#### Vol. 6, No. 6, 2012

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Cusp form motives and admissible $G$-covers

### Dan Petersen

Vol. 6 (2012), No. 6, 1199–1221
##### Abstract

There is a natural ${\mathbb{S}}_{n}$-action on the moduli space $\phantom{\rule{0.3em}{0ex}}{\overline{\phantom{\rule{0.3em}{0ex}}\mathsc{ℳ}}}_{1,n}\left(B{\left(ℤ∕mℤ\right)}^{2}\right)$ of twisted stable maps into the stack $B{\left(ℤ∕mℤ\right)}^{2}$, and so its cohomology may be decomposed into irreducible ${\mathbb{S}}_{n}$-representations. Working over $Specℤ\left[1∕m‘\right]$ 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.

##### Keywords
Chow motive, cusp form, admissible cover, twisted curve, level structure
Primary: 11G18
Secondary: 14H10