Connected Components of Hurwitz Schemes and Malle's Conjecture

TL;DR

This paper extends the heuristic approach relating Hurwitz schemes to Malle's conjecture, addressing non-geometrically connected covers and proposing a modification to the conjecture that avoids known counterexamples.

Contribution

It generalizes Ellenberg-Venkatesh's heuristic to covers of P^1 that are not necessarily geometrically connected, leading to a refined version of Malle's conjecture.

Findings

Heuristic supports a modified Malle's conjecture for non-geometrically connected covers.

Addresses and circumvents known counterexamples to the original conjecture.

Provides a new perspective on counting extensions via Hurwitz schemes.

Abstract

Let Z(X) be the number of degree-d extensions of F_q(t) with bounded discriminant and some specified Galois group. The problem of computing Z(X) can be related to a problem of counting F_q-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 P^1 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.