Sharedwww / talks / 2012-04-28-talk / sqrt5_comp.texOpen in CoCalc
Author: William A. Stein
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\title{Elliptic Curves Over $F=\Q(\sqrt{5})$}
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\author[William Stein]{William Stein (University of Washington)\\
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in Chicago (UIC) at the Atkin Memorial Workshop}
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\institute{University of Washington}
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\date{April 27--29, 2012}
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\begin{document}
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\begin{frame}
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\maketitle
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\end{frame}
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\begin{frame}
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\frametitle{Joint Work...}
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\vfill
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This talk represents joint work with Jonathan Bober, Alyson Deines,
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Joanna Gaski, Ariah Klages-Mundt, Benjamin LeVeque, R. Andrew Ohana,
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Ashwath Rabindranath, and Paul Sharaba.
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\vfill
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{\bf Acknowledgement:} John Cremona, Lassina Dembele, Noam Elkies, Tom
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Fisher, Richard Taylor, and John Voight for helpful conversations and data.
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I used Sage (\url{http://www.sagemath.org}) extensively.
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\vfill
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\end{frame}
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\begin{frame}
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\frametitle{Motivation}
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\vfill
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\begin{quote}
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\Large``The object of numerical computation is theoretical advance.''
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\vspace{1ex}
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\mbox{}\hspace{20ex} - Oliver Atkin
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\end{quote}
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\begin{center}
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\vfill
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\includegraphics[width=1in]{pics/atkin}
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\hspace{3em}
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\includegraphics[width=.5\textwidth]{pics/atkinpile}
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\end{center}
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\vfill
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\end{frame}
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\begin{frame}
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\frametitle{Contents}
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\LARGE
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\vfill
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\begin{enumerate}
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\item Tables
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\item Finding all $E$ attached to a newform $g$
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\item Finding newforms
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\end{enumerate}
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% Overview:
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% \begin{enumerate}
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% \item Find a list of all rational Hilbert modular newforms $f$ in
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% $S_{(2,2)}(\n_E)$. Uses: Quaternion algebras and linear algebra.
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% \item (Stay for Jon Bober's talk right after mine) Find Weierstrass
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% equations for all corresponding elliptic curves. Uses: Many search
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% techniques and computing isogenies.
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% \item Compute invariants of each curve. Uses: descent, Tate's algorithm, $L$-series, etc.
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% \end{enumerate}
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\end{frame}
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\begin{frame}
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\vfill
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\Huge
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\begin{center}
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1: Tables
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\end{center}
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\vfill
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\end{frame}
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\begin{frame}
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Source: These tables and much code were made at a summer
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REU\footnote{Research Experience for Undergraduates} at University
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of Washington last summer.
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\vfill
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See \url{https://github.com/williamstein/sqrt5}.
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\vfill
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{\dred Remark:} If $E/F$ and $\sigma(\sqrt{5})=-\sqrt{5}$, then
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$E^{\sigma}$ is another curve over $F$. All of our tables {\em do}
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include {\em both} $E$ and $E^{\sigma}$! We tried to avoid this redundancy
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but it caused too much confusion.
