\documentclass{amsart}
\input ellc/emac
\title{Homework 3}
\author{Topics in AG: Elliptic Curves}
\begin{document}
\maketitle
Due March 17, 2004
\bigskip
\begin{enumerate}
\item Let $E$ be an elliptic curve in long Weierstrass form.
Show that the tangent at the point $\cO = [0:1:0]$ at
infinity on $E$ is the line at infinity, and that $P$
is a flex (i.e. a point whose tangent intersects $E$
with multiplicity $3$).
\bigskip
\item Let $C$ be a nonsingular cubic curve. Show that if
\begin{enumerate}
\item $\cO = [0:1:0]$ is on $C$,
\item the tangent at $\cO$ is $Z=0$,
\item $\cO$ is a flex,
\end{enumerate}
then $C$ has Weierstrass form.
\bigskip
\item The results of the preceding two exercises allows you
to transform any nonsingular cubic with a rational flex
into a Weierstrass elliptic curve defined over $\Q$ via
some projective transformation (these bijective maps send
$[X:Y:Z]$ to $[X':Y':Z']$, where $X', Y', Z'$ depend
linearly on $X, Y, Z$).
The curve $C: 9x^3 + y^3 + z^3 - 6xyz = 0$ occurs in a
recent elementary proof of Fermat's Last Theorem for
exponent $3$. It has an obvious rational point $P$;
show that $C$ is nonsingular and that $P$ is a flex.
Compute the tangent $\ell$ to $C$ at $P$ and find a
projective transformation that sends $P$ to $[0:1:0]$
and $\ell$ to the line $Z = 0$; finally, give the
cubic in short Weierstrass form.
\bigskip
\item Compute the group structure of $E(\F_5)$ for the elliptic
curves $E_1: y^2 = x^3-x$ and $E_2: y^2 = x^3-2x$.
\end{enumerate}
\end{document}