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arXiv · 2609.18665

Stable Trace Formula for Newton strata of Shimura varieties

Dhruva Kelkar
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Abstract

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 $σ$-conjugacy classes $b\in B(G_{\mathbb{Q}_p},μ_h^{-1})$. For an individual Newton stratum, this paper constructs a stabilized formula for the alternating traces of Frobenius--Hecke correspondences on its compactly supported $\ell$-adic cohomology, with coefficients in an $\ell$-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.

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Dhruva Kelkar. 2026-09-16. Stable Trace Formula for Newton strata of Shimura varieties. https://arxiv.org/abs/2609.18665

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