The stabilization of the Frobenius--Hecke traces on the intersection cohomology of orthogonal Shimura varieties

TL;DR

This paper proves a stabilized formula for Frobenius--Hecke traces on the intersection cohomology of orthogonal Shimura varieties, enabling explicit calculations of their Hasse--Weil zeta functions in certain cases.

Contribution

It extends Morel's formula to a stabilized version for orthogonal Shimura varieties, linking intersection cohomology with automorphic representations.

Findings

Stabilization of Frobenius--Hecke trace formula

Explicit computation of Hasse--Weil zeta functions in special cases

Application of Arthur and Ta"ibi's endoscopic classification

Abstract

We study Shimura varieties associated with special orthogonal groups over the field of rational numbers. We prove a version of Morel's formula for the Frobenius--Hecke traces on the intersection cohomology of the Baily--Borel compactification. Our main result is the stabilization of this formula. As an application, we compute the Hasse--Weil zeta function of the intersection cohomology in some special cases, using the recent work of Arthur and Ta\"ibi on the endoscopic classification of automorphic representations of special orthogonal groups.