\documentclass[12pt]{article}12\newcommand{\doc}{Detailed Research Plan}3\newcommand{\myname}{William A.\ Stein}4\newcommand{\phone}{(510) 883-9938}5\newcommand{\email}{{\tt was@math.berkeley.edu}}6\newcommand{\www}{{\tt http://www.math.berkeley.edu/\~{\mbox{}}was}}7\newcommand{\address}{2041 Francisco Street, \#14\\8Berkeley, CA 94709\\9USA}101112\hoffset=-.1\textwidth13\textwidth=1.2\textwidth14\voffset=-.1\textheight15\textheight=1.2\textheight1617\usepackage{fancyhdr,ifthen}18\pagestyle{fancy}19\cfoot{} % no footers (in pagestyle fancy)20% running left heading21\lhead{\bfseries\LARGE\em \noindent{}\hspace{-.2em}\myname{}22\hfill \doc\vspace{-.2ex}\\}23% running right heading24\newcommand{\spc}{.33ex}25%\rhead{\em {\small{\phone{}}} \hspace{\spc}$\bullet$\hspace{\spc} \email{}26% \hspace{\spc}$\bullet$\hspace{\spc} \www{}}27% adjust, because the header is now taller than usual.28\rhead{}29\setlength{\headheight}{7ex}303132\usepackage{amsmath}33\usepackage{amsthm}34\usepackage{amssymb}3536373839\newcommand{\Adual}{A^{\vee}}40\newcommand{\isom}{\cong}41%\newcommand{\ncisom}{\approx}42%\newcommand{\mysection}[1]{\vspace{12pt}\noindent{\large\bf #1}\vspace{1ex}}43\newcommand{\mysection}[1]{\section{#1}\vspace{-1.5ex}}44%\newcommand{\mysubsection}[1]{\noindent{\bf #1}}45\newcommand{\mysubsection}[1]{\subsection{#1}\vspace{-1ex}}46\newcommand{\Q}{\mathbf{Q}}47\newcommand{\Z}{\mathbf{Z}}48\newcommand{\R}{\mathbf{R}}49\newcommand{\C}{\mathbf{C}}50\newcommand{\PGL}{\mbox{\rm PGL}}51\newcommand{\GL}{\mbox{\rm GL}}52\newcommand{\Div}{\mbox{\rm Div}}53\newcommand{\GQ}{G_\Q}54\newcommand{\ra}{\rightarrow}55\newcommand{\tensor}{\otimes}56\DeclareMathOperator{\tor}{tor}57\font\cyr=wncyr10 scaled \magstep 158\newcommand{\Sha}{\mbox{\cyr X}}59\font\cyrbig=wncyr10 scaled \magstep 260\newcommand{\BigSha}{\mbox{\cyrbig X}}61\newcommand{\intersect}{\cap}62\newcommand{\comment}[1]{}63\theoremstyle{plain}64\newtheorem{theorem}{Theorem}65\newtheorem{corollary}[theorem]{Corollary}66\newtheorem{conjecture}[theorem]{Conjecture}6768\begin{document}697071\begin{center}72{\LARGE \bf \doc}\\73\end{center}7475\mysection{Introduction}76My research program reflects the essential interplay between abstract77theory and explicit machine computation during78the latter half of the twentieth century;79it sits at the intersection of recent work of80B.~Mazur, K.~Ribet, J-P.~Serre, R.~Taylor, and A.~Wiles81on Galois representations attached to modular abelian varieties82(see \cite{ribet:modreps, serre:conjectures,83taylor-wiles:fermat, wiles:fermat})84with work of J.~Cremona, N.~Elkies, and J.-F.~Mestre85on explicit computations involving modular forms86(see \cite{cremona:algs, elkies:ffield}).8788In 1969 B.~Birch~\cite{birch:bsd} described computations that89led to the most fundamental open90conjecture in the theory of elliptic curves:91\begin{quote}92I want to describe some computations undertaken by myself and93Swinnerton-Dyer on EDSAC94by which we have calculated the zeta-functions of certain elliptic95curves. As a result of these computations we have found an analogue96for an elliptic curve of the Tamagawa number of an algebraic group;97and conjectures (due to ourselves, due to Tate, and98due to others) have proliferated.99\end{quote}100The rich tapestry of arithmetic conjectures and theory101we enjoy today would not exist without the ground-breaking102application of computing by Birch and Swinnerton-Dyer.103Computations in the 1980s by Mestre were key in convincing104Serre that his conjectures on modularity of odd105irreducible Galois representations106were worthy of serious consideration (see~\cite{serre:conjectures}).107These conjectures have inspired much recent work;108for example, Ribet's proof of the109$\epsilon$-conjecture, which played an essential role in110Wiles's proof