By  Ken Ono  
From  Department of Mathematics University of Wisconsin 

When  Dec 8, 2005 12:00noon 

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 Lfunctions, 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 halfintegral weight serve as generating functions for the infinite product exponents of other modular forms, thereby greatly generalizing some of the prettiest qseries 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 MaassPoincare series of halfintegral weight. 

Further Information  Rachel Pries 