Motives for modular forms

In [DeFM], Deligne constructs l-adic parabolic cohomology groups attached to holomorphic cusp forms of weight ≥ 2 on congruence subgroups of SL2(Z). These groups occur in the l-adic cohomology of certain smooth projective varieties over Q—the Kuga-Sato varieties— which are suitably compactified families of products of elliptic curves. In view of Grothendieck’s conjectural theory of motives it is natural to hope that the parabolic cohomology groups can be directly constructed as the kernel of some projectors (in a suitable ring of algebraic correspondences) acting on the cohomology of these varieties. In this note we show that this can be done; in fact the projector we use belongs to the group algebra of a finite group of automorphisms of the Kuga-Sato variety.