of Fermat's Last Theorem.111112My work on the Birch and Swinnerton-Dyer113conjecture for modular abelian varieties and search for new114examples of modular icosahedral Galois representations115has led me to discover and implement algorithms for116explicitly computing with modular forms.117My research, which involves finding ways to compute118with modular forms and modular abelian varieties,119is driven by outstanding conjectures in number theory.120121\mysection{Invariants of modular abelian varieties}122Now that the Shimura-Taniyama conjecture has been proved,123the main outstanding problem in the field124is the Birch and Swinnerton-Dyer conjecture (BSD conjecture),125which ties together the arithmetic invariants of an elliptic curve.126There is no general class of elliptic curves for which127the full BSD conjecture is known.128Approaches to the BSD conjecture that rely on congruences129between modular forms are likely to130require a deeper understanding of the analogous131conjecture for higher-dimensional abelian varieties.132As a first step, I have obtained theorems that make possible explicit133computation of some of the arithmetic invariants of modular134abelian varieties.135136\mysubsection{The BSD conjecture}137By~\cite{breuil-conrad-diamond-taylor} we now know138that every elliptic curve over~$\Q$ is139a quotient of the curve~$X_0(N)$ whose complex points140are the isomorphism classes of pairs consisting of a141(generalized) elliptic curve and a cyclic subgroup of order~$N$.142Let~$J_0(N)$ denote the Jacobian of $X_0(N)$; this is an abelian143variety of dimension equal to the genus of~$X_0(N)$ whose points144correspond to the degree~$0$ divisor classes on~$X_0(N)$.145146An {\em optimal quotient} of $J_0(N)$ is a quotient by an abelian subvariety.147Consider an optimal quotient~$A$ such that $L(A,1)\neq 0$.148By~\cite{kolyvagin-logachev:totallyreal},~$A(\Q)$ and~$\Sha(A/\Q)$149are both finite.150The BSD conjecture asserts that151$$\frac{L(A,1)}{\Omega_A} =152\frac{\#\Sha(A/\Q)\cdot\prod_{p\mid N} c_p}153{\# A(\Q)\cdot\#\Adual(\Q)}.$$154Here the Shafarevich-Tate group $\Sha(A/\Q)$ is a measure of the failure155of the local-to-global principle; the Tamagawa numbers~$c_p$ are the156orders of the component groups of~$A$; the real number~$\Omega_A$ is157the volume of~$A(\R)$ with respect to a basis of differentials having158everywhere nonzero good reduction; and~$\Adual$ is the dual of~$A$.159My goal is to verify the full160conjecture for many specific abelian varieties on a case-by-case basis.161This is the first step in a program to verify the above162conjecture for an infinite family of quotients of~$J_0(N)$.163164\mysubsection{The ratio $L(A,1)/\Omega_A$}165Following Y.~Manin's work on elliptic curves,166A.~Agash\'e and I proved the following167theorem in~\cite{stein:vissha}.168\begin{theorem}\label{thm:ratpart}169Let~$m$ be the largest square dividing~$N$.170The ratio $L(A,1)/\Omega_A$ is a rational number that can be171explicitly computed, up to a unit (conjecturally $1$) in $\Z[1/(2m)]$.172\end{theorem}173The proof uses modular symbols combined with an extension of the argument174used by Mazur in~\cite{mazur:rational} to bound the Manin constant.175The ratio $L(A,1)/\Omega_A$ is expressed as the lattice index of176two modules over the Hecke algebra.177I expect the method to give similar results178for special values of twists, and of179$L$-functions attached to eigenforms of higher weight.180I have computed $L(A,1)/\Omega_A$ for all optimal181quotients of level $N\leq 1500$; this table continues to be182of value to number theorists.183184\mysubsection{The torsion subgroup}185I can compute upper and lower bounds on $\#A(\Q)_{\tor}$, but I can not186determine $\#A(\Q)_{\tor}$ in all cases. Experimentally, the deviation187between the upper