\documentclass[landscape,10pt]{slides} \usepackage{fullpage} \newcommand{\page}[1]{\vfill\begin{slide}#1\vfill\end{slide}\vfill} %\renewcommand{\page}[1]{} \newcommand{\apage}[1]{\vfill\begin{slide}#1\vfill\end{slide}\vfill} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb} \newcommand{\defn}[1]{{\em #1}} \newcommand{\an}{{\rm an}} \newcommand{\e}{\mathbf{e}} \DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts \newcommand{\textcyr}[1]{% {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}% \selectfont #1}} \newcommand{\Sha}{{\mbox{\textcyr{Sh}}}} \newcommand{\la}{\leftarrow} \newcommand{\da}{\downarrow} \newcommand{\set}[1]{\{#1\}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\K}{{\mathbb K}} \newcommand{\dual}{\bot} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\h}{\mathfrak{h}} \newcommand{\p}{\mathfrak{p}} \newcommand{\m}{\mathfrak{m}} \newcommand{\pari}{{\sc Pari}} \newcommand{\magma}{{\sc Magma}} \newcommand{\hd}[1]{\vspace{1ex}\noindent{\bf #1} } \renewcommand{\L}{\mathcal{L}} \renewcommand{\l}{\ell} \renewcommand{\t}{\tau} \renewcommand{\O}{\mathcal{O}} \renewcommand{\a}{\mathfrak{a}} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Disc}{Disc} \DeclareMathOperator{\Reg}{Reg} \DeclareMathOperator{\Sel}{Sel} \DeclareMathOperator{\Real}{Re} \renewcommand{\Re}{\Real} \DeclareMathOperator{\new}{new} \DeclareMathOperator{\tor}{tor} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\HH}{H} \renewcommand{\H}{\HH} \newcommand{\eps}[4]{\rput[lb](#1,#2){% \includegraphics[width=#3\textwidth]{#4}}} \usepackage{fancybox} \usepackage{graphicx} \author{\rd{William Stein}\\ Harvard University\\ {\tt http://modular.fas.harvard.edu/talks/bsd2005ucsd/}} \date{\rd{UCSD: February 1, 2005}} %\include{macros} \renewcommand{\dual}{\vee} \usepackage[hypertex]{hyperref} \setlength{\fboxsep}{1em} \setlength{\parindent}{0cm} \usepackage{pstricks} \usepackage{graphicx} \newrgbcolor{dblue}{0 0 0.8} \renewcommand{\hd}[1]{\begin{center}\LARGE\bf\dblue #1\vspace{-2.5ex}\end{center}} \newrgbcolor{dred}{0.7 0 0} \newrgbcolor{dgreen}{0 0.3 0} \newcommand{\rd}[1]{{\bf \dred #1}} \newcommand{\gr}[1]{{\bf \dgreen #1}} \renewcommand{\defn}[1]{{\bf \dblue #1}} \newrgbcolor{purple}{0.3 0 0.3} \renewcommand{\magma}{{\purple\sc Magma}} \newcommand{\bd}[1]{{\bf\dred #1}} \newcommand{\heading}[1]{\begin{center} \Large \sc \dblue #1 \end{center}} \title{\blue\bf Verifying the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves} \begin{document} \page{ \psset{unit=3.0} \pspicture(0,0)(0.1,0.1) \rput[lb](-0.2,-3){\includegraphics[width=10em]{pics/cremona2}} \rput[lb](5,-3){\includegraphics[width=10em]{pics/cremona2mirror}} \endpspicture \vspace{-5ex} \maketitle } \page{ \psset{unit=3.0} \pspicture(0,0)(0.1,0.1) \rput[lb](6,-2){\includegraphics{pics/group2}} \endpspicture This talk reports on a project to verify the Birch\\ and Swinnerton-Dyer conjecture for all elliptic\\ curves over~$\Q$ in John Cremona's book. \vfill \noindent\rd{Joint Work:} Stephen