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Course 428\\[3mm]
Elliptic Curves II}
\author{Dr Timothy Murphy}
\date{Exam Hall\hfil Friday, 7 April 2000\hfil 9:00--11:00}
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\begin{document}
\maketitle
\thispagestyle{empty}
\begin{quotation}\em
\noindent
Attempt 5 questions.
(If you attempt more,
only the best 5 will be counted.)
All questions carry the same number of marks.
\end{quotation}
\begin{enumerate}
\enlargethispage{3cm}
\item %1
What is meant by saying that an elliptic curve
has \emph{good reduction} at the prime $p$?
Sketch the proof that if $p$ is a good prime then the map
\[
\E(\Q) \to \E(\F_p)
\]
is a homomorphism.
What happens if $p$ is bad?
Determine the good primes for the elliptic curve
\[
y^2 + 2xy = x^3 - 1.
\]
\item %2
If the elliptic curve
\[
y^2 = x^3 + ax^2 + bx + c
\]
has good reduction at $p$,
show that the torsion subgroup of $\E(\Q)$
is mapped injectively into $\E(\F_p)$.
Let $\E(\Q)$ be the curve
\[
y^2 = x^3 + 3x + 1.
\]
Determine the group $\E(\F_p)$ for two good primes $p$,
and hence or otherwise determine the torsion group of $\E$.
%\enlargethispage*{1cm}
\item %3
Prove that a finite abelian group $A$ is a direct sum
of cyclic subgroups of prime-power order;
and show that the set of prime-powers depends only on $A$.
\newpage
\item %4
What is a \emph{lattice} $\Lambda \subset \C$;
and what is meant by saying that $f(z)$ is \emph{elliptic}
with respect to $\Lambda$?
Define the Weierstrass elliptic function $\varphi(z)$
associated to $\Lambda$,
and show that $\varphi(z)$ is indeed elliptic with respect to $\Lambda$.
Show that every even elliptic function $f(z)$
with respect to $\Lambda$ is a rational function of $\varphi(z)$, ie
\[
f(z) = \frac{P(\varphi(z))}{Q(\varphi(z))}
\]
where $P(w),Q(w)$ are polynomials.
\item %5
Show that the group $\SL(2,\Z)$ is generated by
\[
S = \begin{pmatrix}
0 & -1\\
1 & 0
\end{pmatrix}
\text{ and }
T = \begin{pmatrix}
1 & 1\\
0 & 1
\end{pmatrix}.
\]
\item %6
Explain how a lattice $\Lambda \subset \C$
gives rise to an elliptic curve
\[
\E(\C): y^2 = x^3 + bx + c;
\]
and sketch the proof that every elliptic curve of this form
arises in this way from a unique lattice $\Lambda$.
\item %7
Prove Fermat's Last Theorem
\emph{either} for the case $n = 3$
\emph{or} for the case $n = 4$.
\end{enumerate}
\end{document}
\end{enumerate}
\end{document}