Singular Moduli

The values of the elliptic modular function j at imaginary quadratic numbers τ are called singular moduli. They are of fundamental importance in the study of elliptic curves and in algebraic number theory, including the study of elliptic curves over finite fields.
The theorem of Gross and Zagier has provided striking congruences satisfied by these numbers. Various aspects of this theorem were generalized in recent years by Jan Bruinier and Tonghai Yang and by Kristin Lauter and the speaker. These may be viewed as concerning the theory of curves of genus 2 and their singular moduli that are obtained by evaluating the Igusa invariants at certain 2×2 complex matrices τ. I will describe some of the recent ideas introduced in this area and, time allowing, describe some current projects.