree $2g-2$, so the index of $X$ divides $2g-2$. When $g=1$ this is no condition at all. {\bf Question:} {\em For every integer $n$, is there a genus one curve with index exactly~$n$?} In \cite{lang-tate}, Lang and Tate prove that for each $n$ there is a genus one curve~$X$ over some number field~$K$ (which depends on $n$) such that~$X$ has index~$n$. In \cite{stein:index}, I prove that if~$K$ is a fixed number field, then for any~$n$ not divisible by~$8$ there is a genus one curve~$X$ over~$K$ of index~$n$. The proof involves reinterpreting genus one curves and the notion of index in terms of Galois cohomology, then finding nontrivial Galois cohomology classes with the requisite properties in the Euler system of Heegner points on $X_0(17)$. \subsubsection{Elliptic curves with full torsion} In \cite{merel-stein}, L.~Merel and I investigate a natural question about fields of definition that is connected with points on modular curves. Let $p$ be a prime. Suppose $E$ is an elliptic curve over a number field~$K$ and all of the $p$ torsion on $E$ is defined over $K$. Properties of the Weil pairing imply that the field $\Q(\zeta_p)$ of $p$th roots of unity is contained in $K$. {\bf Question.} {\em Is there an elliptic curve defined over $\Q(\zeta_p)$ all of whose $p$-torsion is also defined over $\Q(\zeta_p)$?} By combining the significant theory developed in \cite{merel:cyclo} with a nontrivial modular symbols computation, we show that the question has a ``no'' answer for all $p<1000$, except $p=2,3,5,13$. (A student of Merel showed that $13$ also has a ``no'' answer.) \subsubsection{Modularity of icosahedral Galois representations} In \cite{buzzard-stein:artin}, K.~Buzzard and I prove $8$ new cases of the Artin conjecture about modularity of icosahedral Galois representations, only $3$ of which are covered by the subsequent landmark work of Taylor which gave infinitely many new examples. Buzzard and I push through an explicit application of \cite{buzzard-taylor} by combining various theorems with significant modular symbols computations over the finite field of order~$5$. \subsubsection{Approximating $p$-adic modular forms} In \cite{coleman-stein:padicapprox}, R.~Coleman and I consider from a theoretical and computational point of view questions about $p$-adic approximation of infinite slope modular eigenforms by modular eigenforms of finite slope. The slope of an eigenform $f=\sum a_n q^n$ is the $p$-adic valuation of $a_p$, so an eigenform has infinite slope precisely when $a_p=0$. When~$f$ is an eigenform having infinite slope, Naomi Jochnowitz asked if for every~$n$ there is an eigenform~$g$ of finite slope such that $f\equiv g\pmod{p^n}$. We show that the answer in general is no, but prove that if~$f$ is a twist of a finite slope eigenform, then~$f$ can be approximated. We also investigate computationally which forms can be approximated and how the weight of~$g$ grows as a function of~$n$. These computations lead to intriguing unanswered questions. \section{Project Proposal} \subsection{Visibility of Shafarevich-Tate groups at higher level} The following conjecture is the central open problem in visibility theory. \begin{conjecture}[Stein]\label{conj:allvis} Let~$A$ be a modular abelian variety.% \begin{enumerate}% \item Then there is an integer~$N$ and a morphism $f:A\to J_1(N)$ such that every element of $\Sha(f(A))$ is visible in $J_1(N)$.% \item The level $N$ should be determined in some natural way in terms of properties of~$A$. (Part of the conjecture is to give a reasonable interpretation of natural.) \end{enumerate} \end{conjecture} If true, Conjecture~\ref{conj:allvis} would imply finiteness of the Shafarevich-Tate group of~$A$, which would massively strengthen many current results towards the Birch and Swinnerton-Dyer conjecture. In \cite{agashe-stein:visibility}, I proved that each element of the Shafarevich-Tate group of~$A$ is visible in some modular abelian variety~$B$, but in this construction~$B$ depends on the element. As a first step toward Conjecture~\ref{conj:allvis}, I hope to use my result to prove