FI_G-modules and arithmetic statistics

2017/03/21 by Kevin Casto, Casto, Kevin · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1703.07295

openalex publication_date 2017/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting

Cited by

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

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