On the pullback of arithmetic theta functions
by
Stephen Kudla
University of Toronto
Coauthors: Tonghai Yang

In a series of papers with M. Rapoport and T. Yang, we introduced modular forms -which we call arithmetic theta functions -whose Fourier coefficients are constructed from classes in the arithmetic Chow groups of certain moduli spaces. In the simplest case, the arithmetic theta function f_C has weight 1 and is a generating function for the arithmetic degrees of a family of zero cycles on the moduli space C of elliptic curves with CM by the maximal order in a fixed imaginary quadratic field. In another case, the arithmetic theta function f_S has weight 3/2 and is the generating function for the arithmetic degrees of certain divisors on an integral model S of a Shimura curve.

In this talk I will describe natural morphisms j: C --> S and give a formula expressing the pullback j^* f_S of the arithmetic theta function for cycles on S in terms of f_C's and theta functions of weight 1/2. This formula is analogous to factorization formulas for classical theta functions.

Date received: May 15, 2008