that if $\Sha(A)$ is finite then part 1 of the conjecture is true. The main obstruction is that it is unclear how $A$, $B$ and $J_1(N)$ all fit together, and in order to prove the conjecture it is essential to know {\em exactly} how these abelian varieties fit together. I strongly believe resolving this difficulty is within reach and will lead to new ideas. (See Conjecture~\ref{conj:mwallmodular} below for a similar situation.) The first part of Conjecture~\ref{conj:allvis} for a single element of $\Sha(A)$ is analogous to the easy-to-prove assertion that each ideal class in the ring of integers of a number field becomes principal in a suitable extension field, where the extension depends on the ideal class. The second part of the conjecture is reminiscent of the existence of the Hilbert class field of a number field, and deeper investigation into it may prove crucial to understanding Shafarevich-Tate groups. I intend to revisit the computations of \cite{agashe-stein:bsd} and see how far visibility at higher level goes toward constructing the odd part of $\Sha(A/\Q)$ in the remaining $131$ cases not already covered. Suppose $A=A_f$ with $f\in S_2(\Gamma_1(N))$ and $L(A_f,1)\neq 0$. The following discussion illustrates one way in which ideas from visibility have vague unexplored implications for the BSD conjecture, namely for the assertion that if $p\mid \#\Sha(A)$ then $p$ divides the conjectural order of $\Sha(A)$. This is only one of many similar ideas. Suppose $x\in \Sha(A/\Q)[p]$ is an element of prime order $p$ that is visible in $J_1(NM)$ for some $M$. Then in most cases there should be a factor $A_g$ of $J_1(NM)$ that has positive Mordell-Weil rank such that $x$ is in the image of $A_g(\Q)$ under some map. Usually this should imply that~$g$ that is congruent to~$f$ modulo a prime of characteristic~$p$; then by Kato's theorem \cite{kato:secret, rubin:kato} we must have $L(g,1)=0$, since otherwise $A_g$ would have rank $0$. Because congruences between eigenforms usually induce congruences between special values of $L$ functions, this will often imply that \[ L(A_f,1)/\Omega_{A_f} \con L(A_g,1)/\omega = 0 \pmod{p}. \] %%%% I just commented this out because isn't it trivially true by switching parity in functional equation %This discussion also motivates the following conjecture, which may %be viewed as an analytic shadow of visibility: %\begin{conjecture}[Stein] %Suppose $f$ is a newform and $p$ is a prime such that %$L(A_f,1)/\Omega_{A_f}\con 0\pmod{p}$. Then there exists a %newform $g$ that is congruent to $f$ modulo~$p$ such that %$L(g,1)=0$. %\end{conjecture} %Note that the conjecture is trivially true in case $L(f,1)=0$, %since we just take $g=f$. \subsection{Visibility of Mordell-Weil Groups of abelian varieties} Turning Mazur's visibility idea on its head, I introduced the notion of visibility of Mordell-Weil groups. \begin{definition}[Visibility of Mordell-Weil] Suppose that $\pi : J\to A$ is a surjective morphism of abelian varieties with connected kernel~$C$. Let $\delta : A(\Q)\to \H^1(\Q,C)$ be the connecting homomorphism. An element $x\in A(\Q)$ is \emph{$n$-visible} with respect to $\pi$ if $\delta(x)$ has order divisible by~$n$, and $x$ is \emph{$n$-visible in $\Sha$} if moreover $\delta(x)\in \Sha(C/\Q)$. \end{definition} \begin{theorem}\label{thm:allmwvis} If~$A$ is a modular abelian variety and $x\in A(\Q)$, then for every integer~$n$ there is a covering $J\to A$ with connected kernel such that $x$ is $n$-visible in $\H^1(\Q,J)$. \end{theorem} The key idea of the proof is that if $p$ is any prime and $\Q_\infty$ is the cyclotomic $\Z_p$-extension of $\Q$, then by a nonvanishing theorem of Rohrlich \cite{rohrlich:cyclo} and Kato's theorem \cite{kato:secret,rubin:kato}, the group $A(\Q_\infty)$ is finitely generated. From this we deduce that there is an (abelian) extension $K$ of $\Q$ such that $n$ divides the order of the image of $x$ in $A(\Q)/\Tr_{K/\Q}(A(K))$. Trace defines a morphism from the restriction of scalars $\pi:J=\Res_{K/\Q}(A_K)$ to $A$ with connected kernel. Then $x$ is $n$-visible