The eleventh cohomology group of ${\bar{\mathcal{M}}}_{g,n}$

Abstract

We prove that the rational cohomology group $H^{11}({\bar{\mathcal{M}}}_{g,n})$ vanishes unless $g=1$ and $n\ge 11$. We show furthermore that $H^k({\bar{\mathcal{M}}}_{g,n})$ is pure Hodge-Tate for all even $k\le 12$ and deduce that $\#{\bar{\mathcal{M}}}_{g,n}({\mathbb{F}}_q)$ is surprisingly well approximated by a polynomial in $q$. In addition, we use $H^{11}({\bar{\mathcal{M}}}_{1,11})$ 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 non-tautological algebraic cycle classes in Chow cohomology.

arXiv comment18 pages. v2: Final version, to appear in Forum of Mathematics, Sigma

References in corpus (6)

• The motive of a classifying space
• The structure of the tautological ring in genus one
• Cohomology of local systems on the moduli of principally polarized abelian surfaces
• The tautological ring of the space of pointed genus two curves of compact type
• Nontautological Bielliptic Cycles
• Cohomology of moduli spaces via a result of Chenevier and Lannes

Cited by in corpus (3)

• Extensions of tautological rings and motivic structures in the cohomology of ${\overline{\mathcal{M}}}_{g,n}$
• Weight 11 compactly supported cohomology of moduli spaces of curves
• The tautological ring of ${\overline{\mathcal{M}}}_{g,n}$ is rarely Gorenstein