CoCalc Public Fileswww / talks / 2005-02-ucsd-bsd / bsd.texOpen with one click!
Author: William A. Stein
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\author{\rd{William Stein}\\
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Harvard University\\
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{\tt http://modular.fas.harvard.edu/talks/bsd2005ucsd/}}
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\date{\rd{UCSD: February 1, 2005}}
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\title{\blue\bf Verifying the Birch and
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Swinnerton-Dyer Conjecture for Specific Elliptic Curves}
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\begin{document}
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\page{
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\psset{unit=3.0}
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\pspicture(0,0)(0.1,0.1)
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\rput[lb](-0.2,-3){\includegraphics[width=10em]{pics/cremona2}}
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\rput[lb](5,-3){\includegraphics[width=10em]{pics/cremona2mirror}}
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\endpspicture
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\vspace{-5ex}
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\maketitle
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}
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\page{
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\rput[lb](6,-2){\includegraphics{pics/group2}}
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This talk reports on a project to verify the Birch\\
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and Swinnerton-Dyer conjecture for all elliptic\\
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curves over~$\Q$ in John Cremona's book.
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\vfill
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\noindent\rd{Joint Work:} Stephen Donnelly, Andrei Jorza, Stefan Patrikis,
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Michael Stoll.
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\vfill
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\noindent\rd{Thanks:} John Cremona, Ralph Greenberg, Grigor Grigorov,
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Barry Mazur, Robert Pollack, Nick Ramsey, and Tony Scholl.
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}
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\page{
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\heading{Birch and Swinnerton-Dyer (Utrecht, 2000)}
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\begin{center}
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\includegraphics[height=0.86\textheight]{pics/bsd1}
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\end{center}
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}
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\page{
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\heading{The $L$-Function}
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{
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\psset{unit=3.0}
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\pspicture(0,0)(0.1,0.1)
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\rput[lb](6,0){\includegraphics[width=7em]{pics/wiles1}}
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\rput[lb](0,0){\includegraphics[width=7em]{pics/hecke_in_front}}
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\endpspicture
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{\dred Theorem (Wiles et al., Hecke)} The following
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function extends to a holomorphic function on the
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whole complex plane:
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\Large $$
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L(E,s) = \prod_{p\nmid \Delta}
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\left(\frac{1}{1 - a_p \cdot p^{-s} + p \cdot p^{-2s}}\right).
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$$}
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Here
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$ a_p = p+1-\#E(\F_p)$ for all $p\nmid \Delta_E$.
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Note that formally,
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$$
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L(E,1) =
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\prod_{p\nmid \Delta}
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\left(\frac{1}{1-a_p\cdot p^{-1} + p \cdot p^{-2}}\right)
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=
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\prod_{p\nmid \Delta}
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\left(\frac{p}{p-a_p + 1}\right)
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= \prod_{p\nmid \Delta}
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\frac{p}{N_p}
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$$
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} % end page
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%\apage{
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%\heading{The Riemann Zeta Function}
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%The $L$-function of an elliptic curve is analogous to
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%the Riemann Zeta function.
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%} % end page
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\page{
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\heading{Real Graph of the $L$-Series of $y^2+y=x^3-x$}
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\begin{center}
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\psset{unit=1.0}
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\pspicture(0,0)(0,0)
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\eps{-8}{-12}{0.8}{pics/lser}
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} % end page
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\page{
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\heading{More Graphs of Elliptic Curve $L$-functions}
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\vspace{6ex}
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\begin{center}
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\psset{unit=1.0}
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\pspicture(0,0)(0,0)
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\eps{-8}{-12}{0.8}{pics/many_lser}
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\endpspicture
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\end{center}
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} % end page
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\page{
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\heading{The Birch and Swinnerton-Dyer Conjecture}
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\begin{center}
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\psset{unit=1.0}
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\pspicture(0,0)(0,0)
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\eps{-7}{-12}{0.7}{pics/birch_and_swinnerton-dyer}
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\end{center}
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\vspace{-4ex}
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{\dred Conjecture:}
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Let $E$ be any elliptic curve over~$\Q$.
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The order of vanishing of $L(E,s)$ as $s=1$
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equals the rank of $E(\Q)$.
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} % end page
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\page{
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\heading{The Kolyvagin and Gross-Zagier Theorems}
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\begin{center}
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\psset{unit=1.0}
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\pspicture(0,0)(0,0)
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\eps{-11}{-12}{0.3}{pics/koly}
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\eps{-2}{-12}{0.25}{pics/gross}
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\eps{6}{-12}{0.2}{pics/zagier}
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\end{center}
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\vspace{-4ex}
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{\dred Theorem:} If the ordering of vanishing $\ord_{s=1} L(E,s)$ is $\leq 1$,
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then the conjecture is true for $E$.