and lower bound is reflected in congruences with188forms of lower level; I hope to exploit this in a precise way.189I also obtained the following190intriguing corollary that suggests cancellation between191torsion and~$c_p$; it generalizes to higher weight forms, thus192suggesting a geometric explanation for reducibility193of Galois representations.194\begin{corollary}195Let~$n$ be the order of the image of196$(0)-(\infty)$ in $A(\Q)$, and197let~$m$ be the largest square dividing~$N$.198Then $\displaystyle n\cdot L(A,1)/\Omega_A$ is an integer,199up to a unit in $\Z[1/(2m)]$.200\end{corollary}201202\mysubsection{Tamagawa numbers}203\begin{theorem}\label{thm:tamagawa}204When $p^2\nmid N$, the number~$c_p$ can be explicitly computed205(up to a power of~$2$).206\end{theorem}207I prove this in~\cite{stein:compgroup}. Several related problems208remain: when $p^2 \mid N$ it may be possible to compute~$c_p$209using the Drinfeld-Katz-Mazur interpretation210of~$X_0(N)$; it should also be possible to use my211methods to treat optimal quotients of $J_1(N)$.212213I was surprised to find that systematic computations using this214formula indicate the following conjectural refinement of a result of215Mazur~\cite{mazur:eisenstein}.216\begin{conjecture}217Suppose~$N$ is prime and~$A$218is an optimal quotient of $J_0(N)$. Then $A(\Q)_{\tor}$ is219generated by the image of $(0)-(\infty)$220and $c_p = \#A(\Q)_{\tor}$. Furthermore,221the product of the~$c_p$ over all optimal factors222equals the numerator of $(N-1)/12$.223\end{conjecture}224I have checked this conjecture for all $N\leq 997$ and,225up to a power of~$2$, for all $N\leq 2113$.226The first part is known when~$A$ is an elliptic227curve (see~\cite{mestre-oesterle:crelle}).228Upon hearing of this conjecture, Mazur proved it229when all ``$q$-Eisenstein quotients'' are simple.230There are three promising approaches to finding231a complete proof. One involves the explicit232formula of Theorem~\ref{thm:tamagawa};233another is based on Ribet's level lowering theorem,234and a third makes use of a simplicity result of Merel.235236Theorem~\ref{thm:tamagawa}237also suggests a way to compute Tamagawa numbers of238motives attached to eigenforms of higher weight.239These numbers appear in the conjectures of Bloch and Kato,240which generalize the BSD conjecture to motives (see~\cite{bloch-kato}).241242243\mysubsection{Upper bounds on $\#\BigSha$}244V.~Kolyvagin and K.~Kato~\cite{kolyvagin:structureofsha, scholl:kato}245obtained upper bounds on~$\#\Sha(A)$.246To verify the full BSD conjecture for certain abelian247varieties, it is necessary is to make these bounds explicit.248Kolyvagin's bounds involve computations with Heegner points,249and Kato's involve a study of the Galois representations250associated to~$A$. I plan to carry out such251computations in many specific cases.252253\mysubsection{Lower bounds on $\#\BigSha$}254One approach to showing that~$\Sha$ is as large as predicted255by the BSD conjecture is suggested by Mazur's notion of256the visible part of~$\Sha$ (see~\cite{cremona-mazur, mazur:visthree}).257Let~$\Adual$ be the dual of~$A$.258The visible part of $\Sha(\Adual/\Q)$ is the259kernel of $\Sha(\Adual/\Q)\ra \Sha(J_0(N))$.260Mazur observed that if an element of order~$p$261in~$\Sha(\Adual/\Q)$ is visible,262then it is explained by a jump in the rank of Mordell-Weil263in the sense that there is another abelian subvariety $B\subset J_0(N)$264such that $p \mid \#(\Adual\intersect B)$ and the rank of~$B$ is positive.265I think that this observation can be turned around: if there is266another abelian267variety~$B$ of positive rank such that $p\mid \#(\Adual\intersect B)$,268then, under mild hypotheses, there is an element of~$\Sha(\Adual/\Q)$269of order~$p$. Thus the theory of congruences between modular270forms can be used to obtain a lower bound on~$\#\Sha(\Adual/\Q)$.271I am trying to use the cohomological methods of~\cite{mazur:tower}272and suggestions of B.