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\title{Math 168: Homework Assignment 2}
\author{William Stein}
\date{\bf Due: Wednesday, Oct 12, 2005}
\begin{document}
\maketitle

\noindent{\em The SIX problems have equal point value,
and multi-part problems are of the same value.  
You are allowed to use a computer on any problem,
as long as you include the exact code used to
solve the problem with your solution.  Any software
systems (e.g., Magma, SAGE, Mathematica, C) are allowed.}

\section{Announcements}
\begin{enumerate}
\item Office Hours: Tuesdays 3-5 in my office (AP\&M 5111).
\item Section: Thursday 5-6 {\bf in my office}
\end{enumerate}

\section{Problems}

\begin{enumerate}

\item\label{ex:ell2} One rational solution to the equation
$y^2=x^3-2$ is $(3,5)$.  Find a rational solution with $x\neq 3$ by
drawing the tangent line to $(3,5)$ and computing the second point of
intersection.

\item\label{ex:singgrp} Write down an equation $y^2 = x^3 +ax + b$ over a field
$K$ such that $-16(4a^3+27b^2)=0$ (yes equals 0).  Precisely what goes wrong
when trying to endow the set 
$E(K) = \{ (x,y)\in K\cross K : y^2 = x^3 + ax +b \} \union \{\O\}$
with a group structure (in an algebraic way)?

\item\label{ex:ell4} 
Let~$E$ be the elliptic curve over the finite 
field $K=\F_5$ defined by the equation 
$$
y^2 = x^3 + x +1.
$$  
\begin{enumerate}
\item List all~$9$ elements of~$E(K)$.
\item What is the structure of 
$E(K)$, as a product of cyclic groups?
\end{enumerate}

\item\label{ex:ell11} 
Let~$E$ be the elliptic curve 
defined by the equation $y^2 = x^3 +1$.  
For each prime $p\geq 5$, let $N_p$ be the cardinality of the group
$E(\zmod{p})$ of points on this curve having coordinates 
in $\zmod{p}$.  For example, we have that
$N_{5} = 6, N_{7} = 12, N_{11} = 12, N_{13} = 12, 
N_{17} = 18, N_{19} = 12, , N_{23} = 24, \text{ and }N_{29} = 30$
(you do not have to prove this).
\begin{enumerate}
\item  For the set of primes satisfying $p\con 2\pmod{3}$, can you see a
  pattern for the values of $N_p$?  Make a general conjecture for the
  value of $N_p$ when $p\con 2\pmod{3}$.
\item (*) Prove your conjecture.
\end{enumerate}

\item \label{ex:elltrans}
 Suppose $y^2=x^3+ax+b$ with $a,b\in\Q$ defines an elliptic
curve.  Show that there is another equation $Y^2=X^3+AX+B$ with
$A,B\in\Z$ whose solutions are in bijection with the  
solutions to $y^2=x^3+ax+b$ (via a bijection defined by algebraic
formulas).

\item\label{ex:ell13} 
Let~$E$ be an elliptic curve over the real numbers~$\R$.
Prove that $E(\R)$ is not a finitely generated abelian group.




\end{enumerate}
\end{document}
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