Cohomology of Local Systems on the Moduli Space of Curves

Abstract

Let $\M_{g,n}$ and $\overline{\M}_{g,n}$ denote the moduli spaces of smooth and stable curves of genus $g$ with $n$ marked points respectively. We introduce and study these spaces. In particular, we are interested in their cohomology groups. There is an action of $\S_n$ on $\M_{g,n}$ and $\overline{\M}_{g,n}$ permuting their marked points, which extends to an action on the cohomology, making them $\S_n$-representations. Moreover taking advantage of the fact that -depending on the chosen cohomology theory- the cohomology groups are also mixed Hodge structures or $\ell$-adic representations of $\Gal(\overline{\Q}/\Q)$, we can determine several motivic Euler characteristics using point counts over finite fields. We introduce local systems $\V_{\lambda}$ on $\M_g$ given by a partition $\lambda$ of $n$ into at most $g$ parts, and investigate the motivic Euler characteristic given by the sheaf cohomology with coefficients in $\V_{\lambda}$. Calculating the trace of Frobenius on these Euler characteristics then gives information on the existence of non-tautological cohomology on $\M_g$. For $g =4$ and $q=2,3,4$, we attempt to detect non-tautological cohomology using these trace calculations.