Complexes d'intersection des compactifications de Baily-Borel : Le cas des variétés de Siegel

Abstract

In this work, we calculate the trace of a Hecke correspondance composed with a power of the Frobenius endomorphism on the fibre of the intersection complexes of the Baily-Borel compactification of a Siegel modular variety. Our main tool is Pink's theorem about the restriction to the strata of the Baily-Borel compactification of the direct image of a local system on the Shimura variety. To use this theorem, we give a new construction of the intermediate extension of a pure perverse sheaf as a weight truncation of the full direct image. More generally, we are able to define analogs in positive characteristic of the weighted cohomology complexes introduced by Goresky, Harder and MacPherson.

40 pages, in French, final version, appeared in Journal of the American Mathematical Society (2008)