University of Sydney Algebra Seminar

Craig Westerland (University of Melbourne)

Friday 18 March, 12:05-12:55pm, Carslaw 175

Cohen-Lenstra heuristics for function fields and Hurwitz spaces

The Cohen-Lenstra heuristics are a family of (open) conjectures about the distribution of class groups of number fields (recall that the class group of a field measures the failure of its ring of integers to satisfy unique factorization). The analogous question for function fields asks for the distribution of class groups associated to branched coverings of curves in positive characteristic. Joint with Ellenberg and Venkatesh, we resolved this conjecture in the simplest case, corresponding to the setting of quadratic extensions. Remarkably, one can reduce this counting problem to an asymptotic computation of the homology of various Hurwitz moduli spaces of branched coverings. I'll talk about the setup of the problem, and how we went about computing those homology groups.