Averages of Arithmetic Functions over Conductors of Function Fields

Abstract

For a finite group $G$ and a sufficiently large (but fixed) prime power $q$ coprime to $G$ we obtain asymptotics for the number of regular Galois extensions $L/ \mathbb{F}_q(t)$, with $\mathrm{Gal}(L/\mathbb{F}_q(t)) \cong G$, ramified at a single place of $\mathbb{F}_q(t)$, thus making progress on a positive characteristic analog of the Boston--Markin conjecture. We also obtain similar results for other arithmetic functions of the product of places of $\mathbb{F}_q(t)$ ramified in $L$, and for more general one-variable function fields over $\mathbb{F}_q$ in place of $\mathbb{F}_q(t)$. Some of our proofs make crucial use of a series of recent breakthroughs by Landesman--Levy, as well as a new `vanishing of stable homology in a given direction' result for representations of braid groups arising from braided vector spaces. Other inputs include a study of (rings of coinvariants of) braided vector spaces associated to racks with $2$-cocycles, a connection between convolution of arithmetic functions and direct sums of braided vector spaces, and a Goursat lemma for racks.