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\end{frame}
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\begin{frame}
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\frametitle{Counts of Curves over $F$ up to Norm Conductor 1831}
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\begin{center}
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\begin{table}[h]
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\caption{Curves over $\Q(\sqrt{5})$\label{table:total-counts}}
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\begin{tabular}{@{}lcrrcr@{}}\toprule
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\textbf{rank} & \phantom{a} &\textbf{\#isog} & \textbf{\#isom} & \phantom{a} & \textbf{smallest $\Norm(\n)$} \\\cmidrule{1-1}\cmidrule{3-4}\cmidrule{6-6}
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$0$ & & $745$ & $2174$ & & $31$\\
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$1$ & & $667$ & $1192$ & & $199$ \\
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$2$ & & $2$ & $2$ & & $1831$ \\\cmidrule{1-6}
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total & & $1414$ & $3368$ & & -\; \\\bottomrule
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\end{tabular}
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\end{table}
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{Number of Isogeny Classes over $F$ up to Norm Conductor 1831}
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\begin{center}
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\begin{table}[h]
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\caption{Number of Isogeny classes of a given size\label{table:isogeny-sizes}}
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\begin{tabular}{@{}lcrrrrrrrcr@{}}\toprule
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& \phantom{a} & \multicolumn{7}{c}{\textbf{size}} & \phantom{a} & \\\cmidrule{3-9}
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\textbf{bound} & & 1 & 2 & 3 & 4 & 6 & 8 & 10 & & \textbf{total} \\\midrule
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199 & & 2 & 21 & 3 & 20 & 8 & 9 & 1 & & 64 \\
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1831 & & 498 & 530 & 36 & 243 & 66 & 38 & 3 & & 1414 \\\bottomrule
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\end{tabular}
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\end{table}
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{Rank Data}
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\small
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\begin{center}
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\begin{table}[h]
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\caption{Counts of classes and curves with bounded norm conductors and specified ranks\label{table:increasing-counts}}
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\begin{tabular}{@{}lcrrrcrcrrrcr@{}}\toprule
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& \phantom{a} & \multicolumn{5}{c}{\textbf{\#isog}} & \phantom{ab} & \multicolumn{5}{c}{\textbf{\#isom}} \\\cmidrule{3-7}\cmidrule{9-13}
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& & \multicolumn{3}{c}{\textbf{rank}} & & & & \multicolumn{3}{c}{\textbf{rank}} & & \\\cmidrule{3-5}\cmidrule{9-11}
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\textbf{bound} & & 0 & 1 & 2 & & \textbf{total} & & 0 & 1 & 2 & & \textbf{total} \\\midrule
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200 & & 62 & 2 & 0 & & 64 & & 257 & 6 & 0 & & 263 \\
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400 & & 151 & 32 & 0 & & 183 & & 580 & 59 & 0 & & 639 \\
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600 & & 246 & 94 & 0 & & 340 & & 827 & 155 & 0 & & 982 \\
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800 & & 334 & 172 & 0 & & 506 & & 1085 & 285 & 0 & & 1370 \\
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1000 & & 395 & 237 & 0 & & 632 & & 1247 & 399 & 0 & & 1646 \\
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1200 & & 492 & 321 & 0 & & 813 & & 1484 & 551 & 0 & & 2035 \\
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1400 & & 574 & 411 & 0 & & 985 & & 1731 & 723 & 0 & & 2454 \\
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1600 & & 669 & 531 & 0 & & 1200 & & 1970 & 972 & 0 & & 2942 \\
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1800 & & 729 & 655 & 0 & & 1384 & & 2128 & 1178 & 0 & & 3306 \\
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1831 & & 745 & 667 & 2 & & 1414 & & 2174 & 1192 & 2 & & 3368 \\\bottomrule
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\end{tabular}
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\end{table}
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{Isogeny Degrees}
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\small
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\begin{center}
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\begin{table}[h]
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\caption{Isogeny degrees\label{table:degree}}
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\begin{tabular}{@{}lcrrclr@{}}\toprule
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\textbf{degree} & \phantom{a} & \textbf{\#isog} & \textbf{\#isom} & \phantom{a} & \textbf{example curve} & $\Norm(\n)$ \\\cmidrule{1-1}\cmidrule{3-4}\cmidrule{6-7}
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None & & 498 & 498 & & $[\varphi+1,1,1,0,0]$ & 991 \\
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2 & & 652 & 2298 & & $[\varphi,-\varphi+1,0,-4,3\varphi-5]$ & 99 \\
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3 & & 289 & 950 & & $[\varphi,-\varphi,\varphi,-2\varphi-2,2\varphi+1]$ & 1004 \\
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5 & & 65 & 158 & & $[1,0,0,-28,272]$ & 900 \\
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7 & & 19 & 38 & & $[0,\varphi+1,\varphi+1,\varphi-1,-3\varphi-3]$ & 1025 \\\bottomrule
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\end{tabular}
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\end{table}
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\end{center}
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\vfill
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% \begin{itemize}
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% \item Of course as Elkies pointed out, many other isogeny degrees occur
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% for elliptic curves over $F$.
