Connected Components of Hurwitz Schemes and Malle's Conjecture
Abstract
Let Z(X) be the number of degree-d extensions of Fq(t) with bounded discriminant and some specified Galois group. The problem of computing Z(X) can be related to a problem of counting Fq-rational points on certain Hurwitz spaces. Ellenberg and Venkatesh used this idea to develop a heuristic for the asymptotic behavior of Z'(X), the number of -geometrically connected- extensions, and showed that this agrees with the conjectures of Malle for function fields. We extend Ellenberg-Venkatesh's argument to handle the more complicated case of covers of P1 which may not be geometrically connected, and show thatthe resulting heuristic suggests a natural modification to Malle's conjecture which avoids the counterexamples, due to Kl\"uners, to the original conjecture.