\documentclass{amsart}
\input ellc/emac
%\input emac

\title{Homework 3}
\author{Topics in AG: Elliptic Curves}
\begin{document}
\maketitle

Due April 19, 2004

\bigskip
 
\begin{enumerate}

\item Compute the torsion group $E(\Q)_\trs$ for the
      elliptic curve $y^2 = x^3+1$. (Euler proved that
      these are the only rational points on $E$).    

\item Consider the elliptic curve $E: y^2 = x^3 + b$ with $b \in \Z$;
      For any odd prime $p \equiv 2 \bmod 3$ we have $\# E(\F_p) = p+1$.

\item Let $E: y^2 = x^3 + b$ be an elliptic curve, where 
      $b \in \Z$ is not divisible by a sixth power $\ne 1$. 
      Then
      $$ E(\Q)_\trs \simeq \begin{cases} 
               \Z/6\Z, & \text{if} \ b = 1; \\
               \Z/3\Z, & \text{if} \ b = -432 \ \text{or}\ 
                          1 \ne b \ \text{is a square}; \\
                \Z/2\Z  & \text{if $1 \ne b$ is a cube}; \\
                0  & \text{otherwise.} \end{cases} $$

\item Consider the conic $\cP: X^2 - dY^2 = 1$ for squarefree
      integers $d$. 
      \begin{enumerate}
      \item Show that 
         $$ \cP^{(n)} = 
         \{(x,y) \in \cP(\Z_p): x - 1 \equiv y \equiv 0 \bmod p^n\} $$
         are subgroups of $\cP(\Z_p)$ for all $n \ge 0$.  
      \item Show that, for $p > 2$, the map $u: \cP^{(1)} \lra \Z_p$ 
         that sends $P = (x,y)$ to $u(P) = \frac{x-1}y$ is well defined. 
         Give a convincing reason why we should put $u((1,0)) = 0$. Show 
         that for $p=2$, $u$ is defined on $\cP^{(2)}$.
      \item Show that $|u(kP)| = |k| \cdot |u(P)|$ for all 
         $P \in \cP^{(1)}$ and all $k \ge 1$; here $|a|$ denotes the
         $p$-adic absolute value on $\Z_p$. (Try $k = 2$ first;
         I also suspect that 
         $|u(P+Q) - u(P) - u(Q)| < \max \{|u(P)|^2, |u(Q)|^2\}$,
         but haven't found a proof yet). 
      \item Show that $\cP^{(1)}$ is torsion free for $p > 2$,
         and that $\cP^{(2)}$ is torsion free for $p = 2$.
      \end{enumerate}
 
\end{enumerate}      

\end{document}