\documentclass[11pt]{article}1\usepackage{amsmath}2\usepackage{amsthm}3\usepackage{amssymb}4\pagestyle{myheadings}5\markboth{William A. Stein}{William A. Stein}6%\baselineskip=12pt7%\lineskiplimit=3pt8%\lineskip=3pt minus 1pt9\textwidth=6.4in10%\hoffset=-0.75in11\hoffset=-0.5in12%\textheight=9.2in13\textheight=9.25in14\voffset=-.3in1516\newcommand{\Adual}{A^{\vee}}17\newcommand{\isom}{\cong}18%\newcommand{\ncisom}{\approx}19%\newcommand{\mysection}[1]{\vspace{12pt}\noindent{\large\bf #1}\vspace{1ex}}20\newcommand{\mysection}[1]{\section{#1}\vspace{-1.5ex}}21%\newcommand{\mysubsection}[1]{\noindent{\bf #1}}22\newcommand{\mysubsection}[1]{\subsection{#1}\vspace{-1ex}}23\newcommand{\Q}{\mathbf{Q}}24\newcommand{\Z}{\mathbf{Z}}25\newcommand{\R}{\mathbf{R}}26\newcommand{\C}{\mathbf{C}}27\newcommand{\PGL}{\mbox{\rm PGL}}28\newcommand{\GL}{\mbox{\rm GL}}29\newcommand{\Div}{\mbox{\rm Div}}30\newcommand{\GQ}{G_\Q}31\newcommand{\ra}{\rightarrow}32\newcommand{\tensor}{\otimes}33\DeclareMathOperator{\tor}{tor}34\font\cyr=wncyr10 scaled \magstep 135\newcommand{\Sha}{\mbox{\cyr X}}36\font\cyrbig=wncyr10 scaled \magstep 237\newcommand{\BigSha}{\mbox{\cyrbig X}}38\newcommand{\intersect}{\cap}39\newcommand{\comment}[1]{}40\theoremstyle{plain}41\newtheorem{theorem}{Theorem}42\newtheorem{corollary}[theorem]{Corollary}43\newtheorem{conjecture}[theorem]{Conjecture}4445\begin{document}464748\normalsize49\baselineskip=15.9pt5051\begin{center}52{\Large\bf Research Plan}\\53William A.\ Stein\\54\today55\end{center}5657\mysection{Introduction}58My research program reflects the essential interplay between abstract59theory and explicit machine computation during60the latter half of the twentieth century;61it sits at the intersection of recent work of62B.~Mazur, K.~Ribet, J-P.~Serre, R.~Taylor, and A.~Wiles63on Galois representations attached to modular abelian varieties64(see \cite{ribet:modreps, serre:conjectures,65taylor-wiles:fermat, wiles:fermat})66with work of J.~Cremona, N.~Elkies, and J.-F.~Mestre67on explicit computations involving modular forms68(see \cite{cremona:algs, elkies:ffield}).6970In 1969 B.~Birch~\cite{birch:bsd} described computations that71led to the most fundamental open72conjecture in the theory of elliptic curves:73\begin{quote}74I want to describe some computations undertaken by myself and75Swinnerton-Dyer on EDSAC76by which we have calculated the zeta-functions of certain elliptic77curves. As a result of these computations we have found an analogue78for an elliptic curve of the Tamagawa number of an algebraic group;79and conjectures (due to ourselves, due to Tate, and80due to others) have proliferated.81\end{quote}82The rich tapestry of arithmetic conjectures and theory83we enjoy today would not exist without the ground-breaking84application of computing by Birch and Swinnerton-Dyer.85Computations in the 1980s by Mestre were key in convincing86Serre that his conjectures on modularity of odd87irreducible Galois representations88were worthy of serious consideration (see~\cite{serre:conjectures}).89These conjectures have inspired much recent work;90for example, Ribet's proof of the91$\epsilon$-conjecture, which played an essential role in92Wiles's proof of Fermat's Last Theorem.9394My work on the Birch and Swinnerton-Dyer95conjecture for modular abelian varieties and search for new96examples of modular icosahedral Galois representations97has led me to discover and implement algorithms for98explicitly computing with modular forms.99My research, which involves finding ways to compute100with modular forms and modular abelian varieties,101is driven by outstanding conjectures in number