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% \item I'm surprised we see no ``interesting isogeny degrees'' in our data.
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% \end{itemize}
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\end{frame}
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\begin{frame}
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\frametitle{Torsion Subgroups of Elliptic Curves over $F$}
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(I don't trust this table.)
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\begin{center}
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\scriptsize
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\begin{table}[h]
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\caption{Distribution of torsion subgroups up to norm conductor 1831}
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\begin{tabular}{@{}lcrclr@{}}\toprule
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\textbf{structure} & \phantom{a} & \textbf{\#isom} & \phantom{a} & \textbf{example curve} & $\text{Norm}(\mathfrak{n})$ \\\cmidrule{1-1}\cmidrule{3-3}\cmidrule{5-6}
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1 & & 296\footnote{On the previous slide there were 498 with no isogenies, so this or that must be wrong! We will fix this. It should be 790, I think. Sorry.} & & $[0,-1,1,-8,-7]$ & 225 \\
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$\Z/2\Z$ & & 1453 & & $[\vphi,-1,0,-\vphi-1,\vphi-3]$ & 164 \\
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$\Z/3\Z$ & & 202 & & $[1,0,1,-1,-2]$ & 100 \\
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$\Z/4\Z$ & & 243 & & $[\vphi+1,\vphi-1,\vphi,0,0]$ & 79 \\
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$\Z/2\Z\oplus\Z/2\Z$ & & 312 & & $[0,\vphi+1,0,\vphi,0]$ & 256 \\
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$\Z/5\Z$ & & 56 & & $[1,1,1,22,-9]$ & 100 \\
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$\Z/6\Z$ & & 183 & & $[1,\vphi,1,\vphi-1,0]$ & 55 \\
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$\Z/7\Z$ & & 13 & & $[0,\vphi-1,\vphi+1,0,-\vphi]$ & 41 \\
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$\Z/8\Z$ & & 21 & & $[1,\vphi+1,\vphi,\vphi,0]$ & 31 \\
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$\Z/2\Z\oplus\Z/4\Z$ & & 51 & & $[\vphi+1,0,0,-4,-3\vphi-2]$ & 99 \\
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$\Z/9\Z$ & & 6 & & $[\vphi,-\vphi+1,1,-1,0]$ & 76 \\
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$\Z/10\Z$ & & 12 & & $[\vphi+1,\vphi,\vphi,0,0]$ & 36 \\
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$\Z/12\Z$ & & 6 & & $[\vphi,\vphi+1,0,2\vphi-3,-\vphi+2]$ & 220 \\
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$\Z/2\Z\oplus\Z/6\Z$ & & 11 & & $[0,1,0,-1,0]$ & 80 \\
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$\Z/15\Z$ & & 1 & & $[1,1,1,-3,1]$ & 100 \\
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$\Z/2\Z\oplus\Z/8\Z$ & & 2 & & $[1,1,1,-5,2]$ & 45 \\\bottomrule
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\end{tabular}
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\end{table}
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{Comparison: $F$ versus $\Q$}
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\begin{center}
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\scriptsize
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\begin{table}[h]
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\caption{Distribution of torsion subgroups up to (norm) conductor 1831}
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\begin{tabular}{@{}lcrcr@{}}\toprule
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\textbf{structure} & \phantom{a} & \textbf{\#isom over }$F$ & \phantom{a} & \textbf{\#isom over }$\Q$\\
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\cmidrule{1-1}\cmidrule{3-3}\cmidrule{5-5}
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1 & & 296 (*) && 3603\\
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$\Z/2\Z$ & & 1453 && 4580 \\
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$\Z/3\Z$ & & 202 && 523\\
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$\Z/4\Z$ & & 243 && 481\\
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$\Z/2\Z\oplus\Z/2\Z$ & & 312 && 726\\
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$\Z/5\Z$ & & 56 && 54\\
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$\Z/6\Z$ & & 183 && 208\\
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$\Z/7\Z$ & & 13 && 11\\
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$\Z/8\Z$ & & 21 && 16\\
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$\Z/2\Z\oplus\Z/4\Z$ & & 51 && 60\\
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$\Z/9\Z$ & & 6 && 4\\
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$\Z/10\Z$ & & 12 && 8\\
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$\Z/12\Z$ & & 6 && 2\\
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$\Z/2\Z\oplus\Z/6\Z$ & & 11 && 6\\
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$\Z/15\Z$ & & 1 && 0 \\
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$\Z/2\Z\oplus\Z/8\Z$ & & 2 && 1\\
