Vanishing cycles and non-classical parabolic cohomology

The work described here began as an attempt to understand the structure of the cohomology groups associated to a subgroup ? of nite index of SL 2 (Z) which are not congruence subgroups. One knows that to the space of cusp forms of weight w > 2 on ? (whose dimension we denote by d) one can 15] attach a motive M, which is pure of weight w?1 and of rank 2d, deened over some number eld; it is a direct factor of the motive of a suitable compact model of the (w ? 2)-fold bre product of a family of elliptic curves over the modular curve, as in the classical case of a congruence subgroup. The realisations of M are the parabolic cohomology groups. By looking instead at the uncompactiied bre variety (that is, with the divisor over the cusps removed) one gets a mixed motive M 0 , which is a direct factor of the \motive with compact supports" of the noncompact variety; M 0 is an extension of M by an Artin motive of rank equal to the number of cusps of ?. The cohomology classes associated to this Artin motive arise from Eisenstein series. In the case of a congruence subgroup this extension is trivial|it is split by the action of the Hecke algebra (an example of the \Manin-Drinfeld principle"). Already for weight 2 it is known that these extensions can be nontrivial in general, and for rather simple reasons. Namely, Belyi's theorem implies that any connected smooth curve C over Q (not necessarily projective) has a Zariski open subset U whose complex points are isomorphic to the quotient of the upper-half plane by some nite index subgroup ?. In this situation M is the H 1-motive of the compactiication of C, and the mixed motive M 0 is h 1 c (U), the H 1 of U with compact supports. If C is the complement in a projective curve of, say, two points whose diierence is a divisor of innnite order, then the Abel-Jacobi theorem implies that M 0 is a nontrivial extension of motives. The search for algebraic invariants to classify the extension M 0 in general leads naturally to the study of the motivic cohomology of the bre varieties. Beilinson's conjectures give a conjectural description of these groups. In this case the regulator map, which plays an essential role in the Beilinson conjectures, …