with respect to $\pi$. \begin{conjecture}[Stein] Let $A$ be a modular abelian variety, let $x\in A(\Q)$, and let $n$ be a positive integer. Then there is a surjective morphism $\pi:J\to A$ with connected kernel such that $x$ is $n$-visible in $\Sha$ with respect to~$\pi$. \end{conjecture} My attempts so far to prove this conjecture led to the paper \cite{stein:nonsquaresha}, the connection being as follows. Suppose $E$ is an elliptic curve over $\Q$ with $E(\Q)=\Z{}x$, and let $f$ be the newform attached to $f$. Fix a prime~$p$. Suppose $\chi:(\Z/\ell\Z)\to\mu_p$ is a Dirichlet character that satisfies several carefully chosen hypothesis, and let $K$ be the corresponding abelian extension. By chasing the appropriate diagrams and using results about \'etale cohomology and N\'eron models, I show that if $J=\Res_{K/\Q}(E)$ then $x$ is $p$-visible in $\Sha$ with respect to $J\to E$. This means that \[ E(\Q)/pE(\Q)\isom \Z/p\Z\subset \Sha(\ker(J\to E)), \] which is where the nonsquare part of $\Sha$ comes from. \begin{conjecture}[Stein]\label{conj:nonvanishchi} Assume~$p$ is an odd prime and the mod~$p$ Galois representation attached to $E$ is surjective. Then there are infinitely many Dirichlet characters $\chi$ as above that satisfies the following hypothesis: \[ L(E,\chi,1)\neq 0 \qquad\text{and}\qquad a_{\ell}(E) \not\con \ell+1 \pmod{p}. \] \end{conjecture} I have verified this conjecture numerically in thousands of examples, and I hope to prove something about it by assuming it is false and constructing many relations between modular symbols. Analytic methods involving averaging special values of $L$-functions seem incapable of handling twists of high degree. \begin{definition}[Modularity of Mordell-Weil]\label{defn:modmw} If $A$ is a modular abelian variety, and $n$ is an integer, we say that the Mordell-Weil group of $A$ is {\em $n$-modular of level $M$} if there is a quotient $\pi:J_1(M)\to A'$, with connected kernel, such that $A'$ is isogenous to $A$ and% $\pi(J_1(M))\subset n A'(\Q)$. We say that the Mordell-Weil group of $A$ is {\em modular} if it is $n$-modular for every integer~$n$. \end{definition} I think the following conjecture is within reach. \begin{conjecture}[Stein]\label{conj:mwallmodular} The Mordell-Weil group $A(\Q)$ of every modular abelian variety is itself modular, in the sense of Definition~\ref{defn:modmw}. \end{conjecture} This is closely related to Theorem~\ref{thm:allmwvis} since the restriction of scalars of a modular abelian variety is again modular. However it is unclear exactly how $A$, $\Res_{K/\Q}(A)$, and $J_1(M)$ all fit together, and it is essential to understand {\em exactly} how they fit together in order to verify the conjecture. Also, for a given $n$, it would be interesting to decide if the Mordell-Weil group is $n$-visible of level $M$ for some naturally defined~$M$. \subsection{Computing with modular abelian varieties} Bryan Birch once commented to me in reference to computation that ``It is always a good idea to try to prove true theorems.'' To this end, the author proposes to continue developing algorithms and making available tools for computing with modular forms, modular abelian varieties, and motives attached to modular forms. This includes finishing a major new {\sc Magma} \cite{magma} package for computing directly with modular abelian varieties over number fields, extending the Modular Forms Database \cite{mfd}, and searching for algorithms for computing the quantities appearing in the Birch and Swinnerton-Dyer for modular abelian varieties and the Bloch-Kato conjecture for modular motives. The results of this work should give an explicit picture of modular abelian varieties that could never be obtained from general theory. \subsubsection{The Modular forms database} The modular forms database \cite{mfd} contains a large collection of information about objects attached to cuspidal eigenforms. Though greatly appreciated by the many mathematicians who use it, the database currently only scratches the surface of what it should contain. The database is stored using the