 CoCalc Public Fileswww / 168 / notes / 2005-11-16 / projects.tex
Author: William A. Stein
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5\title{Math 168: Project Ideas}
6\author{William Stein}
7\begin{document}
8\maketitle
9
10\begin{enumerate}
11\item How to compute $a_p(E)$ for an elliptic curve $E$ and small $p$
12(see Henri Cohen's first GTM book).
13
14    \item How (e.g., PARI) computes Bernoulli numbers so much faster
15      than anything else
16
17    \item Write discussion about factorization of RSA challenge numbers
18      (e.g., a new \$20000 one was factored a few days ago!) 19 20 \item Write about geometry on the upper half plane (the Poincare metric, 21 the Poincare disk, etc.) 22 23 \item Prove that reduction$\SL_n(\Z) \to \SL_n(\Z/N\Z)$is 24 surjective (following, e.g., the proof in Shimura's book 25 {\em Introduction to Arithmetic Theory of Automorphic Forms}). 26 27 \item A project on the dimension of$S_2(Gamma_0(N))$. This requires 28 more background than you need for the course; in particular, you 29 must know the Riemann-Roch theorem for curves/Riemann surfaces. 30 You would also want to look at the paper by Csirik et al. about 31$d$'s that are not dim$S_2(Gamma_0(N))$for any$N$. 32 33 \item Proof of Manin's theorem that the$2$and$3\$ term relations
34      between Manin symbols are everything.  References: Manin's
35      original 1972 paper; a very very complicated paper by Shokoruv;
36      Tseno's student project (for me) on Shokoruv; Gabor Weise's
37      recent Ph.D. thesis (I have a copy); notes for Math 252 at
38      Harvard.  This is sufficiently broad that it could be a joint
39      project with two people.  (Gabor's Ph.D. claims to have an easier
40      way to do this...)
41
42    \item Write a project about the'' baby-step giant-step
43      algorithm.  How is it used to --- solve discrete log problems?
44      --- find the structure of a group?  etc.?
45
46\end{enumerate}
47\end{document}
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