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\homework{16: Elliptic Curves}{Due Friday, Week 9}

Solve \textbf{one} of the following \textbf{three} problems.    As always, prove your claims/have fun! 


\begin{enumerate}
\item In class, we proved that if $E$ is an elliptic curve and $P, Q$ are two distinct points on $E$ such that the line $L$ through $P, Q$ was not vertical, then $L$ intersects $E$ at some third point $R$.  This problem considers what happens if $P = Q$; that is, if we pick a point $P$ and choose $L$ to be the tangent line to $E$ at $P$!
\begin{prop}
Suppose that $P$ is a point with nonzero $y$-co\"ordinate on an elliptic curve $E$ given by $y^2 = x^3 - ax + b$.  Take the tangent line $L$ to $E$ at $P$.  There are two possibilities:
\begin{itemize}
\item $L$ intersects $E$ at exactly one other point on the curve.  If we graph $L$ by $y = mx+b$, which we can do because at points with nonzero $y$-co\"ordinate we have shown that the slopes of tangent lines exist and are finite, we have that $p(x) = (x^3 - ax + b) - (mx+b)^2$ can be factored into something of the form $(x-r_1)^2(x-r_2)$, where $r_1$ is the $x-$co\"ordinate of $P$, and $r_2$ is the $x$-co\"ordinate of the unique other point on the curve we cross.
\item $L$ never intersects $E$ at any other points on our curve.  If we graph $L$ by $y = mx+b$, we have that $p(x) = (x^3 - ax + b) - (mx+b)^2$ can be factored into something of the form $(x-r_1)^3$, where $r_1$ is the $x-$co\"ordinate of $P$.
\end{itemize}
\end{prop}
Prove this proposition!
\item In class, Connor asked if the names ``elliptical curve'' and ``ellipse'' are related terms.  I said that they were in a sense, but didn't know the full reason off the top of my head.

As it turns out, there's actually a beautiful story here!  To do this problem, go to
\begin{center}
\href{http://www.maa.org/sites/default/files/pdf/upload_library/2/Rice-2013.pdf}{http://www.maa.org/sites/default/files/pdf/upload\_library/2/Rice-2013.pdf}
\end{center}
and read the attached paper, which explains how these terms are related.  Give me a two-three paragraph summary of this paper to solve this problem!
\item Take the elliptic curve $E$ defined by $y^2 = x^3 + 1$ for this problem.
\begin{enumerate}
\item Show that the only points on this curve that have integer co\"ordinates are $(-1,0), (0, \pm 1), (2, \pm 3)$.
\item Take these five points, along with the sixth point $O$ that is the ``point at infinity'' as defined in the notes/in class on Wednesday.  Show that these six points, under the point-addition operation defined in class, form a subgroup of the elliptic curve group.  Give me a group table for these six points.
\end{enumerate}
\end{enumerate}

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