Colorado State University
Modular Forms, Infinite Products, and Singular Moduli
By Ken Ono
From Department of Mathematics
University of Wisconsin
When Dec 8, 2005
Where Room 202, Weber Building
Abstract Modular forms play many roles in mathematics. In number theory, modular forms often arise as generating functions for interesting quantities such as representation numbers of integers by quadratic forms, partition functions, values of L-functions, and also degrees of characters of sporadic simple groups like the Monster.
In his 1994 ICM lecture, Borcherds found a striking new phenomenon. He proved that certain modular forms of half-integral weight serve as generating functions for the infinite product exponents of other modular forms, thereby greatly generalizing some of the prettiest q-series dating back to works of Euler and Jacobi on classical theta functions. His work pertained to an exceptionally rich family of modular forms, those with a ‘Heegner divisor’.
Zagier later found a beautiful number theoretic explanation of the Borcherds phenomenon, one involving singular moduli, complex multiplication, and elliptic curves. In this lecture, we provide a general framework which includes Zagier’s reformulation of Borcherds’ theory as a special case. We show that all of these results follow from beautiful properties of a delightfully rich sequence of modular forms, the weak Maass-Poincare series of half-integral weight.
Further Information Rachel Pries
There will be Refreshments in the lobby at 2.30pm, following the Colloquium and lunch.
The Colloquium counts as Seminar Credit for Mathematics Students.