\documentclass{beamer} %\usepackage{beamerarticle} \definecolor{dblackcolor}{rgb}{0.0,0.0,0.0} \definecolor{dbluecolor}{rgb}{.01,.02,0.7} \definecolor{dredcolor}{rgb}{0.6,0,0} \definecolor{dgraycolor}{rgb}{0.30,0.3,0.30} \newcommand{\dblue}{\color{dbluecolor}} \newcommand{\dred}{\color{dredcolor}} \newcommand{\dblack}{\color{dblackcolor}} \usepackage{listings} \lstdefinelanguage{Sage}[]{Python} {morekeywords={True,False,sage,singular}, sensitive=true} \lstset{frame=none, showtabs=False, showspaces=False, showstringspaces=False, commentstyle={\ttfamily\color{dredcolor}}, keywordstyle={\ttfamily\color{dbluecolor}\bfseries}, stringstyle ={\ttfamily\color{dgraycolor}\bfseries}, language = Sage, basicstyle={\scriptsize \ttfamily}, aboveskip=.3em, belowskip=.1em } \usepackage{url} \usepackage{hyperref} \hypersetup{colorlinks=true, urlcolor=blue} \usepackage{comment} \usepackage{colortbl} \usepackage{fancybox} \usepackage[utf8x]{inputenc} \mode { % \usetheme{Rochester} % \usetheme{Berkeley} \usetheme{PaloAlto} %\usecolortheme{crane} % \usecolortheme{orchid} \usecolortheme{whale} % \usecolortheme{lily} \setbeamercovered{transparent} % or whatever (possibly just delete it) } \usepackage{ngerman} \newcommand{\n}{\mathfrak{n}} \newcommand{\p}{\mathfrak{p}} \newcommand{\Q}{\mathbf{Q}} \newcommand{\Z}{\mathbf{Z}} \DeclareMathOperator{\Norm}{Norm} \DeclareMathOperator{\ord}{ord} \title[Curves over $\Q(\sqrt{5})$]{Modular Elliptic Curves over $\Q(\sqrt{5})$} \date{October 2010} \author[W. Stein]{William Stein (joint work with Aly Deines, Joanna Gaski)} %\newcommand{\todo}[1]{[[#1]]} \newcommand{\todo}[1]{} \begin{document} \begin{frame} \titlepage \end{frame} \begin{frame}{The Problem} \begin{block}{Cremona's Book over $\Q(\sqrt{5})$} Create a document like Cremona's book, but over the real quadratic field $F=\Q(\sqrt{5})$. It will contain a table of every (modular) elliptic curve over $F$ of conductor $\n$ such that $\Norm(\n)\leq 1000$, along with extensive data about every such curve (much more than what is in Cremona's book -- info like Robert Miller is computing about elliptic curves over $\Q$, which is relevant to the BSD conjecture). Then go up to norm {\em one hundred thousand} or more! \end{block} \begin{center} \includegraphics[width=.4\textwidth]{cremona} \end{center} \end{frame} \section{Background} \begin{frame}{Conductor} \begin{block}{What is the ``Conductor'' of an Elliptic Curve?} If $E$, given by $y^2 + a_1 xy + a_3 y = x^3 + a_2x^2 + a_4 x + a_6$ is a Weierstrass equation for an elliptic curve over $F$, it has a discriminant, which is a polynomial in the $a_i$: {\tiny $ \Delta = - a_{1}^{4} a_{2} a_{3}^{2} + a_{1}^{5} a_{3} a_{4} - a_{1}^{6} a_{6} - 8 a_{1}^{2} a_{2}^{2} a_{3}^{2} + a_{1}^{3} a_{3}^{3} + 8 a_{1}^{3} a_{2} a_{3} a_{4} + a_{1}^{4} a_{4}^{2} - 12 a_{1}^{4} a_{2} a_{6} - 16 a_{2}^{3} a_{3}^{2} + 36 a_{1} a_{2} a_{3}^{3} + 16 a_{1} a_{2}^{2} a_{3} a_{4} - 30 a_{1}^{2} a_{3}^{2} a_{4} + 8 a_{1}^{2} a_{2} a_{4}^{2} - 48 a_{1}^{2} a_{2}^{2} a_{6} + 36 a_{1}^{3} a_{3} a_{6} - 27 a_{3}^{4} + 72 a_{2} a_{3}^{2} a_{4} + 16 a_{2}^{2} a_{4}^{2} - 96 a_{1} a_{3} a_{4}^{2} - 64 a_{2}^{3} a_{6} + 144 a_{1} a_{2} a_{3} a_{6} + 72 a_{1}^{2} a_{4} a_{6} - 64 a_{4}^{3} - 216 a_{3}^{2} a_{6} + 288 a_{2} a_{4} a_{6} - 432 a_{6}^{2} $} \end{block} \small Among all models for $E$, there are some that simultaneously minimize $\ord_\p(\Delta)$ for all primes $\p$: the {\em minimal discriminant}. Reducing such a model modulo $\p$ will give either an elliptic curve, or a nodal cubic (e.g., $y^2=x^2(x-1)$), or a cuspidal cubic (e.g., $y^2=x^3$). The {\em conductor} of $E$ is the ideal $$ \n = \prod_{\p\mid \Delta} \p^{f_\p} $$ where $f_\p=1$ if reduction is nodal, and $f_\p\geq 2$ if cuspidal... \end{frame} \begin{frame}{A Concrete Problem} \begin{block}{Concrete Problem} What is the ``simplest'' (smallest conductor) elliptic curve over $F=\Q(\sqrt{5})$ of rank 2? See my recent NSF grant proposal for motivation, at \url{http://wstein.org/grants/2010-ant/}. \end{block} \vfill To the best of my knowledge, this is open. I asked Lassina Dembele, who has a big systematic tables of curves over $F$ of {\em prime} conductor with norm $\leq 5000$ and he didn't know of any. \vfill So we still don't know! \end{frame} \begin{frame}[fragile]{Upper Bound} The following search finds the first curve over $\Q(\sqrt{5})$, which has all $a_i\in \Q$, with rank $2$: \begin{lstlisting} for E in cremona_optimal_curves([1..100]): r = rank_over_F(E) print E.cremona_label(), r if r == 2: break \end{lstlisting} It finds {\bf 61a}: $$ E:\qquad y^2 + xy = x^3 - 2x + 1, $$ for which $E(\Q)$ and $E^{5}(\Q)$ both have rank $1$. The norm of the conductor $\n$ over $F$ is $61^2 = 3721$. \end{frame} \begin{frame}[fragile]{Naive Enumeration} \begin{block}{} In 2004 I had Jennifer Sinnot (a Harvard undergrad) make big tables of elliptic curves (by just running through $a_i$) over various quadratic fields, including $F=\Q(\sqrt{5})$. See \begin{center} \url{http://wstein.org/Tables/e_over_k/} \end{center} \end{block} {\tiny \begin{verbatim} Norm(N) [a4,a6] Torsion j-invariant Conductor N 320 [-7,-6] [2,4] 8 148176/25 (-8*a) 320 [-2,1] [2,4] 8 55296/5 (-8*a) 1024 [-2*a+6,0] [2,1] 2 1728 (32) 1024 [-1,0] [2,2] 4 1728 (32) 1024 [4,0] [4,1] 4 1728 (32) 1280 [-7,6] [2,4] 8 148176/25 (-16*a) 1280 [-a-3,-a-2] [2,2] 4 55296/5 (-16*a) 1280 [-a-3,a+2] [2,2] 4 55296/5 (-16*a) 1280 [-2,-1] [2,2] 4 55296/5 (-16*a) 1296 [0,1] [6,1] 6 0 (36) ... \end{verbatim}} \begin{block}{} \scriptsize Joanni Gaski is doing her masters thesis right now at UW on extending this... \end{block} \end{frame} \begin{frame}[fragile]{Dembele's Ph.D. Thesis $\sim$ 2002} \begin{block}{Dembele's Ph.D. Thesis} \begin{itemize} \item Around 2002, Lassina Dembele did a Ph.D. thesis with Henri Darmon at McGuill university on explicit computation of Hilbert modular forms. \item {\bf Application:} assuming modularity conjectures, enumerate {\em all} elliptic curves over $\Q(\sqrt{5})$ of each conductor! \item Table of curves of prime conductor with norm up to $5000$. \item There are 431 conductors $\n$ with norm $\leq 1000$, and Lassina's tables contain 50 distinct conductors with norm $\leq 1000$ (out of the 163 prime conductors of norm $\leq 1000$). \end{itemize} \end{block} \end{frame} \begin{frame}{Hilbert Modular Forms?} \begin{block}{Hilbert Modular Forms} \begin{itemize} \item These are holomorphic functions on a product of two copies of the upper half plane. \item But there is a purely algebraic/combinatorial reinterpretation of them due to Jacquet and Langlands, which involves quaternion algebras. \item Explicit definition of a finite set $S$ and an action of commuting operators $T_n$ on the free abelian group $X$ on the elements of $S$ such that the systems of Hecke eigenvalues in $\Z$ correspond to elliptic curves over $\Q(\sqrt{5})$, up to isogeny. \item Flip to Dembele's thesis and show the tables of Hecke eigenvalues and explain them. (page 40) \end{itemize} \end{block} \end{frame} \section{The Plan} \begin{frame}[fragile]{The Plan: Finding Curves} \begin{block}{Computing Hilbert Modular Forms} Implement an optimized variant of Dembele's algorithm, which is fast enough to compute a few Hecke operators for any level $\n$ with $\Norm(\n)\leq 10^5$. The dimensions of the relevant space are mostly less than 2000, so the linear algebra is likely very feasible. I did this yesterday {\em modulo bugs...