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\bottomrule
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\end{tabular}
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\end{table}
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\end{center}
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\end{frame}
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\begin{frame}
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\frametitle{Shafarevich-Tate Groups}
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\begin{center}
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\begin{table}[h]
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\caption{$\Sha$\label{table:sha}}
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\begin{tabular}{@{}lcrclr@{}}\toprule
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\textbf{\#\Sha} & \phantom{a} & \textbf{\#isom} & \phantom{a} & \textbf{first curve having \#\Sha}& $\Norm(\n)$ \\\cmidrule{1-1}\cmidrule{3-3}\cmidrule{5-6}
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1 & & 3191 & & $[1,\vphi+1,\vphi,\vphi,0]$ & 31 \\
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4 & & 84 & & $[1, 1, 1, -110, -880]$ & 45 \\
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\multirow{2}{*}{9} & & \multirow{2}{*}{43} & & $[\vphi+1,-\vphi,1,-54686\vphi-35336,$ & \multirow{2}{*}{76} \\
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& & & & \multicolumn{1}{r}{$-7490886\vphi-4653177]$} & \\
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\multirow{2}{*}{16} & & \multirow{2}{*}{16} & & $[1,\vphi,\vphi+1,-4976733\vphi-3075797,$ & \multirow{2}{*}{45} \\
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& & & & \multicolumn{1}{r}{$-6393196918\vphi-3951212998]$} & \\
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25 & & 2 & & $[0, -1, 1, -7820, -263580]$ & 121 \\
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\multirow{2}{*}{36} & & \multirow{2}{*}{2} & & $[1,-\vphi+1,\vphi,1326667\vphi-2146665,$ & \multirow{2}{*}{1580} \\
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& & & & \multicolumn{1}{r}{$880354255\vphi-1424443332]$} & \\\bottomrule
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\end{tabular}
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\end{table}
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\end{center}
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\end{frame}
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\begin{frame}
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\vfill
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\Huge
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\begin{center}
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2: Finding all $E$ attached to a newform $g$
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\end{center}
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\vfill
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\end{frame}
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\begin{frame}
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\frametitle{The Modularity Conjecture}
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Modularity is critical to making systematic tables.
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\vfill
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\begin{conjecture}
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There is a bijection\footnote{We consider $L$-series to be equal only if all
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of their Euler factors are equal!}
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$$
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\left\{L(E,s) : E/F \text{ cond }\n\right\} \xrightarrow{\text{conj }\cong}
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\left\{L(f,s) : \text{ newform } f \in S_{(2,2)}(\Gamma_0(\n);\Q) \right\}
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$$
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\vspace{1ex}
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\end{conjecture}
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\vfill
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{\bf Unpublished Remark (Taylor):} If $E[3]|_{\Gal(\overline{\Q}/F(\zeta_3))}$ is
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absolutely irreducible, then modularity follows from recent work
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of Gee and Kisin.
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\end{frame}
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\begin{frame}
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\frametitle{Finding an $E$ attached to a newform $g$}
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\begin{theorem}
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Assume the modularity conjecture. There is an algorithm that takes
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as input a Hilbert modular newform $g \in
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S_{(2,2)}(\Gamma_0(\n);\Q)$ and outputs an elliptic curve $E/F$ with
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$L(E,s)=L(g,s)$.
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\end{theorem}
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\begin{proof}
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By computing all the rational newforms in
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$S_{(2,2)}(\Gamma_0(\n);\Q)$, find a bound $B$ so that the
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eigenvalues $a_{\p}$ for $N(\p)\leq B$ determine a newform. Enumerate
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the countably many elliptic curves $E/F$ in any way you like; when
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you find one with conductor $\n$, use the bound $B$ to
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determine whether or not $L(E,s)=L(g,s)$. Since $E$ corresponds to
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{\em some} newform, this procedure must terminate with the correct
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answer.
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\end{proof}
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\begin{enumerate}
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\item Similar argument for abelian varieties of $\GL_2$-type.
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\item Cremona: ``this algorithm is not {\em respectable}!''
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\end{enumerate}
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\end{frame}
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\begin{frame}
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\frametitle{Finding an $E$ attached to a newform $g$}
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\begin{enumerate}
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\item {\dblue Naive enumeration} -- previous slide
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\item {\dblue Sieved enumeration} -- use $a_{\p}$ to impose congruence conditions
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\item {\dblue Torsion families} -- use $a_{\p}$ to determine whether $\ell\mid \#E(F)$, and if so search
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over the family of curves with $\ell$-torsion.
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\item {\dblue Congruence families} -- if you know $E'$ and that $E'[\ell]\ncisom E[\ell]$, use
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Tom Fisher's explicit families.
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\item {\dblue Twisting} -- find a minimal conductor twist.
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\item {\dblue Cremona-Lingham} -- find curves with good reduction outside $\n$.
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\item {\dblue Dembele} -- reverse engineer periods from special values of $L$-series.
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\item {\dblue Elkies} -- use the $\lambda$ invariant.
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\end{enumerate}
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\vfill
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Jon Bober's talk next will have a lot more to say about this.
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\end{frame}
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\begin{frame}
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\frametitle{Finding {\em all} $E$ attached to a newform $g$}
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Compute the isogeny class of a curve using the following two steps
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repeatedly on each curve found until we find nothing new.
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\begin{enumerate}
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\item
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Use Billerey (2011) to compute a set $S$ of possible prime degrees of
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isogenies $E\to E'$.
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\item
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For each $\ell \in S$, use formulas (e.g., as in Kohel's thesis) to find
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all $\psi:E \to E'$ of degree $\ell$.
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\end{enumerate}
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\end{frame}
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\begin{frame}[fragile]
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\frametitle{Billerey in Code}
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\tiny
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\begin{verbatim}
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def _plstar1(E, q):
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R.<x> = F[]
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t12 = 2048*x^12 -6144*x^10 + 6912*x^8 -3584*x^6 + 840*x^4 -72*x^2 + 1
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t12p = 2048*x^6 -6144*x^5 + 6912*x^4 -3584*x^3 + 840*x^2 -72*x + 1
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t24 = 2*(t12)^2 - 1
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#this is only for primes that have no ramification and have good reduction
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if len(F.primes_above(q)) == 1:
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w1 = 1 - 2*(q^12)*t12(x/(2*q)) + q^24
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t1 = E.change_ring(F.ideal(q).residue_field()).trace_of_frobenius()
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w = w1(t1)
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m = []
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for zee in factor(ZZ(w)):
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m.append(zee[0])
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return m
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else:
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v = F.primes_above(q)
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t1 = E.change_ring(v[0].residue_field()).trace_of_frobenius()
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t2 = E.change_ring(v[1].residue_field()).trace_of_frobenius()
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w1 = t12p(x^2/(4*q))
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w = 1 - 4*(q^12)*w1(t1)*w1(t2) - 2*(q^24)*(1- 2*(w1(t1)^2 + w1(t2)^2)) \
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- 4*(q^36)*w1(t1)*w1(t2) + q^48
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m = []
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for zee in factor(ZZ(w)):
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m.append(zee[0])
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return m
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def _plstar12(E, q):
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#same caveat, only for unramified and good reduction
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if len(F.primes_above(q)) == 1:
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t1 = E.change_ring(F.prime_above(q).residue_field()).trace_of_frobenius()
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m = [q]
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try:
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for v in factor(t1):
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\end{verbatim}
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\end{frame}
500
501
\begin{frame}[fragile]
502
\tiny
503
\begin{verbatim}
504
m.append(v[0])
505
for v in factor(t1^2 - q^2):
506
m.append(v[0])
507
for v in factor(t1^2 - 4*q^2):
508
m.append(v[0])
509
for v in factor(t1^2 - 3*q^2):
510
m.append(v[0])
511
s1 = set(m)
512
m = list(s1)
513
return m
514
except ArithmeticError:
515
return 0
516
else:
517
t1 = E.change_ring(F.primes_above(q)[0].residue_field()).trace_of_frobenius()
518
t2 = E.change_ring(F.primes_above(q)[1].residue_field()).trace_of_frobenius()
519
m = [q]
520
try:
521
for v in factor((t1^2 + t2^2 - q^2)^2 - 3*(t1^2)*(t2^2)):
522
m.append(v[0])
523
for v in factor(t1^2 - t2^2):
524
m.append(v[0])
525
for v in factor(t1^2 +t2^2 - 4*q^2):
526
m.append(v[0])
527
for v in factor((t1^2 + t2^2 - 3*q^2)^2 - (t1*t2)^2):
528
m.append(v[0])
529
s1 = set(m)
530
m = list(s1)
531
return m
532
except ArithmeticError:
533
return 0
534
535
\end{verbatim}
536
\end{frame}
537
538
\begin{frame}[fragile]
539
\tiny
540
\begin{verbatim}
541
def billerey_primes(E):
542
ans = set([])
543
Bad = [v[0] for v in E.conductor().norm().factor()]
544
Pr = prime_range(1000)
545
num = 0
546
i = 0
547
X = [set([3])]
548
while num < 3:
549
if not Pr[i] in Bad and Pr[i] != 5:
550
try:
551
X.append(set(_plstar1(E, Pr[i]) + _plstar12(E, Pr[i])))
552
num += 1
553
except TypeError:
554
pass
555
i += 1
556
ans = (X[1].intersection(X[2])).intersection(X[3])
557
ans = ans.union(set(Bad)).union(set([2,3,5]))
558
return list(sorted(ans))
559
\end{verbatim}
560
\end{frame}
561
562
563
\begin{frame}
564
\vfill
565
\Huge
566
\begin{center}
567
3: Finding newforms
568
\end{center}
569
\vfill
570
\end{frame}
571
572
573
\begin{frame}
574
\frametitle{Computing Hilbert Modular Forms over $F$}
575
576
\begin{enumerate}
577
578
\item The algorithm is from Lassina Dembele's Ph.D. thesis.
579
See his {\em Explicit computation of Hilbert modular forms on $\Q(\sqrt{5})$} (2005).
580
581
\item Jacquet-Langlands: Computing Hecke module of Hilbert modular
582
forms of level $\n$ over $F$ same as computing Hecke module with
583
basis that right ideal classes in a certain order (of level $\n$) in
584
the Hamilton quaternion algebra over $F$.
585
586
\item Dembele: Computing right ideal classes same as computing
587
$\P^1(R/\n)$, where $R=\Z[\vphi]\subset F$.
588
589
\end{enumerate}
590
591
\end{frame}
592
593
594
\begin{frame}
595
\frametitle{Dembele's Algorithm in One Slide}
596
\begin{enumerate}
597
598
\item Hamiltonian quaternions
599
$F[i,j,k]$ ramified at the infinite places.
600
601
\item Maximal order
602
{\small
603
$$
604
S = R\Bigl[\frac{1}{2}(1-\overline{\vphi} i + \vphi j),\,
605
\frac{1}{2}(-\overline{\vphi} i + j + \vphi k),\,
606
\frac{1}{2}(\vphi i - \overline{\vphi} j + k), \,
607
\frac{1}{2}(i + \vphi j - \overline{\vphi} k)\Bigr].
608
$$}
609
610
\item $\P^1(R/\n)=$ equivalence classes of
611
column vectors with two coprime entries $a,b \in R/\n$ modulo the
612
action of $(R/\n)^*$.
