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\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{\Gal}{Gal} \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{\GL}{GL} \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} \date{\rd{Math 129: May 5, 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 \bf \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 many specific elliptic\\ curves over~$\Q$. \vfill \noindent\rd{Joint Work:} Grigor Grigorov, Andrei Jorza, Corina Patrascu, Stefan Patrikis \vfill \noindent\rd{Thanks:} John Cremona, Stephen Donnelly, Ralph Greenberg, Grigor Grigorov, Barry Mazur, Robert Pollack, Nick Ramsey, Tony Scholl, Micahel Stoll.} \page{ \heading{Elliptic Curves over the Rational Numbers $\Q$} \vspace{-3ex} {\Large An {\dred elliptic curve} is a nonsingular plane cubic curve with a rational point (possibly ``at infinity'').} \vspace{1ex} \psset{unit=1.75} \pspicture(-2.000,-3.000)(2.000,2.000) \newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor %RGB color (0.69999999999999996, 0.69999999999999996, 0.69999999999999996) \newrgbcolor{mycolor}{0.70 0.70 0.70}\mycolor %Grid divided into 2 subdivisions \newrgbcolor{glc}{0.00 0.00 0.00}\psgrid[gridcolor=mycolor, subgriddiv=2, gridlabelcolor=glc] %RGB color (0.20000000000000001, 0.20000000000000001, 0.20000000000000001) 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%RGB color (0.0, 0.0, 0.0) \newrgbcolor{mycolor}{0.00 0.00 0.00}\mycolor %"{\LARGE $y^2+y = x^3-x$}" at position (-1, -3.7000000000000002) \rput[lb](-1.8, -4){{\large\dblue $y^2+y = x^3-x$}} \rput[lb](5,2){{\dgreen EXAMPLES}} \rput[lb](4,1){\Large $y^2+y = x^3-x$} \rput[lb](4,0){{\Large $x^3 + y^3 = z^3$} (projective)} \rput[lb](4,-1){{\Large $y^2 = x^3+ax+b$}} \rput[lb](4,-2){{\Large $3x^3+4y^3+5z^3=0$}} \newrgbcolor{mycolor}{1.00 0.00 0.00}\mycolor \psset{linewidth=0.015} \psline[linecolor=mycolor](2.7,-1.5)(11,-2) \psline[linecolor=mycolor](2.7,-2)(11,-1.5) \endpspicture } % end page \page{ \heading{Mordell's Theorem} \vspace{2ex} \psset{unit=1.0} \pspicture(0,0)(0,0) \eps{17.5}{0.9}{0.25}{pics/mordell1} \endpspicture {\dred Theorem (Mordell).} The group $E(\Q)$ of rational points on an elliptic curve is a {\dgreen finitely generated abelian group}, so $$ E(\Q) \cong \Z^r \oplus T, $$ with $T=E(\Q)_{\rm tor}$ finite. \vspace{2ex} Mazur classified the possibilities for $T$. \vfill \rd{Folklore conjecture:} $r$ can be arbitrary, but the biggest $r$ ever found is (probably) $24$. } % end page \page{ \heading{\dred \mbox{}\hspace{4em}\LARGE Conjectures Proliferated} \psset{unit=1.0} \pspicture(0,0)(0,0) \eps{0}{-1.3}{0.2}{pics/birch1} \endpspicture ``The subject of this lecture is rather a special one. 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 have proliferated. [...] though the associated theory is both abstract and technically complicated, the objects about which I intend to talk are usually simply defined and often machine computable; {\dblue experimentally we have detected certain relations between different invariants}, but we have been unable to approach proofs of these relations, which must lie very deep.'' \hfill -- Birch 1965 } % end page %\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 $$ L(E,s) = \prod_{p\nmid \Delta} \left(\frac{1}{1 - a_p \cdot p^{-s} + p \cdot p^{-2s}}\right) \cdot \prod_{p\mid \Delta} L_p(E,s) $$} Here $ a_p = p+1-\#E(\F_p)$ for all $p\nmid \Delta$, where $\Delta$ is divisible by the primes of bad reduction for~$E$. We do not include the factors $L_p(E,s)$ at bad primes here. } % 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{Graph of $L$-Series of $y^2+y=x^3-x$} \begin{center} \psset{unit=1.0} \pspicture(0,0)(0,0) \eps{-10.5}{-14}{0.9}{pics/elseries37} %\eps{-10.5}{-14}{0.9}{pics/lseries_37a} \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{What about Taylor series of $L(E,s)$ around $s=1$?