~Conrad and Mazur to prove the following conjecture.273\begin{conjecture}274Let~$\Adual$ and~$B$ be abelian subvarieties of~$J_0(N)$.275Suppose that $p\mid \#(\Adual\intersect B)$, that~$p\nmid N$,276and that~$p$ does not divide the277order of any of the torsion subgroups278or component groups of~$A$ or~$B$. Then279$(B(\Q) \oplus \Sha(B/\Q))\tensor\Z/p\Z280\isom (\Adual(\Q)\oplus \Sha(\Adual/\Q) )\tensor\Z/p\Z$.281\end{conjecture}282Unfortunately, $\Sha(\Adual/\Q)$ can fail to be283visible inside~$J_0(N)$.284For example, I found that the BSD conjecture285predicts the existence of invisible elements of odd order in~$\Sha$286for at least~$15$ of the~$37$ optimal quotients of prime287level $\leq 2113$.288For every integer~$M$ (Ribet~\cite{ribet:raising} tells us which~$M$289to choose), we can consider the images of~$\Adual$ in $J_0(NM)$.290There is not yet enough evidence to conjecture the existence of291an integer~$M$ such that all of $\Sha(\Adual/\Q)$ is visible in292$J_0(NM)$.293I am gathering data to determine whether or not294to expect the existence of such~$M$.295296\mysubsection{Motivation for considering abelian varieties}297If $A$ is an elliptic curve, then explaining~$\Sha(A/\Q)$ using298only congruences between elliptic curves is bound to fail.299This is because pairs of nonisogenous elliptic curves300with isomorphic $p$-torsion are, according to301E.~Kani's conjecture, extremely rare.302It is crucial to understand what happens in all dimensions.303304Within the range accessible by computer,305abelian varieties exhibit more richly textured306structure than elliptic curves.307For example, I discovered a visible element of prime order $83341$308in the Shafarevich-Tate group of an abelian variety of309prime conductor~$2333$; in contrast, over all optimal elliptic310curves of conductor up to $5500$, it appears that311the largest order of an element of a Shafarevich-Tate312group is~$7$.313314315\mysection{Conjectures of Artin, Merel, and Serre}316317\mysubsection{Icosahedral Galois representations}318E.~Artin conjectured in~\cite{artin:conjecture}319that the $L$-series associated to any continuous irreducible320representation $\rho:\GQ\ra \GL_n(\C)$, with $n>1$, is entire.321Recent exciting work of Taylor and others suggests that a complete proof322of Artin's conjecture, in the case when $n=2$ and~$\rho$ is odd,323is on the horizon.324This case of Artin's conjecture is known when the image325of~$\rho$ in $\PGL_2(\C)$ is solvable326(see~\cite{tunnell:artin}), and in infinitely327many cases when the image of~$\rho$ is not solvable (see~\cite{bdsbt}).328329In 1998, K.~Buzzard suggested a way to combine the330main theorem of~\cite{buzzard-taylor}, along with331a computer computation, to deduce modularity of certain332icosahedral Galois representations. Buzzard and I recently obtained333the following theorem.334\begin{theorem}335The icosahedral Artin representations of conductor~$1376=2^5\cdot 43$336are modular.337\end{theorem}338We expect our method to yield several more examples.339These ongoing computations are laying a small part of the technical340foundations necessary for a full proof of the Artin conjecture341for odd two dimensional~$\rho$, as well as stimulating342the development of new algorithms for computing with modular forms343using modular symbols in characteristic~$\ell$.344345\mysubsection{Cyclotomic points on modular curves}346If~$E$ is an elliptic curve over~$\Q$347and~$p$ is an odd prime, then the $p$-torsion348on~$E$ can not all lie in~$\Q$; because of349the Weil pairing the $p$-torsion generates a field350that contains~$\Q(\mu_p)$.351For which primes~$p$ does352there exist an elliptic curve~$E$ over $\Q(\mu_p)$353with all of its $p$-torsion rational over $\Q(\mu_p)$?354When $p=2,3,5$ the corresponding moduli space has genus zero355and infinitely many examples exist. Recent work of L.