\documentclass[11pt]{article} \usepackage{url} \newcommand{\thisdocument}{Project Description} \include{macros} \usepackage{natbib} \setlength{\bibsep}{.9ex} \bibpunct{[}{]}{,}{a}{}{;} \begin{document} %\tableofcontents %\newpage \begin{center} \Large\bf Explicit Approaches to the Birch and Swinnerton-Dyer Conjecture \end{center} \section{Introduction} My research reflects the rewarding interplay of theory with explicit computation in number theory, as illustrated by Bryan Birch~\cite{birch:bsd}: \begin{quote} I want to describe some computations undertaken by myself and Swin-nerton-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} The goal of this proposal is to carry out a wide range of computational and theoretical investigations on elliptic curves and abelian varieties motivated by the Birch and Swinnerton-Dyer conjecture (BSD conjecture). This will hopefully improve our practical computational capabilities, extend the data that researchers have available for formulating conjectures, and deepen our understanding of theorems about the BSD conjecture. The PI is one of the more sought after people by the worldwide community of number theorists, for computational confirmation of conjectures, for modular forms algorithms, for data, and for ways of formulating problems so as to make them more accessible to algorithms. The PI has also been successful at involving numerous undergraduate and graduate students at all levels in his research. \subsection{Prior Support} The PI was partly supported by NSF postdoctoral fellowship during 2000--2004 (DMS-0071576) in the amount of \$90,000. The PI was also awarded NSF grant DMS-0555776 (and DMS-0400386) from the ANTC program in the amount of \$177,917 for the period 2004--2007. The PI's work under DMS-0555776 succeeded at improving the modular forms database and resulted in numerous papers on the arithmetic of elliptic curves, modular forms and abelian varieties \cite{stein:vismw,bsdalg1,mazur-stein-tate:padic, stein-watkins:ns, jetchev-stein:higher, agashe-stein:bsd, agashe-ribet-stein:manin, agashe-ribet-stein:modular, stein:bsdmagma}, one completed book \cite{stein:modform} on computing with modular forms, and progress on another book on number theory \cite{stein:ent}. It has also led to a new software initiative (see Section~\ref{sec:sage} below). Funds from DMS-0555776 were used to run a successful workshop at UCSD and to purchase a 16-processor compute server with 64GB RAM. \subsection{The BSD Conjecture} The Birch and Swinnerton-Dyer conjecture is a central problems in number theory, and this proposal is based on a group of ideas related to this conjecture. An {\em elliptic curve} is a projective genus one curve with a distinguished rational point. Every such curve is the projective closure of a nonsingular affine curve given by $ y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6. $ An {\em abelian variety} is a projective group variety---the one dimensional abelian varieties are exactly the elliptic curves. \begin{conjecture}[BSD Conjecture]\label{conj:bsd} Let $A$ be an abelian variety over~$\Q$. (The objects and notation in the formula are discussed below.) \begin{enumerate}% \item The rank $r$ of $A(\Q)$ equals $\ord_{s=1}L(A,s)$.% \item We have $$ \frac{L^{(r)}(A,1)}{r!} =% \frac{\#\Sha(A) \cdot \Omega_{A} \cdot \Reg_A}% {\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}} \cdot {\displaystyle \prod_{\ell\mid N} c_\ell}. $$ \end{enumerate} \end{conjecture} In the conjecture, $L(A,s)$ is the Hasse-Weil $L$-series of $A$. The real volume $\Omega_{A}$ is the measure of $A(\R)$ with respect to a basis of differentials for the N\'eron model of $A$. For each prime $\ell\mid N$, the integer $c_\ell=\#\Phi_{A,\ell}(\F_\ell)$ is the {\em Tamagawa number} of~$A$ at~$\ell$, where $\Phi_{A,\ell}$ denotes the component group of the N\'eron model of~$A$ at $\ell$. The abelian variety dual of $A$ is denoted $A^{\vee}$, and in the conjecture $A(\Q)_{\tor}$ and $A^{\vee}(\Q)_{\tor}$ are the torsion subgroups. The {\em Shafarevich-Tate group} of $A$ is \[ \Sha(A) = \Ker\left(\H^1(\Q,A) \to \bigoplus_{p\leq \infty} \H^1(\Q_p,A)\right), \] which is a group that measures the failure of a local-to-global principle. It is implicit in the statement of the