Mathematics 480a
Senior Seminar
Yale University
Fall 1998

Modular Forms
Professors Ilya Piatetski-Shapiro and Stephen Miller

Overview of the Course

    Modular forms are mathematical objects which are emerging across the discipline in exciting ways.  Most dramatically, Andrew WilesWiles 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
  • L-functions
  • 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, Springer-Verlag, 1990.

    Reference Texts include:

    contact:                                                                                Stephen Miller, August 3, 1998