613
614
\item
615
For each $\p\mid \n$, fix {\em choice} of isomorphism
616
$F[i,j,k]\otimes F_{\p} \ncisom M_2(F_{\p})$, which induces a {\em choice} of left
617
action of $S^*$ on $\P^1(R/\n)$.
618
\item Jacquet-Langlands: There's an isomorphism of $\T$-modules
619
$$
620
\C[S^* \backslash \P^1(R/\n)] \isom M_{(2,2)}(\Gamma_0(\n)).
621
$$
622
\item $S^*$ acts through the {\em octonian} group (which is finite and explicit).
623
\item $T_{\p}([x]) = \sum [\alpha x]$, where sum is over the classes
624
$[\alpha]\in S/S^*$ with $N_{\text{red}}(\alpha)=\pi_{\p}$,
625
where $\pi_{\p}$ is fixed choice of positive generator of~$\p$.
626
\end{enumerate}
627
628
629
\end{frame}
630
631
\begin{frame}
632
\frametitle{Implementation Notes}
633
\begin{enumerate}
634
\item
635
Critical that we can compute with
636
$\P^1(R/\n)$ very, very, very quickly.
637
638
\item Prime power $\n=\p^e$ case: Each $[x:y] \in \P^1(R/\p^e)$ has a
639
unique representative $[1:b]$ or $[a:1]$ with $a$
640
divisible by $\p$.
641
Easy to put any $[x:y]$ in this canonical form.
642
643
\item General case: factor $\n = \prod_{i=1}^m \p_i^{e_i}$.
644
Have a bijection $\P^1(R/\n) \isom \prod_{i=1}^m
645
\P^1(R/\p_i^{e_i})$,
646
thus reducing to the prime power case.
647
Represent elements of $R/\n$ as
648
$m$-tuples in $\prod R/\p_i^{e_i}$, making
649
computation of the bijection trivial.
650
651
\item (Drew Sutherland-style tricks) We minimize dynamic memory
652
allocation speeding up the code by an order of magnitude, by making
653
some arbitrary bounds.
654
655
\item Painful to implement, but it is {\em fast}.
656
Not included in Sage yet: \url{http://trac.sagemath.org/sage_trac/ticket/12465}
657
\end{enumerate}
658
659
\end{frame}
660
661
\begin{frame}
662
\frametitle{What Next?}
663
664
My group's project at the 2012 MRC in Snowbird Utah (June 24--30,
665
2012) will be to compute Hilbert newforms in $S_{(2,2)}(\Gamma_0(\n))$
666
as far as possible, gather {\em arithmetic statistics} about them
667
(e.g., analytic ranks), make conjectures, and perhaps prove something.
668
669
\vfill {\bf Example Goal:} Does the first elliptic curve of rank $3$ have
670
norm conductor $163^2$ or not?
671
672
\vfill
673
{\small
674
\begin{tabular}{|l|l|l|l|}\hline
675
\textbf{rank} & \textbf{norm}$(\n)$ & \textbf{equation} & \textbf{person}\\\hline
676
0 & 31 (prime) & $[1,\vphi+1,\vphi,\vphi,0]$ & Dembele \\
677
1 & 199 (prime) & $[0,-\vphi-1,1,\vphi,0]$ & Dembele \\
678
2 & 1831 (prime) & $[0,-\vphi,1,-\vphi-1,2\vphi+1]$ & Dembele \\
679
3 & 26,569$\,=163^2$ & $[0,0,1,-2,1]$ & Elkies \\
680
4 & 1,209,079 (prime) & $[1, -1, 0, -8-12\vphi, 19+30\vphi]$ & Elkies \\
681
5 & 64,004,329 & $[0, -1, 1, -9-2\vphi, 15+4\vphi]$ & Elkies
682
\\\hline
683
\end{tabular}}
684
685
686
687
688
689
\end{frame}
690
691
692
\begin{frame}[fragile]
693
\frametitle{Epilogue (or Prologue)}
694
\tiny
695
\begin{verbatim}
696
On Wed, Feb 2, 2011 at 12:18 PM, William Stein <wstein@gmail.com> wrote:
697
> Hi John [Voight],
698
>
699
> I'm planning to try to say something about these sorts of things...
