CoCalc Public Fileswww / 168 / syllabus / syllabus.tex
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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28\begin{center}
29\large Math 168: Topics in Applied Math \& Comp. Sci. (Fall 2005):\\
30\Large
31\vspace{1ex}
32
33{\sc Explicit Approaches to Elliptic Curves\\and Modular Forms}\\
34\vspace{1ex}
35
36\large
37
38\vspace{2ex}
39
40{\sf
41Course: MW 3:00-4:20 in 201 Center\\
42Section: Th 5:00pm in 207 Center}\\
43William Stein ({\sf wstein@ucsd.edu})\\
44http://modular.ucsd.edu/168\\
45\large
46\end{center}
47
48\begin{abstract}
49  This course is an introduction to elliptic curve and modular forms,
50  with a special emphasis on how to compute with these objects.
51\end{abstract}
52
53\section{Textbooks}
54The main text are the notes that I've written and will be handing out.
55There are also many books and articles on elliptic curves and modular
56forms, which I will encourage you to look at.  For the first quarter
57of the course, I will closely follow Chapter 6 of {\tt
58  http://modular.ucsd.edu/ent/}.
59
60
61\section{Course Topics}
62\begin{itemize}
63\setlength{\itemsep}{-0.2ex}
64\item{}{[\bf Elliptic Curves]} I will define elliptic curves, explain
65  their two main applications to cryptography, and discuss the Birch
66  and Swinnerton-Dyer conjecture (a million dollar Clay Math prize
67  problem).
68
69\item{}{[\bf Modular Forms]} I will define modular forms of weight
70  $2$, discuss their connection with elliptic curves and Andrew
71  Wiles's celebrated proof of Fermat's Last Theorem.  I will also
72  discuss how to use modular symbols to compute modular forms, and
73  mention open problems.
74\end{itemize}
75
76See the course outline below for more details.
77
78\section{Prerequisites}
79\vspace{-1ex}
80\begin{itemize}
81\setlength{\itemsep}{-0.3ex}
82\item A course on groups, rings and fields.
83\item Ability to follow nontrivial mathematical arguments.
84\item Know how to use a computer.
85\end{itemize}
86
87It will be useful if you know something about algebraic curves,
88complex analysis and have some prior exposure to number theory.
89However, I am not requiring this as a prerequisite.  A few times
90during the course I will give motivation for a topic or a deeper
91explanation for something that assume more background; I will make it
92clear when I am doing this, and it will not be a problem if you don't
93understand it.
94
95
98\begin{itemize}
99\setlength{\itemsep}{-0.3ex}
100\item 20\% midterm
101\item 25\% final exam
102\item 25\% final project
103\item 30\% homework
104\end{itemize}
105If you get 90\% of points you'll get at least an A-, 80\% will give
106you at least a B-, and 70\% at least a C-.
107
108
109\section{Homework}
110There will be one homework assignment per week.  It will be assigned
111by Wednesday, and be due the following Wednesday.  Though I will not
113\begin{center}
114{\bf Please {\em do}{} work together on homework problems!}
115\end{center}
116BUT, write up your solutions individually, and carefully
117acknowledge the people and other sources that you used.
118
119\section{Office Hours}
120My office is AP\&M 5111.   Please come by and chat with me anytime
121I'm there. My official office hours will be announced later.
122
123\section{Course Outline}
124\begin{enumerate}
125\setlength{\itemsep}{-0.2ex}
126\item Overview of elliptic curves and modular forms
127
128\item How to put a natural group structure on the set of points
129  on an elliptic curve
130
131\item Elliptic curves over finite fields
132
133\item How to factor integers using elliptic curves (Application: cracking
134the RSA cryptosystems.)
135
136\item How to make cryptosystems using elliptic curves (Application:
137the best public-key crypotosystems?)
138
139\item Elliptic curves over the rational numbers
140
141\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}).
142
143\item Modular curves as quotients of the upper half plane
144
145\item What does it mean for an elliptic curve to be modular''?
146(What exactly was Andrew Wiles contribution to the proof of
147Fermat?)
148
149\item Introduction to modular symbols (very explicit and
150easy homology'' of modular curves)
151
152\item How to use modular symbols to compute modular forms, I
153
154\item How to use modular symbols to compute modular forms, II
155
156\item SAGE: System for Algebra and Geometry Experimentation
157(on the architecture and design of SAGE).
158
159\item The $L$-series of an elliptic curve over $\mbox{\rm \bf Q}$
160
161\item The Birch and Swinnerton-Dyer conjecture (a million dollar
162prize problem)
163
164\end{enumerate}
165
166
167
168
169\end{document}
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