The leading constant in Malle's conjecture over function fields

TL;DR

This paper proves the analogue of Malle's conjecture for function fields over finite fields with large q, providing a precise formula for the leading constant using advanced algebraic geometry and number theory techniques.

Contribution

It introduces a new approach combining Hurwitz space homology, Brauer group interpretations, and Tamagawa volumes to establish the conjecture's leading constant in the function field setting.

Findings

Confirmed the leading constant matches the predicted formula for large q.

Connected Hurwitz space homology with point counting via Tamagawa measures.

Provided a novel interpretation of Frobenius fixed points in terms of algebraic stacks.

Abstract

We prove the analogue of Malle's conjecture for the global function field $\F_q(t)$ with $q$ sufficiently large, including a precise formula for the leading constant. The main ingredients are the recent breakthrough of Landesman--Levy on the stable homology of Hurwitz spaces, a novel interpretation of the Frobenius fixed components of Hurwitz spaces in terms of the Brauer group of the stack $BG$ and an interpretation of the number of $\F_q$ points of configuration spaces as a certain Tamagawa volume.