\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}