Class invariants in genus 2

Abstract: The main technique to construct class invariants, namely a set of distinguished elements in class fields of a given CM field, is by evaluating modular functions at CM points corresponding to that field. The case of quadratic imaginary fields is classic and well understood; it draws on the existence of modular units and good reduction for CM elliptic curves. The case of quartic fields is much less understood, but has recently seen some new results. Similar to the case of elliptic curves, the reduction of abelian surfaces with CM, and special divisors on the moduli space, play a key role. In this talk, I will report on recent work with Kristin Lauter that provides information about the prime factorization of certain class invariants (first studied by the speaker with DeShalit) and bounds the denominator of the so-called absolute Igusa invariants of genus 2 curves with CM. As will be explained, this last fact has applications to cryptography.