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\title[Research in SMC]{Writing a Research Paper using SageMathCloud}
\author{William Stein}
\institute[UW]
{University of Washington \\
\medskip
\url{http://wstein.org/}}
\date{\today}

\begin{document}

\begin{frame}
\titlepage
\end{frame}

\begin{frame}
\frametitle{``A Databases of Elliptic Curves Ordered by Height and Distributions of Selmer Groups and Ranks''}

\begin{block}{Authors}
Jen Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer, William Stein, Jamie Weigandt
\end{block}

\begin{block}{What's this about?}
\begin{itemize}
\item Past tables are usually ordered by conductor or discriminant.
\item Bhargava--Shankar: new upper bounds on the average algebraic rank ordered by height; got by studying the average sizes of $n$-Selmer groups.
\item Make database ordered by height: compute rank and $2$-Selmer size.
\end{itemize}
\end{block}


\end{frame}


\begin{frame}
\frametitle{Background motivation}
\begin{block}{Order by conductor}
\begin{itemize}
\item Mazur-Stein-Watkins-Bektemerov - 2006 paper: 136,\,832,\,795 curves ordered by conductor; average rank keeps getting bigger!?\\
\begin{center}
\includegraphics[width=.5\textwidth]{bulletins.png}
\end{center}
\item Conjecture: average rank is 1/2

\item Manjul Bharghava: average rank is definitely bounded!
\end{itemize}
\end{block}
\end{frame}



\begin{frame}
\frametitle{Challenge!}

\begin{block}{}
\begin{itemize}
\item Bharghava: order curves by height!

\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).

\item We did this.

\end{itemize}
\end{block}
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\begin{frame}
\frametitle{Avg. Rank: 238764310  Curves of Height $\leq 2.7\cdot 10^{10}$}

\includegraphics[width=.9\textwidth]{plot1}

\end{frame}

\begin{frame}
\frametitle{Order of the torsion subgroup}

\includegraphics[width=\textwidth]{plot2}

\end{frame}

\begin{frame}
\frametitle{Rank of the 2-Selmer Group}

\includegraphics[width=0.9\textwidth]{plot4}

\end{frame}

\begin{frame}
\frametitle{Avg. Sign of the Root Number}

\includegraphics[width=\textwidth]{plot3}

\end{frame}


\begin{frame}
\frametitle{Features of SageMathCloud for writing this paper}

We wrote the paper in \url{https://cloud.sagemath.com}

\begin{block}{Use}
\begin{itemize}
\item Collaborative editing of \LaTeX{} documents

\item Collaborative persistent terminals (e.g., ssh to cluster somewhere)

\item Edit Python code; run from Sage worksheets (or terminals)

\item Use SQLite easily

\item Run project on a 32-core big-memory VM at Google

\item Chat: post comments on the side of any file being edited

\item Following the log

\end{itemize}
\end{block}

\end{frame}



\begin{frame}
\frametitle{Features of Sage for writing this paper}

\begin{block}{Key things we use}
\begin{itemize}
\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.

\item Simon Spicer's Elliptic curve enumeration (coming to Sage soon)

\item MWRANK: the workhorse

\item Magma interface -- helped with a few hard curves at the end.

\end{itemize}

\end{block}

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\end{document}