  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.
