Generalized Borcherds Products and Two number theoretic applications

School of Mathematics Colloquium
Thursday, October 7, 2010 - 11:05
1 hour (actually 50 minutes)
Skiles 249
University of Wisconsin at Madison and Emory University
n his 1994 ICM lecture, Borcherds famously introduced an entirely new conceptin the theory of modular forms. He established that modular forms with very specialdivisors can be explicitly constructed as infinite products. Motivated by problemsin geometry, number theorists recognized a need for an extension of this theory toinclude a richer class of automorphic form. In joint work with Bruinier, the speakerhas generalized Borcherds's construction to include modular forms whose divisors arethe twisted Heegner divisors introduced in the 1980s by Gross and Zagier in theircelebrated work on the Birch and Swinnerton-Dyer Conjecture. This generalization,which depends on the new theory of harmonic Maass forms, has many applications.The speaker will illustrate the utility of these products by resolving open problemson the following topics:  1) Parity of the partition function  2) Birch and Swinnerton-Dyer Conjecture and ranks of elliptic curves.