Complexes d'intersection des compactifications de Baily-Borel : Le cas des vari\'et\'es de Siegel

TL;DR

This paper computes traces of Hecke correspondences combined with Frobenius on intersection complexes of Siegel modular varieties' Baily-Borel compactification, using Pink's theorem and new constructions of intermediate extensions.

Contribution

It introduces a novel construction of intermediate extensions as weight truncations and extends weighted cohomology complexes to positive characteristic.

Findings

Calculated trace formulas for Hecke-Frobenius actions.

Developed a new approach to intermediate extensions of perverse sheaves.

Extended weighted cohomology concepts to positive characteristic.

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.