Mathematics 480a
Senior Seminar
Yale University
Fall 1998
Modular Forms
Professors Ilya PiatetskiShapiro and Stephen
Miller
Overview of the Course
Modular forms are mathematical objects which are emerging
across the discipline in exciting ways. Most dramatically, Andrew Wiles
recently settled Fermat's Last Theorem by linking it to modular forms.
Many other classical number theory problems  some of which have eluded
conquest for centuries  have fresh reformulations in terms of modular
forms. And recently physicists have begun studying modular forms
in connection with string theory (and even black holes!).
We will begin the course with some introductory lectures motivating
the subject of modular forms. In particular, we will open with an
ancient question: how many ways can a number be written as the sum of two
squares? As three? Four? There is a surprisingly simple
answer to this question via modular forms.
During the bulk of the course students will present
material on:
Mobius transformations and Escher paintings
Elliptic curves and lattices
Eisenstein series
Cusp forms
The Petersson inner product
Hecke operators
Lfunctions
Theta functions.
Prerequisites: One semester
of abstract algebra as well as complex analysis.
Course Text: "Modular
Functions and Dirichlet Series in Number Theory," Second Edition,
by Tom M. Apostol, Graduate Texts in Mathematics 41, SpringerVerlag,
1990.
Reference Texts include:

"Introduction to Modular Forms," by Serge Lang, 2nd printing, SpringerVerlag,
1995.

"A Course in Arithmetic," by JeanPierre Serre, Graduate Texts in
Mathematics 7, SpringerVerlag, 1973.

"Lectures on Modular Forms," by R.C. Gunning, Princeton University
Press, 1962.

"Some Applications of Modular Forms," by Peter Sarnak, Cambridge
University Press, 1990.

"Topics in Classical Automorphic Forms," by Henryk Iwaniec, Graduate
Studies in Mathematics 17, Amer. Math. Soc., 1997.
contact: steve@math.yale.edu
Stephen Miller, August 3, 1998