Representation stability and arithmetic statistics of spaces of 0-cycles

TL;DR

This paper investigates the topological and arithmetic properties of spaces of 0-cycles on manifolds, demonstrating stability in cohomology and arithmetic statistics as the degree increases, with implications for rational maps over finite fields.

Contribution

It establishes representation stability for the cohomology of spaces of 0-cycles and connects topological stability to arithmetic statistics over finite fields.

Findings

Proves representation stability for cohomology of 0-cycle spaces.

Establishes subexponential bounds on unstable cohomology growth.

Shows stabilization of arithmetic quantities for rational maps over finite fields.

Abstract

We continue the study of a general class of spaces of 0-cycles on a manifold defined and begun by Farb-Wolfson-Wood. Using work of Gadish on linear subspace arrangements, we obtain representation stability for the cohomology of the ordered version of these spaces. We establish subexponential bounds on the growth of unstable cohomology, and the Grothendieck-Lefschetz trace formula then allows us to translate these topological stability phenomena to stabilization of statistics for spaces of 0-cycles over finite fields. In particular, we show that the average value of certain arithmetic quantities associated to rational maps over finite fields stabilizes as the degree goes to infinity.