database system PostgreSQL, and I wrote the web user interface in Python. During Summer 2003 the Harvard undergraduate Dimitar Jetchev did extensive work on the database, and this pointed out significant deficiencies in how it is currently implemented. It is more difficult than it should be to modify the web interface to the database, the data is not compressed well, and there is no way to submit new data to the database using the web page. I intend to completely rewrite the database using MySQL and PHP, and investigate better algorithms for storing $q$-expansions of modular forms much more efficiently. Currently the limit on the database is not the difficulty of computing modular forms, but the space and time used in storing them. This could be partially remedied by moving the database to a more modern computer (it currently runs on a three year old Pentium III), something I am requesting in this grant. \subsubsection{Example database queries that have not yet been done} \begin{itemize} \item Suppose $d=2,3,4,5$, say. Using the algorithm described in \cite{agashe-stein:bsd}, compute a multiple of the order of the torsion subgroup of $A_f(\Q)$ for each $d$-dimensional $A_f$ in the database. What is the maximum number that occurs? After what level do no new numbers appear? For small $d$ such a computation may suggest a conjectural generalization to modular abelian varieties of Mazur's theorem on torsion points on elliptic curves.% \item Make a conjectural list of all number fields of degree~$d$ (for $d=2,3,4,5$, say) that arise as the field generated by the eigenvalues of a newform in the database. Coleman has conjectured that for each~$d$ only finitely many number fields of degree~$d$ appear. When do new $d$ seem to stop appearing?% \end{itemize} \subsubsection{M{\small AGMA} package for modular abelian varieties} I wrote the modular forms and modular symbols packages that are part of the {\sc Magma} computer algebra system \cite{magma}. I spent June 2003 in Sydney, Australia and did exciting work on designing and implementing a very general package for computing with modular abelian varieties over number fields. Much work is left to be done to finish this package, and several crucial algorithms still need to be developed or refined. When available this package will likely be greatly appreciated by anybody working with modular abelian varieties. The following are problems that arose in work on this package, which must be resolved in order to have a satisfactory system for computing with modular abelian varieties. I need to solve all of these problems. \begin{itemize} \item {\bf Endomorphism ring over $\Qbar$:} {\em Giving a modular abelian variety $A$, explicitly (and efficiently) compute the endomorphism ring $\End(A)$ as a ring of matrices acting $\H_1(A,\Z)$.} I have a modular symbols solution that draw on work of Ribet \cite{ribet:twistsendoalg} and Shimura \cite{shimura:factors} but is too slow to be really useful in practice; however, similar Manin symbols methods must exist and be very efficient.% \item {\bf Decomposition:} {\em Given the endomorphism ring of an abelian variety $A$, compute its decomposition as a product of simple abelian varieties.} This is a standard and difficult problem in general, but it might be possible to combine work of Allan Steel on a ``characteristic~$0$ Meataxe'' with special features of modular abelian varieties to solve it in practice. It is absolutely {\em essential} to solve this problem in order to explicitly enumerate all modular abelian varieties over number fields of given level.% \item {\bf Isomorphism testing:} {\em Given two modular abelian varieties over a number field $K$, represented as explicit quotients of Jacobians $J_1(N)$, decide whether there is an isomorphism between them.} I have solved this problem when the two modular abelian varieties are simple. There are analogues of this problem in other categories, which I intend to investigate.