theory.102103\mysection{Invariants of modular abelian varieties}104Now that the Shimura-Taniyama conjecture has been proved,105the main outstanding problem in the field106is the Birch and Swinnerton-Dyer conjecture (BSD conjecture),107which ties together the arithmetic invariants of an elliptic curve.108There is no general class of elliptic curves for which109the full BSD conjecture is known.110Approaches to the BSD conjecture that rely on congruences111between modular forms are likely to112require a deeper understanding of the analogous113conjecture for higher-dimensional abelian varieties.114As a first step, I have obtained theorems that make possible explicit115computation of some of the arithmetic invariants of modular116abelian varieties.117118\mysubsection{The BSD conjecture}119By~\cite{breuil-conrad-diamond-taylor} we now know120that every elliptic curve over~$\Q$ is121a quotient of the curve~$X_0(N)$ whose complex points122are the isomorphism classes of pairs consisting of a123(generalized) elliptic curve and a cyclic subgroup of order~$N$.124Let~$J_0(N)$ denote the Jacobian of $X_0(N)$; this is an abelian125variety of dimension equal to the genus of~$X_0(N)$ whose points126correspond to the degree~$0$ divisor classes on~$X_0(N)$.127128An {\em optimal quotient} of $J_0(N)$ is a quotient by an abelian subvariety.129Consider an optimal quotient~$A$ such that $L(A,1)\neq 0$.130By~\cite{kolyvagin-logachev:totallyreal},~$A(\Q)$ and~$\Sha(A/\Q)$131are both finite.132The BSD conjecture asserts that133$$\frac{L(A,1)}{\Omega_A} =134\frac{\#\Sha(A/\Q)\cdot\prod_{p\mid N} c_p}135{\# A(\Q)\cdot\#\Adual(\Q)}.$$136Here the Shafarevich-Tate group $\Sha(A/\Q)$ is a measure of the failure137of the local-to-global principle; the Tamagawa numbers~$c_p$ are the138orders of the component groups of~$A$; the real number~$\Omega_A$ is139the volume of~$A(\R)$ with respect to a basis of differentials having140everywhere nonzero good reduction; and~$\Adual$ is the dual of~$A$.141My goal is to verify the full142conjecture for many specific abelian varieties on a case-by-case basis.143This is the first step in a program to verify the above144conjecture for an infinite family of quotients of~$J_0(N)$.145146\mysubsection{The ratio $L(A,1)/\Omega_A$}147Following Y.~Manin's work on elliptic curves,148A.~Agash\'e and I proved the following149theorem in~\cite{stein:vissha}.150\begin{theorem}\label{thm:ratpart}151Let~$m$ be the largest square dividing~$N$.152The ratio $L(A,1)/\Omega_A$ is a rational number that can be153explicitly computed, up to a unit (conjecturally $1$) in $\Z[1/(2m)]$.154\end{theorem}155The proof uses modular symbols combined with an extension of the argument156used by Mazur in~\cite{mazur:rational} to bound the Manin constant.157The ratio $L(A,1)/\Omega_A$ is expressed as the lattice index of158two modules over the Hecke algebra.159I expect the method to give similar results160for special values of twists, and of161$L$-functions attached to eigenforms of higher weight.162I have computed $L(A,1)/\Omega_A$ for all optimal163quotients of level $N\leq 1500$; this table continues to be164of value to number theorists.165166\mysubsection{The torsion subgroup}167I can compute upper and lower bounds on $\#A(\Q)_{\tor}$, but I can not168determine $\#A(\Q)_{\tor}$ in all cases. Experimentally, the deviation169between the upper and lower bound is reflected in congruences with170forms of lower level; I hope to exploit this in a precise way.171I also obtained the following172intriguing corollary that suggests cancellation between173torsion and~$c_p$; it generalizes to higher weight forms, thus174suggesting a geometric explanation