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Author: William A. Stein
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\begin{document}
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\begin{center}
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\large Math 168: Topics in Applied Math \& Comp. Sci. (Fall 2005):\\
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\Large
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\vspace{1ex}
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{\sc Explicit Approaches to Elliptic Curves\\and Modular Forms}\\
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\vspace{1ex}
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\large
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\vspace{2ex}
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{\sf
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Course: MW 3:00-4:20 in 201 Center\\
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Section: Th 5:00pm in 207 Center}\\
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William Stein ({\sf wstein@ucsd.edu})\\
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http://modular.ucsd.edu/168\\
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\large
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\end{center}
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\begin{abstract}
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This course is an introduction to elliptic curve and modular forms,
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with a special emphasis on how to compute with these objects.
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\end{abstract}
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\section{Textbooks}
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The main text are the notes that I've written and will be handing out.
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There are also many books and articles on elliptic curves and modular
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forms, which I will encourage you to look at. For the first quarter
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of the course, I will closely follow Chapter 6 of {\tt
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http://modular.ucsd.edu/ent/}.
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\section{Course Topics}
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\begin{itemize}
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\item{}{[\bf Elliptic Curves]} I will define elliptic curves, explain
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their two main applications to cryptography, and discuss the Birch
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and Swinnerton-Dyer conjecture (a million dollar Clay Math prize
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problem).
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\item{}{[\bf Modular Forms]} I will define modular forms of weight
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$2$, discuss their connection with elliptic curves and Andrew
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Wiles's celebrated proof of Fermat's Last Theorem. I will also
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discuss how to use modular symbols to compute modular forms, and
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mention open problems.
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\end{itemize}
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See the course outline below for more details.
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\section{Prerequisites}
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\begin{itemize}
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\item A course on groups, rings and fields.
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\item Ability to follow nontrivial mathematical arguments.
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\item Know how to use a computer.
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\end{itemize}
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It will be useful if you know something about algebraic curves,
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complex analysis and have some prior exposure to number theory.
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However, I am not requiring this as a prerequisite. A few times
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during the course I will give motivation for a topic or a deeper
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explanation for something that assume more background; I will make it
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clear when I am doing this, and it will not be a problem if you don't
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understand it.
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\section{Grade}
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Your grade will be determined as follows:
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\begin{itemize}
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\item 20\% midterm
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\item 25\% final exam
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\item 25\% final project
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\item 30\% homework
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\end{itemize}
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If you get 90\% of points you'll get at least an A-, 80\% will give
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you at least a B-, and 70\% at least a C-.
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\section{Homework}
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There will be one homework assignment per week. It will be assigned
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by Wednesday, and be due the following Wednesday. Though I will not
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accept late homework, your lowest homework grade will be dropped.
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\begin{center}
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{\bf Please {\em do}{} work together on homework problems!}
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\end{center}
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BUT, write up your solutions individually, and carefully
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acknowledge the people and other sources that you used.
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\section{Office Hours}
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My office is AP\&M 5111. Please come by and chat with me anytime
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I'm there. My official office hours will be announced later.
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\section{Course Outline}
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\begin{enumerate}
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\item Overview of elliptic curves and modular forms
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\item How to put a natural group structure on the set of points
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on an elliptic curve
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\item Elliptic curves over finite fields
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\item How to factor integers using elliptic curves (Application: cracking
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the RSA cryptosystems.)
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\item How to make cryptosystems using elliptic curves (Application:
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the best public-key crypotosystems?)
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\item Elliptic curves over the rational numbers
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\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}).
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\item Modular curves as quotients of the upper half plane
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\item What does it mean for an elliptic curve to be ``modular''?
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(What exactly was Andrew Wiles contribution to the proof of
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Fermat?)
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\item Introduction to modular symbols (very explicit and
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easy ``homology'' of modular curves)
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\item How to use modular symbols to compute modular forms, I
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\item How to use modular symbols to compute modular forms, II
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\item SAGE: System for Algebra and Geometry Experimentation
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(on the architecture and design of SAGE).
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\item The $L$-series of an elliptic curve over $\mbox{\rm \bf Q}$
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\item The Birch and Swinnerton-Dyer conjecture (a million dollar
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prize problem)
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\end{enumerate}
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\end{document}
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