% \end{itemize} \subsection{Some conjectures that were inspired by my computations} \subsubsection{Questions about $J_1(p)$}\label{sec:j1pques} The following conjecture generalizes a famous conjecture of Ogg that $J_0(p)(\Q)_{\tor}$ is cyclic of order the numerator of $(p-1)/12$, a fact that Mazur proved in \cite{mazur:eisenstein}. \begin{conjecture}[Stein] The torsion subgroup of $J_1(p)(\Q)$ is exactly the group generated by the cusps on $X_1(p)$ that lie over $\infty\in X_0(p)$. This is a group of order \[ \frac{p}{2^{p-3}} \cdot\prod_{\eps\neq 1} B_{2,\eps} \] where the product is over the nontrivial even Dirichlet characters~$\varepsilon$ of conductor dividing~$p$, and $B_{2,\eps}$ is a generalized Bernoulli number. \end{conjecture} Mazur's complete proof of the analogue of this statement for $J_0(p)$ is quite deep, though the proof for the prime-to-$2$ part of $J_0(p)(\Q)_{\tor}$ is much easier. I hope to mimic Mazur's method and prove the conjecture above for the prime-to-$2$ part of $J_1(p)(\Q)_{\tor}$. More generally, I would like to investigate the torsion in quotients of $J_1(p)$. \begin{question} If $A_f$ is a quotient of $J_1(p)$ attached to a newform, is the natural map $J_1(p)(\Q)_{\tor}\to A_f(\Q)_{\tor}$ surjective? If so, is the product of the orders of all $A_f(\Q)_{\tor}$ over all classes of newforms $f$ equal to $\# J_1(p)(\Q)_{\tor}$? \end{question} I conjectured that the analogous questions for $J_0(p)$ should have ``yes'' answers, and in \cite{emerton:optimal} M.~Emerton subsequently proved this conjecture. It is still not clear if one should make this conjecture for $J_1(p)$. \subsubsection{Congruences between modular forms of prime level} Ken Ribet enticed me into studying modular forms as a graduate student by asking me the following question: ``Is there a prime $p$ so that $p$ is ramified in the Hecke algebra $\T$ attached to $S_2(\Gamma_0(p))$?'' I answered his question by showing that $p=389$ is the only prime less than $50000$ that ramifies in the associated Hecke algebra of level~$p$. The question remains whether $p=389$ is the only such example, and this seems extremely difficult to say anything useful about. However a related question exhibits a shockingly clear pattern, and this related question is the question Ribet was really interested in for his application to images of Galois representations \cite{ribet:torsion}. Let $p$ be a prime and $k$ a positive even integer. Let $\T$ be the Hecke algebra attached to $S_k(\Gamma_0(p))$ and let $d(k,p)$ be the valuation at $p$ of the index of $\T$ in its normalization. The following conjecture is backed up by significant numerical evidence, and was discovered by staring at tables and looking for a pattern. \begin{conjecture}[Stein, F.~Calegari] Suppose $p\geq k-1$. Then $$ d(k,p) = \left\lfloor\frac{p}{12}\right\rfloor\cdot \binom{m}{2} + a(p,m), $$ where the function $a$ only depends on $p$ modulo $12$ as follows: $$ a(p,m) = \begin{cases} 0 & \text{if $p\equiv 1\pmod{12}$,}\\ 3\cdot\displaystyle\binom{\lceil \frac{m}{3}\rceil}{2} & \text{if $p\equiv 5\pmod{12}$,}\\ 2\cdot\displaystyle\binom{\lceil \frac{m}{2}\rceil}{2} & \text{if $p\equiv 7\pmod{12}$,}\\ a(5,m)+a(7,m) & \text{if $p\equiv 11\pmod{12}$.} \end{cases} $$ \end{conjecture} The situation of interest to Ribet is $k=2$, in which case the conjecture simply asserts that $\T\otimes\Z_p$ is normal, i.e., {\em there or no congruences in characteristic $p$ between non-Galois conjugate newforms in $S_2(\Gamma_0(p))$.} Calegari has given a conjectural interpretation of some of the congruences that the conjecture asserts must exist, which I intend to study further. \section{Summary} This research proposal depicts an intricate network of ongoing investigations into the arithmetic of modular abelian varieties, which unite a theoretical and computational point of view. The basic foundations of visibility theory are nearly complete, but solutions to the questions about visibility outlined in this proposal demand a new level of precision in our understanding of the web of modular abelian varieties. I am determined to advance our understanding in this direction. My work has produced results and tools that are of use to other mathematicians who are exploring the world of modular forms. By supporting my research, you will assure the sustained development of this technology. Let $p$ be a prime and $k$ a positive even integer. Let $\T$ be the Hecke algebra attached to the space $S_k(\Gamma_0(p))$ of cusp forms for $\Gamma_0(p)$ and let $d(k,p)$ be the valuation at $p$ of the index of $\T$ in its normalization. The following conjecture is backed up by significant numerical evidence, and was discovered by staring at tables computed using the PIs {\sc Magma} code and looking for a pattern. \begin{conjecture}[Stein, F.