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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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