Malle’s conjecture for function fields

The inverse Galois problem asks for finite group G, whether G is a finite Galois extension of the rational numbers. Malle’s conjecture is a quantitative version of this problem, giving an asymptotic prediction of how many such extensions exist with bounded discriminant. In joint with Aaron Landesman, we prove Malle’s conjecture for a group G over the function field F_q(t) when q is sufficiently large and relatively prime to the order of G. The key new input to our proof is a general homological stability result for Hurwitz spaces.