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

Abstract:

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 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 group \(S_3\) with conjugacy class transpositions.