Cusp form motives and admissible $G$-covers

TL;DR

This paper links the cohomology of certain moduli spaces of twisted stable maps to cusp forms for modular groups, providing a new geometric approach that replaces classical Kuga-Sato varieties.

Contribution

It demonstrates that the cohomology of a specific moduli space corresponds to cusp form motives, offering an alternative to Scholl's construction and answering Manin's question.

Findings

The alternating part of the cohomology matches cusp form cohomology.

Provides a geometric realization of cusp form motives via moduli spaces.

Offers an alternative to Kuga-Sato varieties for constructing cusp form motives.

Abstract

The moduli space of twisted stable maps into the stack $B(\Z/m\Z)^2$ 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.