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} % end page
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%\page{
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%\heading{The Conjecture of Birch and Swinnerton-Dyer}
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%\bd{BSD Rank:}
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%Let $E$ be an elliptic curve over~$\Q$, and
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%let $r=r_{\an} = \ord_{s=1} L(E, s)$.
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%Then
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%$$
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% r_{\an} = \text{rank}\, E(\Q).
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%$$
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%}
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\page{
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\heading{The Birch and Swinnerton-Dyer \rd{Formula}}
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{\large $$
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\frac{L^{(r)}(E,1)}{r!}
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= \frac{\Omega_{E} \cdot \Reg_{E} \cdot \prod_{p\mid N} c_p }
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{\#E(\Q)_{\tor}^2} \cdot \#\Sha(E)
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$$
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}
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\begin{center}
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\framebox{\begin{minipage}{0.7\textwidth}
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\begin{enumerate}
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\item $L(E,s)$ is an entire $L$-function that encodes $\{\#E(\F_p)\}$, $p$ prime.
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\item $\#E(\Q)_{\tor}$ -- \rd{torsion} order
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\item $c_p$ -- \rd{Tamagawa numbers}
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\item $\Omega_E$ -- \rd{real volume} $\int_{E(\R)} \omega_E$
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\item $\Reg_E$ -- \rd{regulator} of $E$
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\item $\Sha(E) = \Ker(\H^1(\Q,E)\to\bigoplus_v\H^1(\Q_v,E))$ -- \rd{Shafarevich-Tate group}
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\end{enumerate}
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\end{minipage}
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}
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\end{center}
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}
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\page{
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\heading{Motivating Problem 1}
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\vfill
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\bd{Motivating Problem 1.}
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Compute every quantity appearing in the
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BSD conjecture \rd{\em in practice.}
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\vfill
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NOTES:\vspace{2ex}\\
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\noindent{}1. This is \rd{not} meant as a theoretical problem about computability,
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though by compute we mean ``compute with proof.''\\
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\vfill
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\noindent{}2. I am also very interested in the same question but for modular
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abelian varieties.
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\vfill
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}
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\page{
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\heading{Status}
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\begin{enumerate}
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\vfill
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\item When $r_{\an} =\ord_{s=1}L(E,s) \leq 3$, then we can compute $r_{\an}$.\\
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\rd{Open Problem:} Show that $r_{\an}\geq 4$ for some elliptic curve.
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\item Relatively ``easy'' to compute $\#E(\Q)_{\tor}$, $c_p$, $\Omega_E$.
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\item Computing $\Reg_E$ essentially same as computing $E(\Q)$;
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interesting and sometimes very difficult.
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\item Computing $\#\Sha(E)$ is currently \gr{very very difficult}.\\
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\rd{Theorem (Kolyvagin):}\\\mbox{} \hspace{3em}$r_{\an}\leq 1 \, \implies$
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$\Sha(E)$ is finite (with bounds)\\
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\rd{Open Problem:}\\\mbox{} \hspace{3em}Prove that $\Sha(E)$ is finite for
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some $E$ with $r_{\an}\geq 2$.
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\end{enumerate}
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\vfill
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}
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\page{
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\heading{Victor Kolyvagin}
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\vfill
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\begin{center}
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Kolyvagin's work on Euler systems is crucial to our project.
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\vspace{-1ex}
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\includegraphics[height=0.75\textheight]{pics/kolyvagin-ny}
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\end{center}
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}
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\page{
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\heading{Motivating Problem 2: Cremona's Book}
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\bd{Motivating Problem 2.} Prove BSD for
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every elliptic curve over~$\Q$ of conductor at most $1000$,
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i.e., in Cremona's book.
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\begin{enumerate}
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\item
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By Tate's isogeny invariance of BSD,
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it suffices to prove BSD for each \rd{optimal}
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elliptic curve of conductor $N\leq 1000$.
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\vspace{-2ex}
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\item \rd{Rank part}
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of the conjecture has been verified by
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Cremona for all curves with $N\leq 25000$.
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\vspace{-2ex}
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\item All of the quantities in
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the conjecture, \rd{except} for $\#\Sha(E/\Q)$, have been computed by
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Cremona for conductor $\leq 25000$.
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\vspace{-2ex}
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\item \bd{Cremona (Ch.~4, pg.~106):}
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We have
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$\Sha(E)[2]=0$ for \rd{all} optimal curves with conductor $\leq 1000$
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except 571A, 960D, and 960N.