}: {\scriptsize \begin{lstlisting} sage: N = F.factor(100019)[0][0]; N Fractional ideal (65*a + 292) sage: time P = IcosiansModP1ModN(N) CPU times: user 0.19 s, sys: 0.00 s, total: 0.19 s sage: P.cardinality() 1667 sage: time T5 = P.hecke_matrix(F.primes_above(5)[0]) CPU times: user 0.38 s, sys: 0.11 s, total: 0.49 s sage: N.norm() 100019 sage: 10^5 100000 \end{lstlisting}} {\dred Yes, that just took a total less than one second!!!} \end{block} \end{frame} \begin{frame}[fragile]{Magma?} \begin{block}{But Magma can compute Hilbert Modular Forms...} Why not just use Magma, which already has Hilbert modular forms in it, due to the great work of John Voight, Lassina Dembele, and Steve Donnelly? \begin{lstlisting} [wstein@eno ~]$ magma Magma V2.16-13 Fri Nov 5 2010 18:09:32 on eno [Seed = 666889163] Type ? for help. Type -D to quit. > F := QuadraticField(5); > time M := HilbertCuspForms(F, Factorization(Integers(F)*100019)[1][1]); Time: 0.030 > time T5 := HeckeOperator(M, Factorization(Integers(F)*5)[1][1]); Time: 235.730 # 4 minutes \end{lstlisting} The value in Magma's HMF's are that the implementation is {\em very} general. But slow. And the above was just one Hecke operator. We'll need many, and Magma gets {\em much} slower as the subscript of the Hecke operator grows. {\dred A factor of 1000 in speed kind of matters.} \end{block} \end{frame} \begin{frame}{Computing Hilbert Modular Forms} \begin{block}{Overview of Dembele's Algorithm} \begin{enumerate} \item Let $R=$ maximal order in Hamilton quaternion algebra over $F=\Q(\sqrt{5})$. \item Compute the finite set $S = R^* \backslash {\mathbf P}^1(\mathcal{O}_F/\n)$. Let $X=$ free abelian group on $S$. \item To compute the Hecke operator $T_{\p}$ on $X$, compute (and store once and for all) $\#\mathbf{F}_{\p}+1$ elements $\alpha_{\p,i} \in B$ with norm $\p$, then compute $$ T_{\p}(x) = \sum \alpha_{\p,i}(x). $$ \end{enumerate} \end{block} That's it! Now scroll through the 1500 line file I wrote yesterday that implements this in many cases... but still isn't done. Deines-Stein: article about how to do 2-3 above {\em quickly}? \end{frame} \begin{frame}{Computing Equations for Curves and Ranks} \begin{block}{} \begin{enumerate} \item Hilbert modular forms, as explained above, will allow us to find by linear algebra the Hecke eigenforms corresponding to all curves over $\Q(\sqrt{5})$ with conductor of norm $\leq 10^5$. \item Finding the corresponding Weierstrass equations is a whole different problem. Joanni Gaski will have made a huge table by then, and we'll find some chunk of them there. \item Noam Elkies outlined a method to make an even better table, and we'll implement it and try. \item Fortunately, if Sage is super insanely fast at computing Hecke operators on Hilbert modular forms, then it should be possible to compute the {\em analytic ranks} of the curves found above without finding the actual curves. By BSD, this should give the ranks. This should answer the problem that started this lecture, at least assuming standard conjectures (and possibly using a theorem of Zhang). \end{enumerate} \end{block} \end{frame} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: