Cusp form motives and admissible $G$-covers

Abstract

The moduli space of twisted stable maps into the stack $B(\mathbb{Z}/m\mathbb{Z}{)}^2$ carries a natural $S_n$-action and so its cohomology may be decomposed into irreducible $S_n$-representations. Working over $\Spec \Z[1/m]$ we show that the alternating part of the cohomology of one of its connected components is exactly the cohomology associated to cusp forms for $\Gamma (m)$. 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.

24 pages

References in corpus (1)

• A remark on Getzler's semi-classical approximation

Cited by in corpus (2)

• A remark on Getzler's semi-classical approximation
• Moduli of elliptic curves via twisted stable maps