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

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.