The Cohen–Lenstra moments for functions fields and stable homology of Hurwitz spaces

The Cohen--Lenstra heuristics predict the distribution of the odd part of class groups of quadratic fields, and are one of the driving conjectures in arithmetic statistics. I will explain work with Aaron Landesman, where we compute the moments of the Cohen--Lenstra distribution for function fields, when the size of the finite field is sufficiently large (depending on the moment). We follow an approach to this problem due to Ellenberg--Venkatesh--Westerland, and the key new input is the computation of the stable rational homology of Hurwitz spaces associated to certain conjugacy classes in generalized dihedral groups. I will explain the ideas in our computation of the stable homology in the case of the dihedral group of order 6 with conjugacy class transpositions.