for reducibility175of Galois representations.176\begin{corollary}177Let~$n$ be the order of the image of178$(0)-(\infty)$ in $A(\Q)$, and179let~$m$ be the largest square dividing~$N$.180Then $\displaystyle n\cdot L(A,1)/\Omega_A$ is an integer,181up to a unit in $\Z[1/(2m)]$.182\end{corollary}183184\mysubsection{Tamagawa numbers}185\begin{theorem}\label{thm:tamagawa}186When $p^2\nmid N$, the number~$c_p$ can be explicitly computed187(up to a power of~$2$).188\end{theorem}189I prove this in~\cite{stein:compgroup}. Several related problems190remain: when $p^2 \mid N$ it may be possible to compute~$c_p$191using the Drinfeld-Katz-Mazur interpretation192of~$X_0(N)$; it should also be possible to use my193methods to treat optimal quotients of $J_1(N)$.194195I was surprised to find that systematic computations using this196formula indicate the following conjectural refinement of a result of197Mazur~\cite{mazur:eisenstein}.198\begin{conjecture}199Suppose~$N$ is prime and~$A$200is an optimal quotient of $J_0(N)$. Then $A(\Q)_{\tor}$ is201generated by the image of $(0)-(\infty)$202and $c_p = \#A(\Q)_{\tor}$. Furthermore,203the product of the~$c_p$ over all optimal factors204equals the numerator of $(N-1)/12$.205\end{conjecture}206I have checked this conjecture for all $N\leq 997$ and,207up to a power of~$2$, for all $N\leq 2113$.208The first part is known when~$A$ is an elliptic209curve (see~\cite{mestre-oesterle:crelle}).210Upon hearing of this conjecture, Mazur proved it211when all ``$q$-Eisenstein quotients'' are simple.212There are three promising approaches to finding213a complete proof. One involves the explicit214formula of Theorem~\ref{thm:tamagawa};215another is based on Ribet's level lowering theorem,216and a third makes use of a simplicity result of Merel.217218Theorem~\ref{thm:tamagawa}219also suggests a way to compute Tamagawa numbers of220motives attached to eigenforms of higher weight.221These numbers appear in the conjectures of Bloch and Kato,222which generalize the BSD conjecture to motives (see~\cite{bloch-kato}).223224225\mysubsection{Upper bounds on $\#\BigSha$}226V.~Kolyvagin and K.~Kato~\cite{kolyvagin:structureofsha, scholl:kato}227obtained upper bounds on~$\#\Sha(A)$.228To verify the full BSD conjecture for certain abelian229varieties, it is necessary is to make these bounds explicit.230Kolyvagin's bounds involve computations with Heegner points,231and Kato's involve a study of the Galois representations232associated to~$A$. I plan to carry out such233computations in many specific cases.234235\mysubsection{Lower bounds on $\#\BigSha$}236One approach to showing that~$\Sha$ is as large as predicted237by the BSD conjecture is suggested by Mazur's notion of238the visible part of~$\Sha$ (see~\cite{cremona-mazur, mazur:visthree}).239Let~$\Adual$ be the dual of~$A$.240The visible part of $\Sha(\Adual/\Q)$ is the241kernel of $\Sha(\Adual/\Q)\ra \Sha(J_0(N))$.242Mazur observed that if an element of order~$p$243in~$\Sha(\Adual/\Q)$ is visible,244then it is explained by a jump in the rank of Mordell-Weil245in the sense that there is another abelian subvariety $B\subset J_0(N)$246such that $p \mid \#(\Adual\intersect B)$ and the rank of~$B$ is positive.247I think that this observation can be turned around: if there is248another abelian249variety~$B$ of positive rank such that $p\mid \#(\Adual\intersect B)$,250then, under mild hypotheses, there is an element of~$\Sha(\Adual/\Q)$251of order~$p$. Thus the theory of congruences between modular252forms can be used to obtain a lower bound on~$\#\Sha(\Adual/\Q)$.253I am trying to use the cohomological methods of~\cite{mazur:tower}254and suggestions of B.