Donnelly, Andrei Jorza, Stefan Patrikis, Michael Stoll. \vfill \noindent\rd{Thanks:} John Cremona, Ralph Greenberg, Grigor Grigorov, Barry Mazur, Robert Pollack, Nick Ramsey, and Tony Scholl. } \page{ \heading{Birch and Swinnerton-Dyer (Utrecht, 2000)} \begin{center} \includegraphics[height=0.86\textheight]{pics/bsd1} \end{center} } \page{ \heading{The $L$-Function} { \psset{unit=3.0} \pspicture(0,0)(0.1,0.1) \rput[lb](6,0){\includegraphics[width=7em]{pics/wiles1}} \rput[lb](0,0){\includegraphics[width=7em]{pics/hecke_in_front}} \endpspicture {\dred Theorem (Wiles et al., Hecke)} The following function extends to a holomorphic function on the whole complex plane: \Large $$ L(E,s) = \prod_{p\nmid \Delta} \left(\frac{1}{1 - a_p \cdot p^{-s} + p \cdot p^{-2s}}\right). $$} Here $ a_p = p+1-\#E(\F_p)$ for all $p\nmid \Delta_E$. Note that formally, $$ L(E,1) = \prod_{p\nmid \Delta} \left(\frac{1}{1-a_p\cdot p^{-1} + p \cdot p^{-2}}\right) = \prod_{p\nmid \Delta} \left(\frac{p}{p-a_p + 1}\right) = \prod_{p\nmid \Delta} \frac{p}{N_p} $$ } % end page %\apage{ %\heading{The Riemann Zeta Function} %The $L$-function of an elliptic curve is analogous to %the Riemann Zeta function. %} % end page \page{ \heading{Real Graph of the $L$-Series of $y^2+y=x^3-x$} \begin{center} \psset{unit=1.0} \pspicture(0,0)(0,0) \eps{-8}{-12}{0.8}{pics/lser} \endpspicture \end{center} } % end page \page{ \heading{More Graphs of Elliptic Curve $L$-functions} \vspace{6ex} \begin{center} \psset{unit=1.0} \pspicture(0,0)(0,0) \eps{-8}{-12}{0.8}{pics/many_lser} \endpspicture \end{center} } % end page \page{ \heading{The Birch and Swinnerton-Dyer Conjecture} \begin{center} \psset{unit=1.0} \pspicture(0,0)(0,0) \eps{-7}{-12}{0.7}{pics/birch_and_swinnerton-dyer} \endpspicture \end{center} \vspace{-4ex} {\dred Conjecture:} Let $E$ be any elliptic curve over~$\Q$. The order of vanishing of $L(E,s)$ as $s=1$ equals the rank of $E(\Q)$. } % end page \page{ \heading{The Kolyvagin and Gross-Zagier Theorems} \begin{center} \psset{unit=1.0} \pspicture(0,0)(0,0) \eps{-11}{-12}{0.3}{pics/koly} \eps{-2}{-12}{0.25}{pics/gross} \eps{6}{-12}{0.2}{pics/zagier} \endpspicture \end{center} \vspace{-4ex} {\dred Theorem:} If the ordering of vanishing $\ord_{s=1} L(E,s)$ is $\leq 1$, then the conjecture is true for $E$. } % end page %\page{ %\heading{The Conjecture of Birch and Swinnerton-Dyer} %\bd{BSD Rank:} %Let $E$ be an elliptic curve over~$\Q$, and %let $r=r_{\an} = \ord_{s=1} L(E, s)$. %Then %$$ % r_{\an} = \text{rank}\, E(\Q). %$$ %} \page{ \heading{The Birch and Swinnerton-Dyer \rd{Formula}} {\large $$ \frac{L^{(r)}(E,1)}{r!} = \frac{\Omega_{E} \cdot \Reg_{E} \cdot \prod_{p\mid N} c_p } {\#E(\Q)_{\tor}^2} \cdot \#\Sha(E) $$ } \begin{center} \framebox{\begin{minipage}{0.7\textwidth} \begin{enumerate} \item $L(E,s)$ is an entire $L$-function that encodes $\{\#E(\F_p)\}$, $p$ prime. \item $\#E(\Q)_{\tor}$ -- \rd{torsion} order \item $c_p$ -- \rd{Tamagawa numbers} \item $\Omega_E$ -- \rd{real volume} $\int_{E(\R)} \omega_E$ \item $\Reg_E$ -- \rd{regulator} of $E$ \item $\Sha(E) = \Ker(\H^1(\Q,E)\to\bigoplus_v\H^1(\Q_v,E))$ -- \rd{Shafarevich-Tate group} \end{enumerate} \end{minipage} } \end{center} } \page{ \heading{Motivating Problem 1} \vfill \bd{Motivating Problem 1.