Cusp form motives and admissible $G$-covers

Abstract

There is a natural ${\mathbb{S}}_n$-action on the moduli space ${\overline{\mathcal{ℳ}}}_{1,n}\left(B{\left(\mathbb{Z}∕m\mathbb{Z}\right)}^2\right)$ of twisted stable maps into the stack $B{\left(\mathbb{Z}∕m\mathbb{Z}\right)}^2$, and so its cohomology may be decomposed into irreducible ${\mathbb{S}}_n$-representations. Working over $Spec\mathbb{Z}\left[1∕m‘\right]$ we show that the alternating part of the cohomology of one of its connected components is exactly the cohomology associated to cusp forms for $\Gamma \left(m‘\right)$. In particular this offers an alternative to Scholl’s construction of the Chow motive associated to such cusp forms. This answers in the affirmative a question of Manin on whether one can replace the Kuga–Sato varieties used by Scholl with some moduli space of pointed stable curves.