Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves
Jonas Bergström, Carel Faber, Sam Payne
TL;DR
The paper proves that for genus 4 moduli spaces with up to three marked points, the finite-field point counts $\#\overline{\mathcal{M}}_{4,n}(\mathbb{F}_q)$ and $\#\mathcal{M}_{4,n}(\mathbb{F}_q)$ are polynomials in $q$, and uses this to deduce that the rational singular cohomology $H^k(\overline{\mathcal{M}}_{g,n})$ vanishes in all odd degrees $k\le 9$ for all $(g,n)$. The approach blends an inductive Arbarello–Cornalba framework with a Hasse–Weil sieve for counting smooth genus-4 curves on three quadric types, careful control of the boundary via Getzler–Kapranov, and $\mathbb{S}_n$-equivariant point counts to determine the action on cohomology. This yields low-weight Langlands-predicted motives and confirms conjectural correspondences with automorphic representations of conductor 1, resolving a long-standing question of Arbarello–Cornalba. Additionally, the paper computes explicit $\mathbb{S}_n$-equivariant point counts for $n=2,3$ and relates these to cohomology with local-system coefficients, enriching the understanding of the tautological and non-tautological components of $H^*(\overline{\mathcal{M}}_{4,n})$.
Abstract
We compute the number of F_q-points on M_{4,n}, for n less than or equal to 3, and show that it is a polynomial in q, using a sieve based on Hasse-Weil zeta functions. As an application, we prove that the rational singular cohomology groups of moduli spaces of stable curves of genus g with n marked points vanish in all odd degrees less than or equal to 9, for all g and n. Both results confirm predictions of the Langlands program, via the conjectural correspondence with polarized algebraic cuspidal automorphic representations of conductor 1, which are classified in low weight. Our vanishing result for odd cohomology resolves a problem posed by Arbarello and Cornalba in the 1990s.