} Compute every quantity appearing in the BSD conjecture \rd{\em in practice.} \vfill NOTES:\vspace{2ex}\\ \noindent{}1. This is \rd{not} meant as a theoretical problem about computability, though by compute we mean ``compute with proof.''\\ \vfill \noindent{}2. I am also very interested in the same question but for modular abelian varieties. \vfill } \page{ \heading{Status} \begin{enumerate} \vfill \item When $r_{\an} =\ord_{s=1}L(E,s) \leq 3$, then we can compute $r_{\an}$.\\ \rd{Open Problem:} Show that $r_{\an}\geq 4$ for some elliptic curve. \item Relatively ``easy'' to compute $\#E(\Q)_{\tor}$, $c_p$, $\Omega_E$. \item Computing $\Reg_E$ essentially same as computing $E(\Q)$; interesting and sometimes very difficult. \item Computing $\#\Sha(E)$ is currently \gr{very very difficult}.\\ \rd{Theorem (Kolyvagin):}\\\mbox{} \hspace{3em}$r_{\an}\leq 1 \, \implies$ $\Sha(E)$ is finite (with bounds)\\ \rd{Open Problem:}\\\mbox{} \hspace{3em}Prove that $\Sha(E)$ is finite for some $E$ with $r_{\an}\geq 2$. \end{enumerate} \vfill } \page{ \heading{Victor Kolyvagin} \vfill \begin{center} Kolyvagin's work on Euler systems is crucial to our project. \vspace{-1ex} \includegraphics[height=0.75\textheight]{pics/kolyvagin-ny} \end{center} } \page{ \heading{Motivating Problem 2: Cremona's Book} \bd{Motivating Problem 2.} Prove BSD for every elliptic curve over~$\Q$ of conductor at most $1000$, i.e., in Cremona's book. \begin{enumerate} \item By Tate's isogeny invariance of BSD, it suffices to prove BSD for each \rd{optimal} elliptic curve of conductor $N\leq 1000$. \vspace{-2ex} \item \rd{Rank part} of the conjecture has been verified by Cremona for all curves with $N\leq 25000$. \vspace{-2ex} \item All of the quantities in the conjecture, \rd{except} for $\#\Sha(E/\Q)$, have been computed by Cremona for conductor $\leq 25000$. \vspace{-2ex} \item \bd{Cremona (Ch.~4, pg.~106):} We have $\Sha(E)[2]=0$ for \rd{all} optimal curves with conductor $\leq 1000$ except 571A, 960D, and 960N. So we can mostly ignore $2$ henceforth. \end{enumerate} } \page{ \heading{John Cremona} \begin{center} John Cremona's software and book are crucial to our project. \includegraphics[height=0.7\textheight]{pics/cremona} \end{center} } \page{ \heading{The Four Nontrivial $\Sha$'s} \bd{Conclusion:} In light of Cremona's book, the problem is to show that $\Sha(E)$ is {\em trivial} for all but the following four optimal elliptic curves with conductor at most $1000$: \vfill \begin{center} \begin{tabular}{|c|l|c|}\hline Curve & $a$-invariants & $\Sha(E)_?