Automorphic Forms Program
Coxeter Lecture Series

March 10, 11, 12, 2003, 3:30 pm
Stephen S. Kudla (Maryland)
Arithmetic theta series

Audio and Slides of Lectures

The Fields Institute Coxeter Lecture Series (CLS) is intended to bring a leading international mathematician in the field of the thematic program of the Institute to give a series of three lectures. One talk will be an overview to a general mathematical audience including post-doctoral fellows and graduate students. The other two talks will target program participants in their choice of topic(s), in collaboration with the organizers of the related thematic program.

Abstract:

These lectures will survey the construction of modular forms whose Fourier coefficients are quantities from arithmetical algebraic geometry, for example, the heights of certain cycles on arithmetic surfaces. These modular forms can be described as generating series and can be viewed as analogues of theta series. The results to be discussed are joint work with M. Rapoport and T. Yang.

Lecture 1 will give an overview of two parallel examples: modular generating series for curves and 0-cycles on Hilbert modular surfaces and their compact analogues, first studied by Hirzebruch and Zagier, and, in the arithmetic situation, modular generating series for curves and 0-cycles on the arithmetic surfaces attached to Shimura curves.

Lecture 2 will explain the construction and proof of modularity of the generating series for curves on such an arithmetic surface S. The resulting modular form of weight 3/2 is valued in the first arithmetic Chow group of S.

Lecture 3 will explain the construction of a generating series for 0-cycles on S. The main conjecture is that (i) this series coincides with the q-expansion of the central derivative of a Siegel Eisenstein series of genus 2 and weight 3/2, and (ii) the restriction of this function to the `diagonal' SL(2)xSL(2) coincides with the height pairing of two generating functions for curves on S. Recent progress on this conjecture will be discussed.