CoCalc Public Files
Author: William A. Stein
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12\title[Research in SMC]{Writing a Research Paper using SageMathCloud}
13\author{William Stein}
14\institute[UW]
15{University of Washington \\
16\medskip
17\url{http://wstein.org/}}
18\date{\today}
19
20\begin{document}
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22\begin{frame}
23\titlepage
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26\begin{frame}
27\frametitle{A Databases of Elliptic Curves Ordered by Height and Distributions of Selmer Groups and Ranks''}
28
29\begin{block}{Authors}
30Jen Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer, William Stein, Jamie Weigandt
31\end{block}
32
34\begin{itemize}
35\item Past tables are usually ordered by conductor or discriminant.
36\item Bhargava--Shankar: new upper bounds on the average algebraic rank ordered by height; got by studying the average sizes of $n$-Selmer groups.
37\item Make database ordered by height: compute rank and $2$-Selmer size.
38\end{itemize}
39\end{block}
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41
42\end{frame}
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44
45\begin{frame}
46\frametitle{Background motivation}
47\begin{block}{Order by conductor}
48\begin{itemize}
49\item Mazur-Stein-Watkins-Bektemerov - 2006 paper: 136,\,832,\,795 curves ordered by conductor; average rank keeps getting bigger!?\\
50\begin{center}
51\includegraphics[width=.5\textwidth]{bulletins.png}
52\end{center}
53\item Conjecture: average rank is 1/2
54
55\item Manjul Bharghava: average rank is definitely bounded!
56\end{itemize}
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61
62\begin{frame}
63\frametitle{Challenge!}
64
65\begin{block}{}
66\begin{itemize}
67\item Bharghava: order curves by height!
68
69\item Challenge: systematically compute the ranks of {\bf all} elliptic curves $y^2 = x^3 + a_4 x + a_6$ of height $H=\max\{4|a_4|^3, 27 a_6^2\}$ far enough that we can finally clearly see the rank going down (and how).
70
71\item We did this.
72
73\end{itemize}
74\end{block}
75\end{frame}
76
77\begin{frame}
78\frametitle{Avg. Rank: 238764310  Curves of Height $\leq 2.7\cdot 10^{10}$}
79
80\includegraphics[width=.9\textwidth]{plot1}
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82\end{frame}
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84\begin{frame}
85\frametitle{Order of the torsion subgroup}
86
87\includegraphics[width=\textwidth]{plot2}
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89\end{frame}
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91\begin{frame}
92\frametitle{Rank of the 2-Selmer Group}
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94\includegraphics[width=0.9\textwidth]{plot4}
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96\end{frame}
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98\begin{frame}
99\frametitle{Avg. Sign of the Root Number}
100
101\includegraphics[width=\textwidth]{plot3}
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103\end{frame}
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105
106\begin{frame}
107\frametitle{Features of SageMathCloud for writing this paper}
108
109We wrote the paper in \url{https://cloud.sagemath.com}
110
111\begin{block}{Use}
112\begin{itemize}
113\item Collaborative editing of \LaTeX{} documents
114
115\item Collaborative persistent terminals (e.g., ssh to cluster somewhere)
116
117\item Edit Python code; run from Sage worksheets (or terminals)
118
119\item Use SQLite easily
120
121\item Run project on a 32-core big-memory VM at Google
122
123\item Chat: post comments on the side of any file being edited
124
125\item Following the log
126
127\end{itemize}
128\end{block}
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134\begin{frame}
135\frametitle{Features of Sage for writing this paper}
136
137\begin{block}{Key things we use}
138\begin{itemize}
139\item  Simon Spicer's new $L$-function analytic rank bounding code (fully included in Sage by default, very flexible, and well documented!).  Very fast (even for large conductor) code for bounding elliptic curve ranks.
140
141\item Simon Spicer's Elliptic curve enumeration (coming to Sage soon)
142
143\item MWRANK: the workhorse
144
145\item Magma interface -- helped with a few hard curves at the end.
146
147\end{itemize}
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153\end{document}
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