$\\\hline 571A& [0,-1,1,-929,-105954] & 4\\ 681B&[1,1,0,-1154,-15345] & 9\\ 960D& [0,-1,0,-900,-10098] & 4\\ 960N& [0,1,0,-20,-42] & 4\\\hline \end{tabular} \end{center} We first deal with these four. } \page{ \bd{\Large Divisor of Order:} \begin{enumerate} \item Using a $2$-descent we see that $4\mid \#\Sha(E)$ for 571A, 960D, 960N. \item For $E=681B$: Using visibility (or a $3$-descent) we see that $9\mid \#\Sha(E)$. \end{enumerate} } \page{ \bd{\Large Multiple of Order:} \begin{enumerate} \item For $E=681B$, the mod~$3$ representation is surjective, and $3\mid\mid [E(K):y_K]$ for $K=\Q(\sqrt{-8})$, so (our refined) Kolyvagin theorem implies that $\#\Sha(E)=9$, as required. \item Kolyvagin's theorem and computation $\implies$ $\#\Sha(E) = 4^?$ for 571A, 960D, 960N. \item Using MAGMA's {\tt FourDescent} command, we compute $\Sel^{(4)}(E/\Q)$ for 571A, 960D, 960N and deduce that $\#\Sha(E)=4$. (Note: MAGMA Documentation currently misleading.) \end{enumerate} } \page{ \heading{The Eighteen Optimal Curves of Rank $>1$} There are $18$ curves with conductor $\leq 1000$ and rank $>1$ (all have rank~$2$): %was@form:~/people/cremona/data$ awk '$5==2 && $1<=1000 {print $1$2" & "$4"\\\\"}' curves.1-8000 \vfill \begin{center} 389A, 433A, 446D, 563A, 571B, 643A, 655A, 664A, 681C,\\ 707A, 709A, 718B, 794A, 817A, 916C, 944E, 997B, 997C \end{center} \vfill For these~$E$ \rd{nobody} currently knows how to show that $\Sha(E)$ is finite, let alone trivial. (But mention, e.g., $p$-adic $L$-functions.) \vfill \bd{Motivating Problem 3:} Prove the BSD Conjecture for all elliptic curve over~$\Q$ of conductor at most $1000$ and rank $\leq 1$. \vfill \bd{SECRET MOTIVATION:} Our actual motivation is to unify and extend results about BSD and algorithms for elliptic curves. The computational challenge is there to see what interesting phenomena occur in the data. } \page{ \heading{Our Goal} \begin{itemize} \item There are $2463$ optimal curves of conductor at most $1000$. \item Of these, $18$ have rank~$2$, which leaves~$2445$ curves. \item Of these, $2441$ are conjectured to have trivial $\Sha$. \end{itemize} \begin{center} Thus our \rd{goal} is to prove that $$\#\Sha(E)=1$$ for these $2441$ elliptic curves. \end{center} } \page{ \heading{Our Strategy} \begin{enumerate} \item{}[\rd{Refine}] \label{step:refine} Prove a refinement of \underline{Kolyvagin's bound} on $\#\Sha(E)$ that is suitable for computation. \vspace{-2ex} \item{}[\rd{Algorithm}] \label{step:alg}\\ \mbox{}\hspace{1em}\rd{Input:} An elliptic curve over $\Q$ with $r_{\an}\leq 1$.\\ \mbox{}\hspace{1em}\rd{Output:} Odd $B \geq 1$ such that if $p\nmid 2B$, then $p\nmid \#\Sha(E)$. \vspace{-4ex} \item{}[\rd{Compute}] \label{step:implement} Run the algorithm on our $2441$ curves. \vspace{-2ex} \item{}[\rd{Descent}] \label{step:analysis} If $p\mid B$ and $E[p]$ is reducible: Use $p$-descent? \vspace{-2ex} \item{}[\rd{New Methods}] If $p\mid B$ and $E[p]$ irreducible: Try Kato when $r_{\an}=0$. When $r_{\an}=1$, use Schneider's theorem, Kato's work, explicit computations with $p$-adic heights and $p$-adic $L$-functions. Also, visibility and level lowering? Further refinement of Kolyvagin's theorem? \end{enumerate} } \page{ \heading{Our Algorithm to Bound $\Sha(E)$} \rd{INPUT:} An elliptic curve~$E$ over $\Q$ with $r_{\an} \leq 1$.