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Cusp form motives and admissible \(G\)-covers. (English) Zbl 1303.14025

Summary: There is a natural \(S_n\)-action on the moduli space \(\overline{\mathcal M}_{1,n}(B(\mathbb Z/m\mathbb Z)^2)\) of twisted stable maps into the stack \(B(\mathbb Z/m\mathbb Z)^2\), and so its cohomology may be decomposed into irreducible \(S_n\)-representations. Working over \(\mathrm{Spec}\,\mathbb Z[1/m]\) 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.

MSC:

14D23 Stacks and moduli problems
11G18 Arithmetic aspects of modular and Shimura varieties
14H10 Families, moduli of curves (algebraic)
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