~Calegari] Suppose $p\geq k-1$. Then $$ d(k,p) = \left\lfloor\frac{p}{12}\right\rfloor\cdot \binom{m}{2} + a(p,m), $$ where the function $a$ only depends on $p$ modulo $12$ as follows: $$ a(p,m) = \begin{cases} 0 & \text{if $p\equiv 1\pmod{12}$,}\\ 3\cdot\displaystyle\binom{\lceil \frac{m}{3}\rceil}{2} & \text{if $p\equiv 5\pmod{12}$,}\\ 2\cdot\displaystyle\binom{\lceil \frac{m}{2}\rceil}{2} & \text{if $p\equiv 7\pmod{12}$,}\\ a(5,m)+a(7,m) & \text{if $p\equiv 11\pmod{12}$.} \end{cases} $$ \end{conjecture} The conjecture is of interest to Ribet in the case $k=2$, because it is a hypotheses to the main argument of \cite{ribet:torsion}. For a long time it was unclear what to conjecture when $k=2$; finally, investigation into what happens at higher weight suggested the above conjectural formula, which specializes in weight $2$ to the assertion that $\T\otimes \Z_p$ is normal. The PI has no idea how to prove this conjecture when $k=2$, but intends to at least find similar conjectures when $\Gamma_0(p)$ is replaced by $\Gamma_1(p)$ and when~$p$ is replaced by a composite number. \newpage \newcommand{\etalchar}[1]{$^{#1}$} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } % \MRhref is called by the amsart/book/proc definition of \MR. \providecommand{\MRhref}[2]{% \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \begin{thebibliography}{BCDT01} \bibitem[AS]{agashe-stein:bsd} A.~Agashe and W.\thinspace{}A. Stein, \emph{Visible {E}vidence for the {B}irch and {S}winnerton-{D}yer {C}onjecture for {M}odular {A}belian {V}arieties of {A}nalytic {R}ank~$0$}, To appear in Mathematics of Computation. \bibitem[AS02]{agashe-stein:visibility} \bysame, \emph{Visibility of {S}hafarevich-{T}ate groups of abelian varieties}, J. Number Theory \textbf{97} (2002), no.~1, 171--185. \bibitem[BCDT01]{breuil-conrad-diamond-taylor} C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor, \emph{On the modularity of elliptic curves over {$\bold Q$}: wild 3-adic exercises}, J. Amer. Math. Soc. \textbf{14} (2001), no.~4, 843--939 (electronic). \bibitem[BCP97]{magma} W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}. {T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4, 235--265, Computational algebra and number theory (London, 1993). \bibitem[Bir71]{birch:bsd} B.\thinspace{}J. Birch, \emph{Elliptic curves over \protect{${\mathbf{Q}}$: {A}} progress report}, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, R.I., 1971, pp.~396--400. \bibitem[BS02]{buzzard-stein:artin} K.~Buzzard and W.\thinspace{}A. Stein, \emph{A mod five approach to modularity of icosahedral {G}alois representations}, Pacific J. Math. \textbf{203} (2002), no.~2, 265--282. \bibitem[BT99]{buzzard-taylor} K.~Buzzard and R.~Taylor, \emph{Companion forms and weight one forms}, Ann. of Math. (2) \textbf{149} (1999), no.~3, 905--919. \bibitem[CES03]{conrad-edixhoven-stein:j1p} B.~Conrad, S.~Edixhoven, and W.\thinspace{}A. Stein, \emph{${J}_1(p)$ {H}as {C}onnected {F}ibers}, To appear in Documenta Mathematica (2003). \bibitem[CM00]{cremona-mazur} J.\thinspace{}E. Cremona and B.~Mazur, \emph{Visualizing elements in the {S}hafarevich-{T}ate group}, Experiment. Math. \textbf{9} (2000), no.~1, 13--28. \bibitem[Col03]{coleman-stein:padicapprox} R.~Coleman, \emph{Approximation of infinite-slope modular eigenforms by finite-slope eigenforms}, To appear in the Dwork Proceedings (2003). \bibitem[CS02]{stein:compgroup} B.~Conrad and W.\thinspace{}A. 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