\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}
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