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

Abstract

We compute the number of (Formula presented)-points on (Formula presented) for n ≤ 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 (Formula presented) vanishes for all odd k ≤ 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.