Moduli spaces of curves with polynomial point counts

TL;DR

This paper establishes that the count of genus g curves over finite fields is polynomial in field size for g ≤ 8, and identifies the minimal number of marked points where this fails, using cohomology computations.

Contribution

It proves the polynomial count property for moduli spaces of curves up to genus 8 and determines the minimal marked points where polynomiality breaks down.

Findings

Polynomial point count holds for genus g ≤ 8.

Identifies minimal n where polynomial count fails for each g.

Computes the 13th cohomology group of moduli spaces for all g and n.

Abstract

We prove that the number of curves of a fixed genus g over finite fields is a polynomial function of the size of the field if and only if g is at most 8. Furthermore, we determine for each positive genus g the smallest n such that the moduli space of curves of genus g with n marked points does not have polynomial point count. A key ingredient in the proofs, which is also a new result of independent interest, is the computation of the thirteenth cohomology group of the moduli spaces of stable curves of genus g with n marked points, for all g and n.