\\ \rd{OUTPUT:} Odd $B\geq 1$ such that if $p\nmid 2B$, then $\Sha(E/\Q)[p]=0$. \begin{enumerate} \item{} [\rd{Choose $K$}] Choose a quadratic imaginary field $K=\Q(\sqrt{D})$ that satisfy the Heegner hypothesis, such that $E/K$ has analytic rank~1, and $\Disc(K)$ is divisible by \rd{two primes}. (Or two such $K$ each divisible by a single prime.) \item{}[\rd{Find $p$-torsion}] Decide for which primes $p$ there is a curve $E'$ that is $\Q$-isogenous to $E$ such that $E'(\Q)[p]\neq 0$. Let $A$ be the product of these primes. \newpage \item{}[\rd{Compute Mordell-Weil}] \begin{enumerate} \item If $r_{\an}=0$, compute generator $z$ for $E^D(\Q)$ mod torsion. \item If $r_{\an}=1$, compute generator $z$ for $E(\Q)$ mod torsion. \end{enumerate} \item{}[\rd{Height of Heegner point}] Compute the height $h_K(y_K)$, e.g., using the Gross-Zagier formula (and/or directly). \item{}[\rd{Index of Heegner point}] Compute\\\mbox{}\hspace{3em} $I_K = \sqrt{h_K(y_K)/h_K(z)} = [E(K)_{/\tor} : \Z y_K].$ \item{}[\rd{Refined Kolyvagin}] Output $B = A \cdot I_K$. \end{enumerate} \vfill \gr{Theorem (refinement of Kolyvagin):} $p\nmid 2B \implies p\nmid \#\Sha(E/\Q).$ \vfill } \page{ \heading{First Attempt to Run the Algorithm} % It appears that the one case in which $p\mid B$ but there is no % rational $p$-isogeny and $\Sha(E/\Q)[p]=0$ is when $p$ divides some % Tamagawa number and $E$ has rank $1$ (when $E$ has rank $0$, a % theorem of Kato applies). \vfill \begin{itemize} \item Using \magma{} and the MECCAH cluster, I implemented and ran the algorithm on the curves of conductor $\leq 1000$, but stopped runs if they took over 30 minutes. \item The computation for $318$ curves didn't finish. We do not include them below. Also, I don't trust some of \magma{}'s elliptic curves functions, since the documentation is unclear. However, we assume correctness for the rest of this talk. \item \rd{Future plan:} run each computation without timeouts using {\tt mwrank} and {\tt PARI}. \end{itemize} } \page{ \heading{Results of the First Attempt} \vspace{-2ex} \begin{enumerate} \item For $1363$ curves we have $B=1$. For these curves we have proved the full BSD conjecture! \vspace{-2ex} \item There are $94$ curves for which $B\geq 11$. Of these, $6$ have rank~$0$ (so we can likely use Kato's theorem). \vspace{-2ex} \item There are $39$ curves for which $B\geq 19$. For {\em all} of these curves the rank is $1$. \vspace{-2ex} \item The largest $B$ is $77$, for the rank~$1$ curves 618F and 894G. \vspace{-2ex} \item The largest prime divisor of any $B$ is $31$, for the rank~$1$ curve 674C. \vspace{-2ex} \item When $E$ has rank $0$, the algorithm is much more difficult, so more likely to time out. \end{enumerate} } \page{ \heading{Major Obstruction: Tamagawa Numbers}\label{sec:level} \bd{Serious Issue:} The Gross-Zagier formula and the BSD conjecture together