Averages of Arithmetic Functions over Conductors of Function Fields
Jordan Ellenberg, Mark Shusterman
TL;DR
The paper develops a robust framework combining homological stability for braid-group actions, braided-vector-space methods, and étale cohomology to study averages of arithmetic functions over conductors of regular Galois extensions of function fields. It proves a key vanishing theorem that yields slope-controlled stability and uses it to derive asymptotics for counts of Galois extensions with prescribed ramification, as well as power-saving cancellations for various arithmetic functions (including irreducibility of conductors, Möbius-type sums, and discriminant characters) in large-genus and large-n regimes. By linking configuration and Hurwitz spaces to trace functions of sheaves, the authors translate arithmetic questions into cohomological bounds, enabling explicit error terms and, in many cases, explicit q-dependence. The results have implications for random-profinite-group models, factorization statistics over finite fields, and refinements of function-field Cohen–Lenstra heuristics, with broader potential impact on the study of arithmetic statistics in positive characteristic. Overall, the work advances the positive-characteristic analogs of classical arithmetic-statistics problems through a unifying topology–arithmetic approach grounded in braided-geometry.
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.