conjecture that $\Sha(A)$ is finite, though this is only known in some cases. %when $L(A,1)\neq 0$ and in some cases when $L(A,1)=0$ %(see \cite{kato:secret} and \cite{kolyvagin-logachev:finiteness}). The regulator $\Reg_A$ is the absolute value of the discriminant of the N\'eron-Tate canonical height pairing $A(\Q)_{/\tor} \times A(\Q)_{/\tor}\to \R$. If $A$ is an elliptic curve then $\#A(\Q)_{\tor}$, $\# A^{\vee}(\Q)_{\tor}$, $\Omega_A$, and $c_\ell$ are relatively easy to compute; none of the other quantities are known to be computable in general, even when $A$ is an elliptic curve, though many can in practice be computed. \begin{conjecture}[$\BSD(A,p)$]\label{conj:bsdp} Let $A$ be an abelian variety over $\Q$ of rank $r$ and let $p$ be a prime. Then $$ \ord_p\left(\frac{L^{(r)}(A,1)}{r!\cdot \Reg_A \cdot \Omega_A}\right) =% \ord_p\left(\frac{\#\Sha(A)}% {\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}} \cdot {\displaystyle \prod_{\ell\mid N} c_\ell}\right). $$ \end{conjecture} In Conjecture~\ref{conj:bsdp} the fraction on the left side is not known to be a rational number (except when $r\leq 1$), so its rationality is part of the conjecture. Tate \cite{tate:bsd} formulated the BSD conjecture for any abelian variety over a global field~$K$, and proved (generalizing work of Cassels) that if $A$ and $B$ are related by an isogeny, then $\BSD(A,p)$ is true if and only if $\BSD(B,p)$ is true. \subsection{Modular Abelian Varieties} Modular abelian varieties are a special class of abelian varieties over $\Q$ that have been studied intensively. Computation with modular abelian varieties is attractive because they are easier to describe than arbitrary abelian varieties, have extra hidden structure, and their $L$-functions are reasonably well understood. We recall Shimura's construction \cite{shimura:factors} of modular abelian varieties. Let~$f=\sum a_n q^n$ be a weight~$2$ newform on $\Gamma_1(N)$. Then~$f$ corresponds to 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 kernel of the homomorphism $\T\to \Z[a_1,a_2,a_3, \ldots]$ that sends $T_n$ to $a_n$, and attach to~$f$ the quotient $$A_f=J_1(N)/I_f J_1(N).$$ Then $A_f$ is a simple 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~$f$. We also sometimes consider a similar construction with $J_1(N)$ replaced by the Jacobian $J_0(N)$ of the modular curve $X_0(N)$ that parametrizes isomorphism classes of pairs $(E,C)$, where $C$ is a cyclic subgroup of $E$ of order~$N$. \begin{definition}[Modular abelian variety] An abelian variety over a number field is a {\em modular abelian varieties} if it is a quotient of $J_1(N)$ for some~$N$. \end{definition} %Newform abelian varieties are simple (but not necessarily %absolutely simple), and every modular %abelian variety is isogenous to a product of copies of newform %abelian varieties. Newform abelian varieties need not be %absolutely simple. Algorithms for computing many of the invariants in the BSD conjecture for modular abelian varieties has been a major part of the PI's research program. The celebrated modularity theorem of C.~Breuil, B.~Conrad, F.~Diamond, R.~Taylor, and A.~Wiles \cite{breuil-conrad-diamond-taylor} asserts that every elliptic curve over~$\Q$ is a modular abelian variety. Also, it is now known (due to very recent work of Khare, Wintenberger, and Dieulefait) that every abelian variety of $\GL_2$-type (see \cite{ribet:abvars}) and odd conductor is a modular abelian variety. The PI recently completed a book on computing with modular forms \cite{stein:modform} that will be published by the AMS. He is working on a graduate textbook with Ken Ribet on modular abelian varieties, and an undergraduate text on number theory, both intended for publication by Springer-Verlag. He led a 2-week high-school student workshop on the Birch and Swinnerton-Dyer conjecture (see \cite{simuw}). \subsection{Software for Algebra and Geometry Experimentation}\label{sec:sage} The PI is the principal author of SAGE---Software for Algebra and Geometry Experimentation (see \cite{sage}). Substantial work on SAGE has been done jointly with students (7 undergraduates and 5 graduate students). The goal of SAGE is to create an optimal {\em open source} software environment for research in algebra, geometry, number