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So we can mostly ignore $2$ henceforth.
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\end{enumerate}
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}
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\page{
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\heading{John Cremona}
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\begin{center}
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John Cremona's software and book are crucial to our project.
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\includegraphics[height=0.7\textheight]{pics/cremona}
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\end{center}
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}
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\page{
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\heading{The Four Nontrivial $\Sha$'s}
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\bd{Conclusion:} In light of Cremona's book, the
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problem is to show that $\Sha(E)$ is {\em trivial}
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for all but the following four
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optimal elliptic curves with conductor at most $1000$:
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\vfill
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\begin{center}
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\begin{tabular}{|c|l|c|}\hline
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Curve & $a$-invariants & $\Sha(E)_?$\\\hline
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571A& [0,-1,1,-929,-105954] & 4\\
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681B&[1,1,0,-1154,-15345] & 9\\
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960D& [0,-1,0,-900,-10098] & 4\\
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960N& [0,1,0,-20,-42] & 4\\\hline
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\end{tabular}
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\end{center}
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We first deal with these four.
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}
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\page{
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\bd{\Large Divisor of Order:}
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\begin{enumerate}
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\item Using a $2$-descent we see
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that $4\mid \#\Sha(E)$ for 571A, 960D, 960N.
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\item For $E=681B$: Using visibility
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(or a $3$-descent) we see that $9\mid \#\Sha(E)$.
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\end{enumerate}
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}
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\page{
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\bd{\Large Multiple of Order:}
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\begin{enumerate}
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\item For $E=681B$, the mod~$3$ representation is surjective,
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and $3\mid\mid [E(K):y_K]$ for $K=\Q(\sqrt{-8})$, so (our refined)
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Kolyvagin theorem implies that $\#\Sha(E)=9$, as required.
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\item Kolyvagin's theorem and computation $\implies$ $\#\Sha(E) = 4^?$
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for 571A, 960D, 960N.
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\item
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Using MAGMA's {\tt FourDescent} command,
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we compute $\Sel^{(4)}(E/\Q)$ for 571A, 960D, 960N
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and deduce that $\#\Sha(E)=4$. (Note: MAGMA Documentation currently
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misleading.)
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\end{enumerate}
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}
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\page{
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\heading{The Eighteen Optimal Curves of Rank $>1$}
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There are $18$ curves with conductor $\leq 1000$ and rank $>1$
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(all have rank~$2$):
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%[email protected]:~/people/cremona/data$ awk '$5==2 && $1<=1000 {print $1$2" & "$4"\\\\"}' curves.1-8000
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\vfill
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\begin{center}
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389A,
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433A,
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446D,
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563A,
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571B,
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643A,
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655A,
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664A,
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681C,\\
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707A,
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709A,
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718B,
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794A,
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817A,
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916C,
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944E,
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997B,
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997C
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\end{center}
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\vfill
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For these~$E$ \rd{nobody} currently knows how to show that
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$\Sha(E)$ is finite, let alone trivial. (But mention, e.g., $p$-adic
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$L$-functions.)
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\vfill
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\bd{Motivating Problem 3:}
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Prove the BSD Conjecture for all elliptic
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curve over~$\Q$ of conductor at most $1000$ and rank $\leq 1$.
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\vfill \bd{SECRET MOTIVATION:} Our actual motivation is to
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unify and extend results about BSD and
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algorithms for elliptic curves. The computational challenge is there
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to see what interesting phenomena occur in the data.
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}
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\page{
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\heading{Our Goal}
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\begin{itemize}
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\item
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There are $2463$ optimal curves of conductor at most $1000$.
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\item Of these,
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$18$ have rank~$2$, which leaves~$2445$ curves.
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\item Of these, $2441$ are conjectured to have trivial $\Sha$.
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\end{itemize}
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\begin{center}
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Thus our \rd{goal}
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is to prove that $$\#\Sha(E)=1$$ for these $2441$ elliptic curves.
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\end{center}
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}
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\page{
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\heading{Our Strategy}
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\begin{enumerate}
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\item{}[\rd{Refine}] \label{step:refine} Prove a refinement of
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\underline{Kolyvagin's
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bound} on $\#\Sha(E)$ that is
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suitable for computation.
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\vspace{-2ex}
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\item{}[\rd{Algorithm}] \label{step:alg}\\
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\mbox{}\hspace{1em}\rd{Input:} An elliptic curve over $\Q$ with $r_{\an}\leq 1$.\\
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\mbox{}\hspace{1em}\rd{Output:} Odd $B \geq 1$ such that if
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$p\nmid 2B$, then $p\nmid \#\Sha(E)$.
