Polynomial point counts and odd cohomology vanishing on moduli spaces of stable curves

Abstract

We compute the number of $\mathbb{F}_q$-points on $\overline{\mathcal{M}}_{4,n}$ for $n \leq 3$ 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 group $H^k (\overline{\mathcal{M}}_{g,n})$ vanishes for all odd $k \leq 9$. 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.