theory, and related areas. The PI intends to make all data and algorithms developed as part of the proposed research freely available online and from SAGE. The PI is the author of the modular forms and modular abelian varieties components of Magma \cite{magma}. When possible many of the proposed computations will be independently verified using Magma. When we describe a result that relies on computation below, there is an implicit assumption that certain software produced correct output. Also, the ranges of computations, e.g., ``all curves of conductor up to 1000'', are in many cases arbitrary. Our {\em primary} motivation for doing these computations is to motivate the development of new conjectures and computational and theoretical tools. \section{The BSD Formula: Computing $\Sha$} Much of this research proposal is about computing $\Sha$, which is the most difficult to compute invariant appearing in the BSD conjecture. \subsection{Applying Theorems of Kato and Kolyvagin} The PI, 3 undergraduates and a graduate student proved the following in \cite{bsdalg1}: \begin{theorem}[Stein et al.]\label{thm:main} Suppose that $E$ is a non-CM elliptic curve of rank $\leq 1$, conductor $\leq 1000$ and that $p$ is a prime. If $p$ is odd, assume further that the mod~$p$ representation $\rhobar_{E,p}$ is irreducible and~$p$ does not divide any Tamagawa number of~$E$. Then $\BSD(E,p)$ is true. \end{theorem} The proof involves an application of results of Kato and Kolyvagin, new refinements of Kolyvagin's theorem, explicit 2-descent and 3-descent and much explicit calculation. This is a first step toward the following goals: \begin{goal}\label{prob:bsd01} Verify the BSD Conjecture for every elliptic curve over $\Q$ of conductor $<1000$, except for the $18$ curves of rank $2$. \end{goal} \begin{goal}\label{prob:bsd2} For each curve $E$ over $\Q$ of conductor $<1000$ and rank $2$, prove that $\Sha(E)[p]=0$ for all $p<1000$. \end{goal} We hope to make further progress toward Goal~\ref{prob:bsd2} using $p$-adic methods (see Section~\ref{sec:padic} below), since unconditional computation of $\Sel^{(p)}(E/\Q)$ directly using standard algebraic number theory techniques for $p>5$ appears to not be practical. Another approach is to improve on work of Kolyvagin---Theorem~\ref{thm:main} excludes divisors of Tamagawa numbers because Kolyvagin's theorem is not sufficiently precise at such primes. \begin{goal}\label{kolybsd} Refine Kolyvagin's bound when a prime $p$ divides a Tamagawa number. \end{goal} Dimitar Jetchev (who began working with the PI as an undergraduate) has been working on Goal~\ref{kolybsd} in consultation with the PI. %So far he has proved the following, which will allow %the PI to improve Theorem~\ref{thm:main}. Let~$E$ be an elliptic curve over~$\Q$, let~$K$ be a field that satisfies the {\em Heegner hypothesis} for~$E$, i.e., such that each prime dividing $N$ splits in $K$. Assume that~$E$ has analytic rank~$1$ over~$K$, and let $y_K\in E(K)$ be the Heegner point. Suppose $p\geq 5$ and that $\rhobar_{E,p}$ is surjective. Jetchev has made great progress toward the following: \begin{goal}\label{goal:jetchev} If $\ord_p(\prod c_\ell)\geq 1$, where the $c_\ell$ are the Tamagawa numbers of $E$, prove that $ \ord_p(\#\Sha(E/K)) \leq \ord_p([E(K): \Z y_K]) - 2. $ \end{goal} Kolyvagin gives a formula (see, e.g., \cite{kolyvagin:structure_of_selmer,gross:kolyvagin,mccallum:kolyvagin}) for $\#\Sha(E/K)$ which involves global divisibility of Heegner points and Jetchev links those global divisibility exponents to Tamagawa numbers. Work toward Goal~\ref{goal:jetchev} uses Poitou-Tate global duality and the Chebotarev density theorem to show that mod $p$ there are no nontrivial Kolyvagin systems; this is equivalent to showing that all Heegner points $P_n$ on $E$ over ring class fields are globally divisible by $p$. \subsection{Complex Multiplication Curves} A {\em complex multiplication} (CM) elliptic curve $E$ over a number field is one such that $\End(E_{\C})\neq \Z$. The PI and Aron Lum proved the following result: \begin{theorem}[Stein, Lum]\label{thm:cm} Suppose $E$ is a CM elliptic curve over $\Q$ with rank at most~$1$ and conductor at most $5000$. Then $\BSD(E,p)$ is true for all primes $p\geq 5$ of good reduction for $E$. \end{theorem} This is an application of Rubin \cite{rubin:main-conjectures} for rank 0 curves and Kolyvagin \cite[Cor.