The eleventh cohomology group of

Abstract

AbstractWe prove that the rational cohomology group$H^{11}(\overline {\mathcal {M}}_{g,n})$vanishes unless$g = 1$and$n \geq 11$. We show furthermore that$H^k(\overline {\mathcal {M}}_{g,n})$is pure Hodge–Tate for all even$k \leq 12$and deduce that$\# \overline {\mathcal {M}}_{g,n}(\mathbb {F}_q)$is surprisingly well approximated by a polynomial inq. In addition, we use$H^{11}(\overline {\mathcal {M}}_{1,11})$and its image under Gysin push-forward for tautological maps to produce many new examples of moduli spaces of stable curves with nonvanishing odd cohomology and nontautological algebraic cycle classes in Chow cohomology.