700
> mainly that I'm ignorant in each case. But I'm curious what thoughts
701
> you might have about these...
702
>
703
> -- stein-watkins style search
704
> -- elkies approach: Q(sqrt(5)) curves
705
> -- rank info
706
> -- gens (simon 2-descent output)
707
> -- L-function (fast computation of a_p?)
708
> -- congruence number
709
> -- isogeny class (enumerate)
710
> -- root number
711
> -- torsion subgroup
712
> -- tamagawa numbers
713
> -- all integral points
714
> -- Kodaira symbols
715
> -- zeros of L(E/F,s) in critical strip
716
> -- notion of "canonical" minimal weierstrass model
717
> -- picture
718
> -- height pairing / regulator
719
> -- heegner points
720
> -- #Sha(E/F) -- when hypo of Zhang's work satisfied, there is hope.
721
> -- images of Galois reps?
722
> -- as much as possible of the above for modular abelian varieties A_f.
723
\end{verbatim}
724
\end{frame}
725
726
727
% \begin{frame}
728
% \frametitle{BSD Invariants over $F$}
729
% $$
730
% \frac{L^{(r)}(E/F,1)}{r!} =
731
% \frac{\Omega_E \cdot \prod_{v} c_{E,v} \cdot \Reg_E \cdot \Sha(E)}
732
% {\#E(F)_{\tor}^2 \cdot \sqrt{5}}
733
% $$
734
% [[todo: is $\sqrt{5}$ definitely in numerator????]]
735
736
% \begin{enumerate}
737
% \item $L$-series:
738
% \item Torsion subgroup:
739
% \item Generators for Mordell-Weil group:
740
% \item Regulator:
741
% \item Tamagawa numbers:
742
% \item Period lattice $\Omega_E$:
743
% \end{enumerate}
744
745
% \end{frame}
746
747
% \begin{frame}
748
% \frametitle{BSD Invariants of First Rank 0 Curve}
749
750
% [[like above slide with numbers filled in]]
751
752
% \end{frame}
753
754
% \begin{frame}
755
% \frametitle{BSD Invariants of First Rank 1 Curve}
756
757
% [[like above slide with numbers filled in]]
758
759
% \end{frame}
760
761
% \begin{frame}
762
% \frametitle{BSD Invariants of First Rank 2 Curve}
763
764
% [[like above slide with numbers filled in]]
765
766
% \end{frame}
767
768
% \begin{frame}
769
% \frametitle{BSD Invariants of First\footnote{?} Rank 3 Curve}
770
771
% [[like above slide with numbers filled in]]
772
773
% \end{frame}
774
775
776
777
% \begin{frame}
778
% \frametitle{Computing Shafarevich-Tate Group of Curves over $F$}
779
% \end{frame}
780
781
782
% \begin{frame}
783
% \frametitle{Modular Degrees of Elliptic Curves over $F$}
784
% \end{frame}
785
786
787
% \begin{frame}
788
% \frametitle{Heegner Points on Elliptic Curves over $F$}
789
790
791
% [[Also mention Chow-Heegner points.]]
792
793
% \end{frame}
794
795
796
% \begin{frame}
797
% \frametitle{Modular Abelian Varieties over $F$}
798
% \end{frame}
799
800
% \begin{frame}
801
% \frametitle{Bigger Tables of all Elliptic Curves over $F$ of Given Conductor}
802
803
% [[estimate of how far we can go finding newforms.]]
804
805
% \end{frame}
806
807
808
% \begin{frame}
809
% \frametitle{Large Tables of Elliptic Curves over $F$ with Bounded Discriminant}
810
811
% [[work toward generalizing stein-watkins]]
812
813
% [[time to compute $L$-series, etc.]]
814
815
% \end{frame}
816
817
818
819
\end{document}