~D]{kolyvagin:euler_systems} for rank 1 curves. The PI to compute to much higher conductor. %A number of interesting issues arise in translating %the theoretical results of \cite{blah} into practical %algorithms. \begin{goal}\label{prob:bsdcmbad} Suppose $E$ is a complex multiplication elliptic curve over $\Q$ and that $p\geq 5$ is a prime of bad reduction for $E$. Find and implement a {\em practical} algorithm to verify $\BSD(E,p)$, then apply it to all $E$ of conductor up to $5000$. \end{goal} Goal~\ref{prob:bsdcmbad} is difficult mainly because the mod $p$ representation is not ``as surjective as possible'', so the methods used for Theorem~\ref{thm:cm} do not apply. One approach to Goal~\ref{prob:bsdcmbad} is to try do computations using Theorem~\ref{thm:schneider} below. \subsection{$p$-adic Methods}\label{sec:padic} Suppose $E$ is an elliptic curve over a number field $K$ and $p\geq 5$ is a prime (of $\mathcal{O}_K$) of good ordinary reduction for $E$. In \cite{mazur-stein-tate:padic} the PI, Mazur, and Tate give a new approach to computing $p$-adic heights $$h_p:E(K) \to \Q_p$$ that leverages Kedlaya's fast algorithm \cite{kedlaya:counting_mw, kedlaya:mw2} for explicit computation of Monsky-Washnitzer cohomology groups. We made our algorithm explicit only when $K=\Q$ or $K$ is a quadratic imaginary field, though the key ideas for creating a general explicit algorithm are given in \cite{mazur-stein-tate:padic}. The PI, David Harvey, Jen Balakrishnan, and Liang Xiao have implemented and optimized this algorithm over $\Q$ and quadratic imaginary $K$ in both SAGE and Magma (this was a major project at an MSRI graduate student workshop that the PI ran in August 2006). \begin{goal} Design and implement a general algorithm for computing $p$-adic heights (for all primes $p$) on elliptic curves over arbitrary number fields. \end{goal} The natural height pairing here is not very well understood; indeed, it still only conjectural that is nondegenerate. It can be verified in particular cases via computation: \begin{conjecture}[Schneider] Suppose $E$ is an elliptic curve over $\Q$ and $p$ is a prime of good ordinary reduction. Then the $p$-adic cyclotomic height pairing on $E(\Q)$ is nondegenerate. \end{conjecture} There are very few general results toward this conjecture (except in the CM case). It can be verified in particular cases: \begin{goal}\label{prob:padicreg} Create a table of $p$-adic regulators to precision $O(p^{5})$ for every elliptic curve of conductor up to $120000$ and the first five primes $p\geq 5$ of good ordinary reduction for $E$. %In particular, %verify for five primes $p$ the nondegeneracy of the $p$-adic height %pairing. \end{goal} The data from Goal~\ref{prob:padicreg} will also be of interest in investigations about congruences between algebraic parts of $p$-adic $L$-functions. %Computation of $p$-adic heights is unclear when %$p=2,3$, so we propose: %\begin{goal} %Find and implement an algorithm for computing $p$-adic %heights when $p=2,3$ is prime of good ordinary reduction. %\end{goal} \begin{theorem}[Schneider \cite{schneider:iwasawa} and Perrin-Riou]\label{thm:schneider} Suppose $p$ is an odd prime of good ordinary reduction for $E$, and let $\Reg_E^{(p)}\in \Q_p$ be the $p$-adic regulator of $E$, i.e., the discriminant of the $p$-adic height pairing on $E(\Q)$. If the $p$-primary part $\Sha(E/\Q)(p)$ of $\Sha(E/\Q)$ is finite, then the leading term of the algebraic $p$-adic $L$-function of $E$ has the same $p$-adic valuation as $$ \frac{\#\Sha(E) \cdot \Reg_E^{(p)}}% {\#E(\Q)_{\tor}^2} \cdot \#E(\F_p)^2 \cdot \prod_{\ell \mid N} c_\ell. $$ \end{theorem} %As explained in Section~\ref{sec:padic}, we %can compute $p$-adic regulators quickly. The following is needed in order to apply Theorem~\ref{thm:schneider}: \begin{goal}\label{goal:padicl} Create software and better algorithms for computing with $p$-adic $L$-functions of elliptic curves and modular abelian varieties. \end{goal} Much work toward Goal~\ref{goal:padicl} in Magma has already been done by the PI, Robert Pollack, and Christian Wuthrich. Substantial work on the main conjecture of Iwasawa theory connects the algebraic and analytic (computable) $L$-functions. \begin{remark} There are other cases where we can apply analogues of Theorem~\ref{thm:schneider}, e.g., when $p$ is a prime of good supersingular reduction (see \cite{pr:exp}). % See \cite{stein-wuthrich} %for a survey of such results in the context of computing $\#\Sha$. \end{remark} \subsection{Constructing Nonzero Elements of $\Sha(E)$} There are 6581 optimal curves in \cite{cremona:onlinetables} of conductor up to $120000$ for which $p$ divides the BSD conjectural order of $\Sha(E)$ for some $p\geq 3$. Of these, 1387 have conjectural order divisible by a prime $p\geq 5$ and $339$ have conjectural order divisible by a prime $p\geq 7$. Of these $1387$ curves, the mod~$p$ representation is surjective except in $11$ cases. %33825o1 25 [1] %40455k1 25 [1] %52094m1 25 [1] %59643a1 25 CM %61446de1 49 [1] %62400dt1 25 [1] %82418u1 25 [1] %97104cp1 25 [1] %108489m1 25 [1] %119025ck1 25 [-1] %119025a1 25 CM \begin{goal}\label{prob:constructsha} For the $1387$ curves of conductor up to $120000$ for which a prime $p\geq 5$ divides the conjectured $\#\Sha(E)$, construct an element of order~$p$ in $\Sha(E)$. \end{goal} One approach to Goal~\ref{prob:constructsha} for small $p$, e.g., $p=5$, is to construct $\Sha(E)$ using visibility, i.e., by finding an elliptic curve $F$ of rank $2$ such that $E[p]\isom F[p]$, and using the visibility techniques of \cite{cremona-mazur, agashe-stein:visibility, agashe-stein:bsd}. A second approach, which works in many cases (because the mod~$p$ representation is often surjective), is to use results of \cite{grigor:phd}, which gives an explicit criterion in terms of modular symbols to construct nonzero elements of $\Sha(E)[p]$. Initial computations of the PI and Grigorov (discussed in \cite{grigor:phd}) suggest that this approach will succeed in many cases. A third approach is to use Theorem~\ref{thm:schneider} and explicit calculation of $p$-adic regulators and $p$-adic $L$-functions to at least prove that $\Sha(E)(p)$ has the conjectured order. This approach doesn't give an explicit construction of $\Sha(E)(p)$. A fourth approach is to explicitly compute Kolyvagin cohomology classes $c_{n,p} \in \H^1(K,E)[p]$ and show that they are in fact elements of $\Sha(E)[p]$. Jetchev, Lauter, and the author have done such a computation in a few cases (unpublished). Recent exciting work of Skinner and Urban also addresses the question of providing very general explicit lower bounds on $\#\Sha(E)$. \subsection{Tamagawa numbers} The following is a consequence of the BSD conjecture. \begin{conjecture}\label{conj:tamprod} Suppose $E$ is an elliptic curve and that $p\geq 5$ is odd prime such that $\rhobar_{E,p}$ is irreducible. Then $$ \ord_p\left(\prod_{\ell\mid N} c_\ell\right) \leq \ord_{p}\left(\frac{L(E,1)}{\Omega_E}\right) $$ \end{conjecture} Let $f = f_E$ be the newform corresponding to $E$. Under the hypotheses of Conjecture~\ref{conj:tamprod}, \cite{ribet:lowering} implies that there is a newform $g$ of level a proper divisor of the conductor of $E$ whose coefficients (of index coprime to the conductor) are congruent to the coefficients of $f$. The PI showed in unpublished work how to use this congruence, in some cases, to prove that $L(E,1)/\Omega_E \con 0\pmod{p}$. The PI proposes to write up the details of this argument (jointly with Jetchev), then attempt to refine the argument to yield the congruence of Conjecture~\ref{conj:tamprod}. The PI also hopes to find connections between the components group of $E$ and the $\T$-module (defined using modular symbols) whose order is the $p$-part of $L(E,1)/\Omega_E$. This approach gives a conceptual explanation of part of the BSD conjecture. \section{The Rank} Let $E$ be an elliptic curve over $\Q$. If $r_{\an} = \ord_{s=1} L(E,s) \leq 1$, then the rank part of the BSD conjecture is known for $E$; moreover, in principle, and often in practice (as explained elsewhere in this proposal) one can verify the full BSD conjecture. There isn't a single $E$ with $r_{\an}\geq 2$ for which the PI is aware of even a plausible strategy for proving that $\Sha(E)$ is finite, let alone verifying the full BSD conjecture---it is even unknown that $L''(E,1)/(\Omega_E\cdot \Reg_E)\in\Q$ for any~$E$. One can verify in particular cases the rank part of the BSD conjecture if $r_{\an} \leq 3$; the PI is aware of no strategy to verify the rank statement for even a single example when $r_{\an}\geq 4$. Morever, if $E$ has analytic rank~$2$ or larger, then the PI is aware of no conjectural construction of $E(\Q)$ analogous to that of Gross-Zagier in the rank~$1$ case. Thus new ideas are needed when $r_{\an}\geq 2$, and the PI hopes the computations he proposes might play a role in finding them. \subsection{Kolyvagin's Cohomology Classes}\label{sec:kolyclass} Let $E$ be an elliptic curve over $\Q$ with conductor $N$, and to simplify the discussion assume that $E$ does not have complex multiplication (there are analogues of everything below even if $E$ has CM). Let $K$ be a quadratic imaginary field that satisfies the Heegner hypothesis for $E$. Fix a prime $p$ such that $E[p]$ is irreducible (we could also replace $p$ by an integer $n$ and remove the requirement that the representation be irreducible). Kolyvagin defined (see, e.g., \cite{gross:kolyvagin}) classes $c_{n, p} \in \H^1(K,E[n])$ for infinitely many squarefree integers $n$ satisfying a Chebotarev condition. Let $d_{n,p}$ be the image in $\H^1(K,E)[p]$ of $c_{n,p}$. Kolyvagin proved that $\res_{v}(d_{n,p})=0$ for all $v\nmid n$ and computed the order of $\res_{\ell}(d_{n,p})$ in terms of local properties of the point $y_K$. He used these classes to prove his celebrated results toward the BSD conjecture. Dimitar Jetchev, Kristin Lauter and the PI have solved the following problem in a handful of cases, and intend to continue working on refining our methods. \begin{goal} Find and implement a practical algorithm to compute the order of $c_{n,p}$ and the order of its image $d_{n,p}$. \end{goal} For simplicity, assume that $n=\ell$ is a prime. By the modularity theorem there is a surjective homomorphism $\pi: X_0(N)\to E$. The two degeneracy maps $X_0(N\ell)\to X_0(N)$ induce maps $\delta_1$ and $\delta_\ell$ from $J_0(N)$ to $J_0(N\ell)$. We say that $d_{\ell,p}$ is {\em visible of level $N\ell$} if $d_{\ell, p}$ maps to $0$ under the map on cohomology induced by $$ E \xrightarrow{\,\,\,\,\pi^*\,\,} J_0(N) \xrightarrow{\,\,\,\,(\delta_1 \pm \delta_\ell)^*\,\,} J_0(N\ell) $$ for either choice of sign. \begin{goal}\label{prob:kolyvisalg} Find and implement a practical algorithm to determine whether or not a Kolyvagin class $d_{\ell,p}$ is visible at level $N\ell$. \end{goal} \begin{goal} Based on the data obtained from Goal~\ref{prob:kolyvisalg} formulate a conjecture about visibility of the classes $d_{\ell,p}$. \end{goal} %Carry out large-scale verification of Kolyvagin's conjecture %(data gathering problem). Suppose $E$ is an elliptic curve over $\Q$ with analytic rank $\geq 2$. %, so by \cite{gross-zagier} the Heegner point $y_K$ is torsion. Fix an odd prime~$p$ such that $E[p]$ is irreducible. Then the Kolyvagin classes $d_{n,p}$ are elements of $\Sha(E)[p]$ and the classes $c_{n,p}$ lie in $\Sel^{(p)}(E)$. \begin{conjecture}[Kolyvagin] Let $E$ and $p$ be as above. Then $\Sel^{(p)}(E)$ is generated by the classes $c_{n,p}$. \end{conjecture} The PI intends to develop a theory for computing with the subgroup of $\Sel^{(p)}(E)$ generated by a given finite collection of classes $c_{n,p}$. Kolyvagin hints at doing this in \cite[Pg.~120]{kolyvagin:structureofsha}. For example, let $E$ be the rank $2$ elliptic curve $y^2 + y = x^3 + x^2 - 2x$ of conductor $389$. The PI, Jetchev, and Lauter computed the class $c_{5,3}$ and showed that it defines a nonzero element of $\Sel^{(3)}(E)$---this computation did not require computing $E(\Q)$. This computation thus gives an explicit construction using Heegner points of the image in $\Sel^{(3)}(E)$ of a nonzero element of the rank~$2$ Mordell-Weil group $E(\Q)$. Making this computation practical has already involved interesting techniques (e.g., using $p$-adic methods to verify global non-divisibility of Heegner points). Moreover, we are computationally verifying Kolyvagin's conjecture, which has interesting consequences (unpublished work of Cornut, Nekovar). \subsection{Mazur and Rubin's Shadow Lines} When~$E$ is an elliptic curve over $\Q$ of rank~$2$ there is a construction of Mazur and Rubin, using $p$-adic