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\vspace{-4ex}
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\item{}[\rd{Compute}] \label{step:implement} Run the algorithm on our $2441$ curves.
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\vspace{-2ex}
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\item{}[\rd{Descent}] \label{step:analysis}
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If $p\mid B$ and $E[p]$ is reducible:
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Use $p$-descent?
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\vspace{-2ex}
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\item{}[\rd{New Methods}] If $p\mid B$ and $E[p]$ irreducible:
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Try Kato when $r_{\an}=0$. When $r_{\an}=1$,
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use Schneider's theorem, Kato's work,
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explicit computations with $p$-adic heights and
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$p$-adic $L$-functions.
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Also, visibility and level lowering? Further refinement of Kolyvagin's
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theorem?
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\end{enumerate}
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}
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\page{
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\heading{Our Algorithm to Bound $\Sha(E)$}
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\rd{INPUT:} An elliptic curve~$E$ over $\Q$ with $r_{\an} \leq 1$.\\
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\rd{OUTPUT:} Odd $B\geq 1$ such that if $p\nmid 2B$, then
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$\Sha(E/\Q)[p]=0$.
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\begin{enumerate}
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\item{} [\rd{Choose $K$}] Choose a
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quadratic imaginary field $K=\Q(\sqrt{D})$ that
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satisfy the Heegner hypothesis, such that $E/K$
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has analytic rank~1, and $\Disc(K)$
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is divisible by \rd{two primes}.
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(Or two such $K$ each divisible by a single prime.)
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\item{}[\rd{Find $p$-torsion}] Decide for which primes $p$ there is a
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curve $E'$ that is $\Q$-isogenous to $E$ such that $E'(\Q)[p]\neq 0$.
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Let $A$ be the product of these primes.
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\newpage
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\item{}[\rd{Compute Mordell-Weil}]
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\begin{enumerate}
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\item If $r_{\an}=0$, compute generator $z$ for $E^D(\Q)$ mod torsion.
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\item If $r_{\an}=1$, compute generator $z$ for $E(\Q)$ mod torsion.
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\end{enumerate}
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\item{}[\rd{Height of Heegner point}] Compute the
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height $h_K(y_K)$, e.g., using the Gross-Zagier formula
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(and/or directly).
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\item{}[\rd{Index of Heegner point}]
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Compute\\\mbox{}\hspace{3em}
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$I_K = \sqrt{h_K(y_K)/h_K(z)} = [E(K)_{/\tor} : \Z y_K].$
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\item{}[\rd{Refined Kolyvagin}]
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Output $B = A \cdot I_K$.
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\end{enumerate}
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\vfill
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\gr{Theorem (refinement of Kolyvagin):}
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$p\nmid 2B \implies p\nmid \#\Sha(E/\Q).$
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\vfill
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}
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\page{
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\heading{First Attempt to Run the Algorithm}
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% It appears that the one case in which $p\mid B$ but there is no
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% rational $p$-isogeny and $\Sha(E/\Q)[p]=0$ is when $p$ divides some
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% Tamagawa number and $E$ has rank $1$ (when $E$ has rank $0$, a
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% theorem of Kato applies).
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\vfill
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\begin{itemize}
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\item Using \magma{} and the MECCAH cluster,
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I implemented and ran the algorithm on the
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curves of conductor $\leq 1000$, but stopped
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runs if they took over 30 minutes.
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\item
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The computation
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for $318$ curves didn't finish. We
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do not include them below. Also, I don't trust some of
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\magma{}'s elliptic curves functions, since the documentation
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is unclear. However, we assume correctness
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for the rest of this talk.
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\item
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\rd{Future plan:} run each computation without timeouts using
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{\tt mwrank} and {\tt PARI}.
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\end{itemize}
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}
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\page{
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\heading{Results of the First Attempt}
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\vspace{-2ex}
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\begin{enumerate}
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\item For $1363$ curves we have $B=1$. For these curves
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we have proved the full BSD conjecture!
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\vspace{-2ex}
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\item There are $94$ curves for which $B\geq 11$. Of
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these, $6$ have rank~$0$ (so we can likely use Kato's theorem).
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\vspace{-2ex}
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\item There are $39$ curves for which $B\geq 19$.
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For {\em all} of these curves the rank is $1$.
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\vspace{-2ex}
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\item The largest $B$ is $77$, for the rank~$1$
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curves 618F and 894G.
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\vspace{-2ex}
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\item The largest prime divisor of any $B$ is $31$,
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for the rank~$1$ curve 674C.
