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.

Index of
Fields Distinguished and Coxeter Lectures.