Modular Forms
Professors Ilya Piatetski-Shapiro and Stephen
Miller
Overview of the Course
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:
Prerequisites: One semester of abstract algebra as well as complex analysis.Mobius transformations and Escher paintings Elliptic curves and lattices Eisenstein series Cusp forms The Petersson inner product Hecke operators L-functions Theta functions.
Course Text: "Modular Functions and Dirichlet Series in Number Theory," Second Edition, by Tom M. Apostol, Graduate Texts in Mathematics 41, Springer-Verlag, 1990.
Reference Texts include:
