\documentclass[10pt]{article}1\voffset=-0.07\textheight2\textheight=1.14\textheight3\hoffset=-0.07\textwidth4\textwidth=1.14\textwidth5%\usepackage{fullpage}6\pagestyle{empty}7% bulleted list environment8\newenvironment{mylist}9{10\begin{list}11{$\cdot$}12{13\setlength{\itemsep}{-.2ex}14% \setlength{\parsep}{0ex}15\setlength{\leftmargin}{4em}16% \setlength{\parskip}{0ex}17% \setlength{\topsep}{0ex}18}19}20{21\end{list}22}23%end newenvironment242526\begin{document}27\begin{center}28\large Math 168: Topics in Applied Math \& Comp. Sci. (Fall 2005):\\29\Large30\vspace{1ex}3132{\sc Explicit Approaches to Elliptic Curves\\and Modular Forms}\\33\vspace{1ex}3435\large3637\vspace{2ex}3839{\sf40Course: MW 3:00-4:20 in 201 Center\\41Section: Th 5:00pm in 207 Center}\\42William Stein ({\sf wstein@ucsd.edu})\\43http://modular.ucsd.edu/168\\44\large45\end{center}4647\begin{abstract}48This course is an introduction to elliptic curve and modular forms,49with a special emphasis on how to compute with these objects.50\end{abstract}5152\section{Textbooks}53The main text are the notes that I've written and will be handing out.54There are also many books and articles on elliptic curves and modular55forms, which I will encourage you to look at. For the first quarter56of the course, I will closely follow Chapter 6 of {\tt57http://modular.ucsd.edu/ent/}.585960\section{Course Topics}61\begin{itemize}62\setlength{\itemsep}{-0.2ex}63\item{}{[\bf Elliptic Curves]} I will define elliptic curves, explain64their two main applications to cryptography, and discuss the Birch65and Swinnerton-Dyer conjecture (a million dollar Clay Math prize66problem).6768\item{}{[\bf Modular Forms]} I will define modular forms of weight69$2$, discuss their connection with elliptic curves and Andrew70Wiles's celebrated proof of Fermat's Last Theorem. I will also71discuss how to use modular symbols to compute modular forms, and72mention open problems.73\end{itemize}7475See the course outline below for more details.7677\section{Prerequisites}78\vspace{-1ex}79\begin{itemize}80\setlength{\itemsep}{-0.3ex}81\item A course on groups, rings and fields.82\item Ability to follow nontrivial mathematical arguments.83\item Know how to use a computer.84\end{itemize}8586It will be useful if you know something about algebraic curves,87complex analysis and have some prior exposure to number theory.88However, I am not requiring this as a prerequisite. A few times89during the course I will give motivation for a topic or a deeper90explanation for something that assume more background; I will make it91clear when I am doing this, and it will not be a problem if you don't92understand it.939495\section{Grade}96Your grade will be determined as follows:97\begin{itemize}98\setlength{\itemsep}{-0.3ex}99\item 20\% midterm100\item 25\% final exam101\item 25\% final project102\item 30\% homework103\end{itemize}104If you get 90\% of points you'll get at least an A-, 80\% will give105you at least a B-, and 70\% at least a C-.106107108\section{Homework}109There will be one homework assignment per week. It will be assigned110by Wednesday, and be due the following Wednesday. Though I will not111accept late homework, your lowest homework grade will be dropped.112\begin{center}113{\bf Please {\em do}{} work together on homework problems!}114\end{center}115BUT, write up your solutions individually, and carefully116acknowledge the people and other sources that you used.117118\section{Office Hours}119My office is AP\&M 5111. Please come by and chat with me anytime120I'm there. My official office hours will be announced later.121122\section{Course Outline}123\begin{enumerate}124\setlength{\itemsep}{-0.2ex}125\item Overview of elliptic curves and modular forms126127\item How to put a natural group structure on the set of points128on an elliptic curve129130\item Elliptic curves over finite fields131132\item How to factor integers using elliptic curves (Application: cracking133the RSA cryptosystems.)134135\item How to make cryptosystems using elliptic curves (Application:136the best public-key crypotosystems?)137138\item Elliptic curves over the rational numbers139140\item The group $\mbox{\rm SL}_2(\mbox{\rm \bf Z})$ and the complex upper half plane (Ch.~7 of Serre's {\em A course in arithmetic}).141142\item Modular curves as quotients of the upper half plane143144\item What does it mean for an elliptic curve to be ``modular''?145(What exactly was Andrew Wiles contribution to the proof of146Fermat?)147148\item Introduction to modular symbols (very explicit and149easy ``homology'' of modular curves)150151\item How to use modular symbols to compute modular forms, I152153\item How to use modular symbols to compute modular forms, II154155\item SAGE: System for Algebra and Geometry Experimentation156(on the architecture and design of SAGE).157158\item The $L$-series of an elliptic curve over $\mbox{\rm \bf Q}$159160\item The Birch and Swinnerton-Dyer conjecture (a million dollar161prize problem)162163\end{enumerate}164165166167168\end{document}169170171