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

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.

arXiv commentFixed some calculation errors pointed out by Zeev Rudnick

References in corpus (4)

• Representations of categories of G-maps
• Categories of FI type: a unified approach to generalizing representation stability and character polynomials
• Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting
• ${\mathrm{FI}}_G$-modules, orbit configuration spaces, and complex reflection groups

Cited by in corpus (1)

• Representation stability and arithmetic statistics of spaces of 0-cycles