Stable Trace Formula for Newton strata of Shimura varieties

Let $(G,X)$ be a Shimura datum of abelian type satisfying the hypotheses of the main theorem, and suppose that the associated Shimura varieties have hyperspecial good reduction at $p$. Their special fibres are stratified by the $\sigma$-conjugacy classes $b\in B(G_{{\mathbb{Q}}_p},{\mu }_h^{-1})$. For an individual Newton stratum, this paper constructs a stabilized formula for the alternating traces of Frobenius–Hecke correspondences on its compactly supported $\mathrm{\ell }$-adic cohomology, with coefficients in an $\mathrm{\ell }$-adic local system. The formula, obtained by modifying the Langlands–Kottwitz method, expresses these Lefschetz numbers as elliptic stable geometric distributions on endoscopic groups of $G$, placing the cohomological traces in a form suitable for comparison with automorphic spectral data. We also give a group-theoretic criterion determining which elliptic endoscopic groups can contribute to a given Newton stratum. For certain unitary Shimura varieties, we exhibit an intermediate Newton stratum with no non-trivial endoscopic contribution, although there are non-trivial endoscopic contributions in the cohomology of the ambient Shimura variety.