} \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{\bf Taylor Series} {\large For $y^2+y=x^3-x$, the {\bf Taylor series} about $1$ is} {\Large $$ L(E,s) = 0 + ( s-1)0.3059997\ldots $$ $$ \qquad\qquad + (s-1)^20.18636\ldots + \cdots$$ } \vfill In general, it's $$ L(E,s) = c_0 + c_1 s + c_2 s^2 + \cdots. $$ \dblue{Big Mystery:} Do these Taylor coefficients $c_n$ have any deep arithmetic meaning? } \page{ \heading{BSD \rd{Formula} Conjecture} Let $r=\ord_{s=1} L(E,s)$. Then Birch and Swinnerton-Dyer made a famous guess for the first nonzero coefficient $c_r$: {\large $$ c_r = \frac{\Omega_{E} \cdot \Reg_{E} \cdot \prod_{p\mid N} t_p } {\#E(\Q)_{\tor}^2} \cdot \#\Sha(E) $$ } \begin{center} \framebox{\begin{minipage}{0.7\textwidth} \begin{itemize} %\item $L(E,s)$ is an entire $L$-function that encodes $\{\#E(\F_p)\}$, $p$ prime. \item $\#E(\Q)_{\tor}$ -- \rd{torsion} order \item $t_p$ -- \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{itemize} \end{minipage} } \end{center} } \page{ \heading{What about $c_{r+1}$, $c_{r+2}$, etc?} \vfill {\Large I think nobody has even a \rd{wild and crazy} guess for an interpretation of these. \vfill \large They are probably not ``periods'' like $c_r$ is, so perhaps should not have any nice interpretation... } \vfill } %\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{Motivating Problem 1} \vfill \bd{Motivating Problem 1.} For specific curves, compute every quantity appearing in the BSD formula conjecture \rd{\em in practice.} \vfill NOTE:\vspace{2ex}\\ \noindent{}This is \rd{not} meant as a theoretical problem about computability, though by compute we mean ``compute with proof.''\\ \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 $N\leq 40000$. \vspace{-2ex} \item All of the quantities in the conjecture, \rd{except} for $\#\Sha(E/\Q)$, have been computed by Cremona for conductor $\leq 40000$. \vspace{-2ex} \item \bd{Cremona (Ch.~4, pg.~106):} We have $2\nmid \#\Sha(E)$ 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 and the above results, 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 Kolyvagin's 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. Also, the computations give rise to many surprising and interesting examples. } \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} \vfill \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{Find an Algorithm}] Based on deep work of Kolyvagin, Kato, et al. \label{step:alg}\\ \mbox{}\hspace{1em}\rd{Input:} An elliptic curve over $\Q$ with $r_{\an}\leq 1$.\\ \mbox{}\hspace{1em}\rd{Output:} $B \geq 1$ such that if $p\nmid B$, then $p\nmid \#\Sha(E)$.\\ \vspace{-4ex} \item{}[\rd{Compute}] \label{step:implement} Run the algorithm on our $2441$ curves. \vspace{-2ex} \item{}[\rd{Reducible}] \label{step:analysis} If $E[p]$ is reducible say nothing. \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} \vfill } \page{ \heading{Kolyvagin Bound on $\#\Sha(E)$} \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 $p\nmid \#\Sha(E/\Q)$. \begin{enumerate} \item{} [\rd{Choose $K$}] Choose a quadratic imaginary field $K=\Q(\sqrt{D})$ with certain properties, such that $E/K$ has analytic rank~1. Assume $\Q(E[p])$ has degree $\#\GL_2(\F_p)$. \item{}[\rd{Compute Mordell-Weil}] \begin{enumerate} \item If $r=0$, compute generator $z$ for $E^D(\Q)$ mod torsion. \item If $r=1$, compute generator $z$ for $E(\Q)$ mod torsion. \end{enumerate} \item{}[\rd{Index of Heegner point}] Compute the ``Heegner point'' $y_K\in E(K)$ associated to~$K$. This is a point that comes from the ``modularity'' map $X_0(N)\to E$. \item{}[\rd{Finished}] Output $B = I \cdot A$, where $A$ is the product of primes such that $\Q(E[p])$ has degree less than $\#\GL_2(\F_p)$. \end{enumerate} \vfill \gr{Theorem (Kolyvagin):} $p\nmid 2B \implies p\nmid \#\Sha(E/\Q).$ \vfill } \end{document}