imply that if an odd prime $p$ divides a Tamagawa number, then $p\mid [E(K) : \Z y_K]$. \begin{itemize} \item If $E$ has $r_{\an}=0$, and $p\geq 5$, and $\rho_{E,p}$ is surjective, then Kato's theorem (and Mazur, Rubin, et al.) imply that {\dred\large $$\ord_p(\#\Sha(E)) \leq \ord_p(L(E,1)/\Omega_E),$$} so squareness of $\#\Sha(E)$ frequently saves us. \item Unfortunately, in many cases there is a big Tamagawa number and $r_{\an}=1$, so Kato doesn't apply. \end{itemize} } \page{ \heading{An Example} \vfill The elliptic curve $E$ called 141A and given by $y^2 + y = x^3 + x^2 - 12x + 2$ has rank 1 and $c_3 = 7$. We compute that $$ \#\Sha(E) = 49^{???}. $$ The representation $\rho_{E,7}$ is surjective, but~$E$ has rank~$1$.\\ \vfill %\begin{itemize} %\item{}[Visibility?] %The Jacobian $J_0(47)$ is of rank %$0$ and is simple of dimension $4$, and we find that $E[7]$ sits in %the old subvariety of $J_0(3\cdot 47)$. %Hope: Proving %something about the Shafarevich-Tate group of the simple rank $0$ abelian %variety $J_0(47)$ will imply something about $\Sha(E)[7]$. %Note that $L(J_0(47),1)/\Omega = 16/23$. %\vfill %\item{}[$p$-Adic Approach?] \rd{Ralph Greenberg's suggestion:} Compute a $p$-adic $L$-function, a $p$-adic regulator, and use theorems of Kato and Peter Schneider to show that $7\nmid \#\Sha(E)$. I hope to do this soon. %\end{itemize} } \page{ \heading{What Next?} \vspace{-2ex} \begin{enumerate} \item{}[\rd{Efficiency}] Make the algorithm more efficient. {\small The reason the discriminant must be divisible by two primes, or we choose two fields is so we can weaken the surjectivity hypothesis that Kolyvagin imposed. However, in many cases we have surjectivity and could directly use Kolyvagin's theorem. Also \rd{Byungchul Cha's} 2003 Johns Hopkins Ph.D. thesis weakens Kolyvagin's hypothesis in another way. Combining all this should speed up the algorithm.} \vspace{-2ex} \item{}[\rd{Finish!}] Run the algorithm to completion on all curves of conductor up to $1000$. Hard part is finding $E^D(\Q)$ for rank~$1$ $E^D$, where $D$ has $3$ digits (so the conductor has $\sim 12$ digits). \vspace{-4ex} \item{}[\rd{New Theory}] Find a strategy that works when $r_{\an}=1$ and $E$ has a Tamagawa number $\geq 5$. Either refine Kolyvagin, use visibility and level lowering, or Schneider and Kato's results on the $p$-adic main conjecture. \end{enumerate} } \end{document} \page{ \heading{More Examples} \begin{itemize} \item 190A1: We have $190=2\cdot 5\cdot 19$ and $c_{2}=11$. There is a $4$-dimensional abelian variety over rank $0$ and level $95$ with $\Sha[11]$ trivial that contains $E[11]$. \item 214A1: We have $214=2\cdot 107$ and $c_{2}=7$. There is a rank $0$ simple abelian variety over level $107$ and dimension $7$ that contains $E[7]$. \item 674C1: We have $214=2\cdot 337$ and $c_{2}=31$. For this one, there is a rank $0$ simple abelian variety of level $337$ and dimension $15$ that contains $E[31]$ and according to BSD has trivial $\Sha[31]$. \end{itemize} }