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\vspace{-2ex}
599
600
\item When $E$ has rank $0$, the algorithm is much more
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difficult, so more likely to time out.
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\end{enumerate}
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}
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\page{
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\heading{Major Obstruction: Tamagawa Numbers}\label{sec:level}
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\bd{Serious Issue:} The Gross-Zagier formula and the BSD conjecture
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together imply that if
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an odd prime $p$ divides a Tamagawa number, then
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$p\mid [E(K) : \Z y_K]$.
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615
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\begin{itemize}
617
\item
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If $E$ has $r_{\an}=0$, and $p\geq 5$, and $\rho_{E,p}$ is surjective,
619
then Kato's theorem (and Mazur, Rubin, et al.) imply that
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{\dred\large $$\ord_p(\#\Sha(E)) \leq \ord_p(L(E,1)/\Omega_E),$$}
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so squareness of $\#\Sha(E)$ frequently saves us.
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623
\item
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Unfortunately, in many cases there is a big
625
Tamagawa number and $r_{\an}=1$, so Kato doesn't apply.
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\end{itemize}
627
}
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629
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\page{
631
\heading{An Example}
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\vfill
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The elliptic curve $E$ called
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141A and given by $y^2 + y = x^3 + x^2 - 12x + 2$ has rank 1
635
and $c_3 = 7$.
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We compute that
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$$
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\#\Sha(E) = 49^{???}.
639
$$
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The representation $\rho_{E,7}$ is surjective, but~$E$ has rank~$1$.\\
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\vfill
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%\begin{itemize}
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%\item{}[Visibility?]
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%The Jacobian $J_0(47)$ is of rank
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%$0$ and is simple of dimension $4$, and we find that $E[7]$ sits in
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%the old subvariety of $J_0(3\cdot 47)$.
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%Hope: Proving
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%something about the Shafarevich-Tate group of the simple rank $0$ abelian
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%variety $J_0(47)$ will imply something about $\Sha(E)[7]$.
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%Note that $L(J_0(47),1)/\Omega = 16/23$.
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%\vfill
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%\item{}[$p$-Adic Approach?]
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\rd{Ralph Greenberg's suggestion:}
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Compute a $p$-adic $L$-function, a $p$-adic regulator, and use
657
theorems of Kato and Peter Schneider to show that $7\nmid
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\#\Sha(E)$. I hope to do this soon.
659
660
%\end{itemize}
661
}
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664
\page{
665
\heading{What Next?}
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\vspace{-2ex}
667
\begin{enumerate}
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\item{}[\rd{Efficiency}] Make the algorithm more efficient.
669
{\small The reason
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the discriminant must be divisible by two primes, or we choose
671
two fields is so we can weaken the surjectivity hypothesis that
672
Kolyvagin imposed. However,
673
in many cases we have surjectivity and could directly use
674
Kolyvagin's theorem. Also \rd{Byungchul Cha's} 2003
675
Johns Hopkins Ph.D. thesis weakens Kolyvagin's
676
hypothesis in another way. Combining all this should speed up
677
the algorithm.}
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\vspace{-2ex}
679
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\item{}[\rd{Finish!}] Run the algorithm to completion on all curves of conductor
681
up to $1000$. Hard part is finding $E^D(\Q)$ for
682
rank~$1$ $E^D$, where $D$ has $3$ digits (so the conductor
683
has $\sim 12$ digits).
684
\vspace{-4ex}
685
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\item{}[\rd{New Theory}] Find a strategy that works when $r_{\an}=1$ and $E$ has a
687
Tamagawa number $\geq 5$. Either refine Kolyvagin, use visibility and level lowering,
688
or Schneider and Kato's results on the $p$-adic main conjecture.
689
690
691
\end{enumerate}
692
693
}
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\end{document}
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\page{
697
\heading{More Examples}
698
699
\begin{itemize}
700
\item 190A1: We have $190=2\cdot 5\cdot 19$ and $c_{2}=11$. There
701
is a $4$-dimensional abelian variety over rank $0$ and level $95$
702
with $\Sha[11]$ trivial that contains $E[11]$.
703
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\item 214A1: We have $214=2\cdot 107$ and $c_{2}=7$. There is
705
a rank $0$ simple abelian variety over level $107$ and dimension $7$
706
that contains $E[7]$.
707
708
\item 674C1: We have $214=2\cdot 337$ and $c_{2}=31$. For this one,
709
there is a rank $0$ simple abelian variety of level $337$ and
710
dimension $15$ that contains $E[31]$ and according to BSD has
711
trivial $\Sha[31]$.
712
\end{itemize}
713
}
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716