Steve Kudla (University of Toronto)

Title: Arithmetic Theta Series

Abstract: I will recount a family history of theta series through several generations. Theta series for positive definite integral quadratic forms provide some of the most classical examples of elliptic modular forms and their Siegel modular variants. Analogous series were defined by Siegel and Maass for lattices with indefinite quadratic forms say with signature (p,q). These series are no longer holomorphic and depend on an additional variable in the Grassmannian of negative q-planes, i.e., the symmetric space for the orthogonal group O(p,q). Motivated by work of Hirzebruch and Zagier on the generating series for curves on Hilbert modular surfaces, Millson and I constructed a theory of theta series valued in the cohomology of certain locally symmetric spaces -- geometric theta series. More recently, a theory of arithmetic theta series has been emerging, theta series valued in the Chow groups or arithmetic Chow groups of the integral models of certain Shimura varieties.