heights, that attaches to appropriate quadratic imaginary fields $K$ a certain line in the $p$-adic completion of the Mordell-Weil group; these are sometimes referred to as ``shadow lines''. Conjecturally, for every quadratic imaginary field~$K$ such that the Mordell-Weil group of the twist of~$E$ by the quadratic character of~$K$ has rank one (and such that the discriminant of~$K$ is prime to the conductor of~$E$) we have such a ``shadow line''. On the one hand the shadow line is the image of universal norms in the $p$-adic completion $E(\Q)\tensor\Z_p$ of the Mordell-Weil group of the elliptic curve over layers of the $p$-adic anticyclotomic tower attached to $K$, and on the other hand the shadow line is the null space of the $p$-adic anti-cyclotomic height pairing. \begin{goal}\label{prob:shadow1} Gather extensive numerical data about Mazur-Rubin shadow lines. In particular, are they uniformly distributed as we vary $K$? \end{goal} The PI has done computations in this direction using slow methods (see \cite{mazur-rubin:pairings_arith}). More recently, he and Liang Xiao have been applying the methods of Section~\ref{sec:padic}. %\begin{goal} %Study connections between shadow lines and %the Kolyvagin cohomology classes $d_{n,p}$, which we can %explicitly compute as in Section~\ref{sec:kolyclass}. %\end{goal} \section{Databases} The modular forms database (see \cite{mfd}) is a freely-available collection of data about objects attached to modular forms. It is analogous to Neil Sloane's tables of integer sequences, and generalizes John Cremona's tables of elliptic curves \cite{cremona:onlinetables} to dimension bigger than one and weight bigger than two. The database is used by many prominent number theorists. The PI proposes to greatly expand the databases with more information about modular forms, elliptic curves, and modular abelian varieties. In each enumeration problem below, we intend to store the result of the computation in two forms: \begin{enumerate} \item A plain text file that can be easily parsed, similar to \cite{cremona:onlinetables}. \item SAGE provides robust support for saving nearly arbitrary individual objects. Since SAGE is free and every version of SAGE is archived in multiple locations, this data will not be lost because of changes to SAGE. \end{enumerate} \subsection{Elliptic Curves} The elliptic curve over $\Q$ with rank $4$ and smallest known conductor is the curve $ y^2 + xy = x^3 - x^2 - 79x + 289, $ with conductor $234446 = 2 \cdot 117223$. Nobody knows if there is a curve with smaller conductor and rank $4$, though the PI showed that any such curve has composite conductor by finding all curves of prime conductor up to $234446$ (see \cite{rank4}). \begin{goal}\label{prob:enum234446} Enumerate every elliptic curve up to isogeny of conductor up to 234446. \end{goal} Cremona \cite{cremona:onlinetables} has spent years methodically enumerating all curves of conductor up to $130000$ and computing the invariants in the BSD conjecture (except $\#\Sha$). The PI believes an independent computation whose goal is just to find all curves would be extremely valuable, as the following illustrates: \begin{quote} Bad news: Bill Allombert found an elliptic curve not in my database, conductor $97200$ (it was $[0,0,0,0,15]$ !!!!) and I just found that there are $40$ conductors in the range 90k--100k where there are curves in Stein-Watkins and not in Cremona. Misery! \\ \mbox{}\hfill -- email to me from John Cremona, 2006-09-07 \end{quote} The Stein-Watkins table \cite{stein-watkins:ants5, bmsw:bulletins} mentioned above is a massive table by the PI and Mark Watkins of over 100 million elliptic curves, which was made by systematically enumerating Weierstrass equations. Though it does not contain every curve of a given conductor, a substantial fraction are there. Cremona enumerates elliptic curves by computing the matrix of the Hecke operator $T_2$ on the space of weight $2$ modular symbols for $\Gamma_0(N)$, then finding the kernels of $T_2-2$, $T_2-1$, $T_2$, $T_2+1$, $T_2+2$. Next, he computes the Hecke operator $T_3$ restricted to each of the kernels, and decomposes those kernels under $T_3$, and likewise for $T_5$, etc. The crucial point is that for the purposes of finding elliptic curves