Stable cohomology for Hurwitz spaces and arithmetic applications
Jordan Ellenberg, University of Wisconsin

Abstract: We will discuss a theorem in topology (or maybe even group theory) with a motivation from number theory. Hurwitz spaces are moduli spaces of finite branched covers of P^1 -- for instance, the moduli space of trigonal curves is a Hurwitz space. We will discuss the stabilization of the cohomology of these spaces (or, what is the same, for certain congruence subgroups of Artin braid groups) as the number of branch points (resp. number of strands) grows, with the Galois group of the cover being fixed; this can be thought of as a "Harer theorem" for this family of moduli spaces. It turns out that the function field analogues of many popular conjectures in analytic number theory (due to Cohen-Lenstra, Bhargava, etc.) reduce to topological questions about Hurwitz spaces. We will discuss the arithmetic consequences of the stabilization theorem, and of a geometrically natural conjecture about the stable cohomology classes of Hurwitz spaces. (joint work with Akshay Venkatesh and Craig Westerland)