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

There is a natural Sn-action on the moduli space ¯1,n(B(m)2) of twisted stable maps into the stack B(m)2, and so its cohomology may be decomposed into irreducible Sn-representations. Working over Spec [1m] we show that the alternating part of the cohomology of one of its connected components is exactly the cohomology associated to cusp forms for Γ(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.

Chow motive, cusp form, admissible cover, twisted curve, level structure
Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 14H10
Received: 18 March 2011
Accepted: 18 October 2011
Published: 12 August 2012
Dan Petersen
Department of Mathematics
KTH Royal Institute of Technology
Institutionen för matematik
Kungliga Tekniska Högskolan
100 44 Stockholm