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\begin{document}
\begin{center}
\large Math 168: Topics in Applied Math \& Comp. Sci. (Fall 2005):\\
\Large
\vspace{1ex}

{\sc Explicit Approaches to Elliptic Curves\\and Modular Forms}\\
\vspace{1ex}

\large

\vspace{2ex}

{\sf 
Course: MW 3:00-4:20 in 201 Center\\
Section: Th 5:00pm in 207 Center}\\
William Stein ({\sf wstein@ucsd.edu})\\
http://modular.ucsd.edu/168\\
\large 
\end{center}

\begin{abstract}
  This course is an introduction to elliptic curve and modular forms,
  with a special emphasis on how to compute with these objects.
\end{abstract}

\section{Textbooks}
The main text are the notes that I've written and will be handing out.
There are also many books and articles on elliptic curves and modular
forms, which I will encourage you to look at.  For the first quarter
of the course, I will closely follow Chapter 6 of {\tt
  http://modular.ucsd.edu/ent/}.


\section{Course Topics}
\begin{itemize}
\setlength{\itemsep}{-0.2ex}
\item{}{[\bf Elliptic Curves]} I will define elliptic curves, explain
  their two main applications to cryptography, and discuss the Birch
  and Swinnerton-Dyer conjecture (a million dollar Clay Math prize
  problem).

\item{}{[\bf Modular Forms]} I will define modular forms of weight
  $2$, discuss their connection with elliptic curves and Andrew
  Wiles's celebrated proof of Fermat's Last Theorem.  I will also
  discuss how to use modular symbols to compute modular forms, and
  mention open problems.
\end{itemize}

See the course outline below for more details.

\section{Prerequisites}
\vspace{-1ex}
\begin{itemize}
\setlength{\itemsep}{-0.3ex}
\item A course on groups, rings and fields.
\item Ability to follow nontrivial mathematical arguments.
\item Know how to use a computer.
\end{itemize}

It will be useful if you know something about algebraic curves,
complex analysis and have some prior exposure to number theory.
However, I am not requiring this as a prerequisite.  A few times
during the course I will give motivation for a topic or a deeper
explanation for something that assume more background; I will make it
clear when I am doing this, and it will not be a problem if you don't
understand it.


\section{Grade}
Your grade will be determined as follows:
\begin{itemize}
\setlength{\itemsep}{-0.3ex}
\item 20\% midterm
\item 25\% final exam
\item 25\% final project 
\item 30\% homework
\end{itemize}
If you get 90\% of points you'll get at least an A-, 80\% will give
you at least a B-, and 70\% at least a C-.


\section{Homework}
There will be one homework assignment per week.  It will be assigned
by Wednesday, and be due the following Wednesday.  Though I will not
accept late homework, your lowest homework grade will be dropped.
\begin{center}
{\bf Please {\em do}{} work together on homework problems!}
\end{center}
BUT, write up your solutions individually, and carefully
acknowledge the people and other sources that you used.

\section{Office Hours}
My office is AP\&M 5111.   Please come by and chat with me anytime
I'm there. My official office hours will be announced later. 

\section{Course Outline}
\begin{enumerate}
\setlength{\itemsep}{-0.2ex}
\item Overview of elliptic curves and modular forms

\item How to put a natural group structure on the set of points
  on an elliptic curve

\item Elliptic curves over finite fields

\item How to factor integers using elliptic curves (Application: cracking
the RSA cryptosystems.)

\item How to make cryptosystems using elliptic curves (Application:
the best public-key crypotosystems?)

\item Elliptic curves over the rational numbers

\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}).

\item Modular curves as quotients of the upper half plane

\item What does it mean for an elliptic curve to be ``modular''?
(What exactly was Andrew Wiles contribution to the proof of 
Fermat?)

\item Introduction to modular symbols (very explicit and
easy ``homology'' of modular curves)

\item How to use modular symbols to compute modular forms, I

\item How to use modular symbols to compute modular forms, II

\item SAGE: System for Algebra and Geometry Experimentation 
(on the architecture and design of SAGE).

\item The $L$-series of an elliptic curve over $\mbox{\rm \bf Q}$

\item The Birch and Swinnerton-Dyer conjecture (a million dollar
prize problem)

\end{enumerate}




\end{document}

