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\title{Math 168: Project Ideas}
\author{William Stein}
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\begin{enumerate}
\item How to compute $a_p(E)$ for an elliptic curve $E$ and small $p$
(see Henri Cohen's first GTM book).
\item How (e.g., PARI) computes Bernoulli numbers so much faster
than anything else
\item Write discussion about factorization of RSA challenge numbers
(e.g., a new \$20000 one was factored a few days ago!)
\item Write about geometry on the upper half plane (the Poincare metric,
the Poincare disk, etc.)
\item Prove that reduction $\SL_n(\Z) \to \SL_n(\Z/N\Z)$ is
surjective (following, e.g., the proof in Shimura's book
{\em Introduction to Arithmetic Theory of Automorphic Forms}).
\item A project on the dimension of $S_2(Gamma_0(N))$. This requires
more background than you need for the course; in particular, you
must know the Riemann-Roch theorem for curves/Riemann surfaces.
You would also want to look at the paper by Csirik et al. about
$d$'s that are not dim $S_2(Gamma_0(N))$ for any $N$.
\item Proof of Manin's theorem that the $2$ and $3$ term relations
between Manin symbols are everything. References: Manin's
original 1972 paper; a very very complicated paper by Shokoruv;
Tseno's student project (for me) on Shokoruv; Gabor Weise's
recent Ph.D. thesis (I have a copy); notes for Math 252 at
Harvard. This is sufficiently broad that it could be a joint
project with two people. (Gabor's Ph.D. claims to have an easier
way to do this...)
\item Write a project about ``the'' baby-step giant-step
algorithm. How is it used to --- solve discrete log problems?
--- find the structure of a group? etc.?
\end{enumerate}
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