it is not necessary to compute the minimal polynomial of $T_2$, which is extremely difficult when $N$ is large, e.g., if $N=200003$ then this dimension is $16667$. Another key fact is that Cremona computes Hecke operators using modular symbols, which yield dense matrices, so the linear algebra is difficult. %goal would be to find the curves up to isogeny, %and not necessary the equations, finding even just the Hecke eigenvalues %(using modular symbols, the method of graphs, or quaternion algebras) %would be enough. \begin{goal}\label{prob:enum_mestre} Use the Mestre method of graphs \cite{mestre:graphs} to find all elliptic curves of conductor $N\leq 234446$, for all integers $N$ that are either prime or of the form $pM$ with $p$ prime and $M\leq 10$ or $M=12,13,16,18$. \end{goal} The method of graphs, when applicable, very quickly produces {\em extremely sparse matrices} of the Hecke operators $T_p$ on $S_2(\Gamma_0(N))_{p-\new}$ for small $p$ (e.g., $p=2,3,5,7,11$, say). Sparse linear algebra then yields an upper bound on the number of isogeny classes of elliptic curves of conductor $N$. Consulting the tables of Stein-Watkins and Cremona yields a lower bound. If these disagree, it is likely that we have found a new curve; we then prove this by sparse linear algebra computations over $\Q$, which we do via a multimodular algorithm (see \cite[Ch.~7]{stein:modform}). Ifti Burhanuddin did partial work toward Goal~\ref{prob:enum_mestre} at the MSRI summer graduate student workshop that the PI organized. David Kohel, Lassina Dembele, and the PI have been working on a strategy to carry out the following computation: \begin{goal}\label{prob:enum_quat} Use quaternion algebra arithmetic over~$\Q$ (much as in \cite{dembele:hilbert5}) to compute sparse matrices for several Hecke operators, then proceed as above to find all curves of conductor $\le 234446$ for which some prime exactly divides~$N$. \end{goal} Computing the quaternion algebra presentation and the Hecke operators in the first place can be time consuming---however, for large levels linear algebra with matrices of Hecke operators dominates the runtime, so obtaining sparse matrices using quaternion algebras or the method of graphs is extremely helpful. There are only $960$ integers up to $234446$ that do not satisfy the conditions of Goal~\ref{prob:enum_quat} (which is about 0.4\%). For these remaining conductors we might use modular symbols to compute all rational eigenforms: \begin{goal}\label{enum:modsym} Compute using modular symbols all elliptic curves of the $960$ conductors up to $234446$ not covered by Goal~\ref{prob:enum_quat}. \end{goal} %We emphasize that in Goals~\ref{prob:enum_mestre}--\ref{enum:modsym} %we are concerned with finding %data that determines all curves of given conductor. These calculations are easy to parallelize, and the linear algebra involved in some cases requires a huge amount of memory (which is one reason the PI is requesting a memory upgrade to 128GB for his main server). \subsection{Modular Abelian Varieties} In each case the elliptic curves enumeration in Goals~\ref{prob:enum_mestre}--\ref{enum:modsym} starts by computing matrices of Hecke operators. The PI will store these matrices and also use them to enumerate modular abelian varieties (and, equivalently, newforms in $S_2(\Gamma_0(N))$). The linear algebra involved in searching for higher-degree factors is much more prohibitive than in searching for elliptic curves. \begin{goal}\label{prob:j0n} Compute every quotient $A_f$ of $J_0(N)$ for all $N \leq 10000$. \end{goal} Note, for example, that $\dim S_2(\Gamma_0(10000)) = 1411$, so computing and factoring the relevant minimal polynomials should be possible. \begin{goal}\label{prob:j1n} Compute every quotient $A_f$ of $J_1(N)$ for all $N\leq 1000$. \end{goal} For Goal~\ref{prob:j1n}, we directly compute the spaces $S_2(\Gamma_1(N), \eps)$ for each Dirichlet character $\eps$ of modulus $N$. The main difficulty is linear algebra over large cyclotomic fields. Recent work in progress of the PI on algorithms for multimodular linear algebra over cyclotomic fields will be very helpful (see \cite[Ch.~7]{stein:modform}). \newpage \bibliographystyle{amsalpha} \addtolength{\itemsep}{-3pt} \bibliography{biblio} \end{document}