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Author: William A. Stein
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\documentclass[11pt]{article}
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\input{macros}
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\title{Math 168: Project Ideas}
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\author{William Stein}
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\begin{document}
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\maketitle
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\begin{enumerate}
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\item How to compute $a_p(E)$ for an elliptic curve $E$ and small $p$
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(see Henri Cohen's first GTM book).
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\item How (e.g., PARI) computes Bernoulli numbers so much faster
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than anything else
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\item Write discussion about factorization of RSA challenge numbers
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(e.g., a new \$20000 one was factored a few days ago!)
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\item Write about geometry on the upper half plane (the Poincare metric,
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the Poincare disk, etc.)
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\item Prove that reduction $\SL_n(\Z) \to \SL_n(\Z/N\Z)$ is
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surjective (following, e.g., the proof in Shimura's book
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{\em Introduction to Arithmetic Theory of Automorphic Forms}).
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\item A project on the dimension of $S_2(Gamma_0(N))$. This requires
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more background than you need for the course; in particular, you
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must know the Riemann-Roch theorem for curves/Riemann surfaces.
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You would also want to look at the paper by Csirik et al. about
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$d$'s that are not dim $S_2(Gamma_0(N))$ for any $N$.
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\item Proof of Manin's theorem that the $2$ and $3$ term relations
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between Manin symbols are everything. References: Manin's
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original 1972 paper; a very very complicated paper by Shokoruv;
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Tseno's student project (for me) on Shokoruv; Gabor Weise's
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recent Ph.D. thesis (I have a copy); notes for Math 252 at
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Harvard. This is sufficiently broad that it could be a joint
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project with two people. (Gabor's Ph.D. claims to have an easier
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way to do this...)
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\item Write a project about ``the'' baby-step giant-step
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algorithm. How is it used to --- solve discrete log problems?
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--- find the structure of a group? etc.?
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\end{enumerate}
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\end{document}
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