/ Craig Westerland: Homology of Hurwitz spaces and Malle’s conjecture for function fields

Craig Westerland: Homology of Hurwitz spaces and Malle’s conjecture for function fields

October 5, 2019
2:30 pm - 3:20 pm

Abstract: In 2002, Malle conjectured an asymptotic (in discriminant) enumeration of number fields with prescribed Galois data; very few non-abelian cases of Malle’s conjecture have been established. Translating this conjecture into the setting of function fields, it becomes a question about enumeration of points over finite fields on certain Hurwitz moduli schemes of branched covers. In recent joint work with Ellenberg and Tran, we established an upper bound for this asymptotic (consonant with Malle’s prediction) using some coarse bounds on the Betti numbers of the complex points of these schemes. I’ll explain some of this work, along with some recent ongoing work which aims to refine these cohomological computations, leading to better understanding of the first and second order terms in the point count.