~Merel,356combined with computations he enlisted me to do,357suggest that these are the only primes~$p$ for358which such elliptic curves exist.359In~\cite{merel:cyclo}, Merel exploits360cyclotomic analogues of the techniques used361in his proof of the uniform boundedness conjecture362to obtain an explicit criterion that can363be used to answer the above question for many primes~$p$,364on a case-by-case basis.365Theoretical work of Merel, combined with my computations of366twisted $L$-values and character groups of tori,367give the following result (see~\cite[\S3.2]{merel:cyclo}):368\begin{theorem}369Let $p \equiv 3\pmod{4}$ be a prime satisfying370$7 \leq p < 1000$. There are no elliptic curves371over $\Q(\mu_p)$ all of whose $p$-torsion is rational372over $\Q(\mu_p)$.373\end{theorem}374The case in which~$p$ is congruent to~$1$ modulo~$4$ presents375additional difficulties that involve showing that~$Y(p)$ has no376$\Q(\sqrt{p})$-rational points. Merel and I hope to tackle these377difficulties in the near future.378379380\mysubsection{Serre's conjecture modulo $pq$}381Let~$p$ and~$q$ be primes, and consider a continuous382representation $\rho:\GQ\ra \GL(2,\Z/pq\Z)$383that is irreducible in the sense that its reductions384modulo~$p$ and modulo~$q$ are both irreducible.385Call~$\rho$ {\em modular} if there is a modular386form~$f$ such that a mod~$p$ representation attached to~$f$387is the mod~$p$ reduction of~$\rho$, and ditto for~$q$.388I have carried out specific computations suggested by Mazur389in hopes of determining when one should expect390that such mod~$pq$ representations are modular; the computation391suggests that the right conjectures are elusive.392Ribet's theorem (see~\cite{ribet:raising})393produces infinitely many levels $pq\ell$ at which there is394a form giving rise to $\rho$~mod~$p$ and another395giving rise to $\rho$~mod~$q$; we hope to determine if for some~$\ell$396there is a single form giving rise to both reductions.397398\mysection{Genus one curves}399The index of an algebraic curve~$C$ over~$\Q$ is the400order of the cokernel of the degree map $\Div_\Q(C)\ra\Z$;401rationality of the canonical divisor implies that the index402divides $2g-2$, where~$g$ is the genus of~$C$.403When $g=1$ this is no condition at all; Artin conjectured, and404Lang and Tate~\cite{lang-tate} proved, that for every integer~$m$405there is a genus one curve of index~$m$ over some number field.406Their construction yields genus one curves over~$\Q$ only for a few407values of~$m$, and they ask whether one can find genus one curves408over~$\Q$ of every index. I have answered409this question for odd~$m$.410\begin{theorem}411Let~$K$ be any number field. There are genus one curves over~$K$412of every odd index.413\end{theorem}414The proof involves showing that enough cohomology classes in415Kolyvagin's Euler system of Heegner points do not vanish416combined with explicit Heegner point computations.417I hope to show that curves of every index occur, and418to determine the consequences of my419nonvanishing result for Selmer groups. This can be viewed420as a contribution to the problem of understanding $H^1(\Q,E)$.421422423424\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}425\newcommand{\closer}{\vspace{-1.5ex}}426\begin{thebibliography}{10}427428\bibitem{agashe}429A.~Agash\'{e}, \emph{On invisible elements of the {T}ate-{S}hafarevich group},430C. R. Acad. Sci. Paris S\'er. I Math. \textbf{328} (1999), no.~5, 369--374.431\closer432433\bibitem{stein:vissha}434A.~Agash\'{e} and W.\thinspace{}A. 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