$\mathrm{FI}_G$-modules and arithmetic statistics

TL;DR

This paper extends FI_G-module methods to study the stability of Galois representations in etale cohomology, establishing bounds and applying trace formulas to derive asymptotic arithmetic statistics over finite fields.

Contribution

It introduces new stability results for equivariant Galois representations using FI_G-modules and applies these to arithmetic statistics of orbit configuration spaces.

Findings

Stability of equivariant Galois representations established.

Subexponential bounds on unstable cohomology growth.

Average Gauss sums over polynomials stabilize as degree increases.

Abstract

This is a sequel to the paper [Cas]. Here, we extend the methods of Farb-Wolfson using the theory of FI_G-modules to obtain stability of equivariant Galois representations of the etale cohomology of orbit configuration spaces. We establish subexponential bounds on the growth of unstable cohomology, and then use the Grothendieck-Lefschetz trace formula to obtain results on arithmetic statistics for orbit configuration spaces over finite fields. In particular, we show that the average value, across polynomials over F_q, of certain Gauss sums over their roots, stabilizes as the degree goes to infinity.