Averages of Arithmetic Functions over Conductors of Function Fields

TL;DR

This paper establishes asymptotic formulas for counting Galois extensions over function fields with specific ramification properties, advancing the understanding of arithmetic functions in positive characteristic and building on recent breakthroughs in braid group representations.

Contribution

It provides new asymptotic results for Galois extensions with prescribed ramification over function fields, extending previous work and employing novel homological and algebraic techniques.

Findings

Asymptotics for the number of regular Galois extensions with one ramified place.

Results for arithmetic functions of ramified places in function fields.

Application of recent homological breakthroughs to number theory problems.

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.