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

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