~Conrad and Mazur to prove the following conjecture.255\begin{conjecture}256Let~$\Adual$ and~$B$ be abelian subvarieties of~$J_0(N)$.257Suppose that $p\mid \#(\Adual\intersect B)$, that~$p\nmid N$,258and that~$p$ does not divide the259order of any of the torsion subgroups260or component groups of~$A$ or~$B$. Then261$(B(\Q) \oplus \Sha(B/\Q))\tensor\Z/p\Z262\isom (\Adual(\Q)\oplus \Sha(\Adual/\Q) )\tensor\Z/p\Z$.263\end{conjecture}264Unfortunately, $\Sha(\Adual/\Q)$ can fail to be265visible inside~$J_0(N)$.266For example, I found that the BSD conjecture267predicts the existence of invisible elements of odd order in~$\Sha$268for at least~$15$ of the~$37$ optimal quotients of prime269level $\leq 2113$.270For every integer~$M$ (Ribet~\cite{ribet:raising} tells us which~$M$271to choose), we can consider the images of~$\Adual$ in $J_0(NM)$.272There is not yet enough evidence to conjecture the existence of273an integer~$M$ such that all of $\Sha(\Adual/\Q)$ is visible in274$J_0(NM)$.275I am gathering data to determine whether or not276to expect the existence of such~$M$.277278\mysubsection{Motivation for considering abelian varieties}279If $A$ is an elliptic curve, then explaining~$\Sha(A/\Q)$ using280only congruences between elliptic curves is bound to fail.281This is because pairs of nonisogenous elliptic curves282with isomorphic $p$-torsion are, according to283E.~Kani's conjecture, extremely rare.284It is crucial to understand what happens in all dimensions.285286Within the range accessible by computer,287abelian varieties exhibit more richly textured288structure than elliptic curves.289For example, I discovered a visible element of prime order $83341$290in the Shafarevich-Tate group of an abelian variety of291prime conductor~$2333$; in contrast, over all optimal elliptic292curves of conductor up to $5500$, it appears that293the largest order of an element of a Shafarevich-Tate294group is~$7$.295296297\mysection{Conjectures of Artin, Merel, and Serre}298299\mysubsection{Icosahedral Galois representations}300E.~Artin conjectured in~\cite{artin:conjecture}301that the $L$-series associated to any continuous irreducible302representation $\rho:\GQ\ra \GL_n(\C)$, with $n>1$, is entire.303Recent exciting work of Taylor and others suggests that a complete proof304of Artin's conjecture, in the case when $n=2$ and~$\rho$ is odd,305is on the horizon.306This case of Artin's conjecture is known when the image307of~$\rho$ in $\PGL_2(\C)$ is solvable308(see~\cite{tunnell:artin}), and in infinitely309many cases when the image of~$\rho$ is not solvable (see~\cite{bdsbt}).310311In 1998, K.~Buzzard suggested a way to combine the312main theorem of~\cite{buzzard-taylor}, along with313a computer computation, to deduce modularity of certain314icosahedral Galois representations. Buzzard and I recently obtained315the following theorem.316\begin{theorem}317The icosahedral Artin representations of conductor~$1376=2^5\cdot 43$318are modular.319\end{theorem}320We expect our method to yield several more examples.321These ongoing computations are laying a small part of the technical322foundations necessary for a full proof of the Artin conjecture323for odd two dimensional~$\rho$, as well as stimulating324the development of new algorithms for computing with modular forms325using modular symbols in characteristic~$\ell$.326327\mysubsection{Cyclotomic points on modular curves}328If~$E$ is an elliptic curve over~$\Q$329and~$p$ is an odd prime, then the $p$-torsion330on~$E$ can not all lie in~$\Q$; because of331the Weil pairing the $p$-torsion generates a field332that contains~$\Q(\mu_p)$.333For which primes~$p$ does334there exist an elliptic curve~$E$ over $\Q(\mu_p)$335with all of its $p$-torsion rational over $\Q(\mu_p)$?336When $p=2,3,5$ the corresponding moduli space has genus zero337and infinitely many examples exist. Recent work of L.~Merel,338combined with computations he enlisted me to do,339suggest that these are the only primes~$p$ for340which such elliptic curves exist.341In~\cite{merel:cyclo}, Merel exploits342cyclotomic analogues of the techniques used343in his proof of the uniform boundedness conjecture344to obtain an explicit criterion that can345be used to answer the above question for many primes~$p$,346on a case-by-case basis.347Theoretical work of Merel, combined with my computations of348twisted $L$-values and character groups of tori,349give the following result (see~\cite[\S3.2]{merel:cyclo}):350\begin{theorem}351Let $p \equiv 3\pmod{4}$ be a prime satisfying352$7 \leq p < 1000$. There are no elliptic curves353over $\Q(\mu_p)$ all of whose $p$-torsion is rational354over $\Q(\mu_p)$.355\end{theorem}356The case in which~$p$ is congruent to~$1$ modulo~$4$ presents357additional difficulties that involve showing that~$Y(p)$ has no358$\Q(\sqrt{p})$-rational points. We hope to tackle these in359the near future.360361362\mysubsection{Genus one curves}363The index of an algebraic curve~$C$ over~$\Q$ is the364order of the cokernel of the degree map $\Div_\Q(C)\ra\Z$;365rationality of the canonical divisor implies that the index366divides $2g-2$, where~$g$ is the genus of~$C$.367When $g=1$ this is no condition at all; Artin conjectured, and368Lang and Tate~\cite{lang-tate} proved, that for every integer~$m$369there is a genus one curve of index~$m$ over some number field.370Their construction yields genus one curves over~$\Q$ only for a few371values of~$m$, and they ask whether one can find genus one curves372over~$\Q$ of every index. I have answered373this question for odd~$m$.374\begin{theorem}375Let~$K$ be any number field. There are genus one curves over~$K$376of every odd index.377\end{theorem}378The proof involves showing that enough cohomology classes in379Kolyvagin's Euler system of Heegner points do not vanish380combined with explicit Heegner point computations.381I hope to show that curves of every index occur, and382to determine the consequences of my383nonvanishing result for Selmer groups. This can be viewed384as a contribution to the problem of understanding $H^1(\Q,E)$.385386\mysubsection{Serre's conjecture modulo $pq$}387Let~$p$ and~$q$ be primes, and consider a continuous388representation $\rho:\GQ\ra \GL(2,\Z/pq\Z)$389that is irreducible in the sense that its reductions390modulo~$p$ and modulo~$q$ are both irreducible.391Call~$\rho$ {\em modular} if there is a modular392form~$f$ such that a mod~$p$ representation attached to~$f$393is the mod~$p$ reduction of~$\rho$, and ditto for~$q$.394I have carried out specific computations suggested by Mazur395in hopes of determining when one should expect396that such mod~$pq$ representations are modular; the computation397suggests that the right conjectures are elusive.398Ribet's theorem (see~\cite{ribet:raising})399produces infinitely many levels $pq\ell$ at which there is400a form giving rise to $\rho$~mod~$p$ and another401giving rise to $\rho$~mod~$q$; we hope to determine if for some~$\ell$402there is a single form giving rise to both reductions.403404\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}405\newcommand{\closer}{\vspace{-1.5ex}}406\begin{thebibliography}{10}407408\bibitem{agashe}409A.~Agash\'{e}, \emph{On invisible elements of the {T}ate-{S}hafarevich group},410C. 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