Stable Trace Formula for Newton strata of Shimura varieties
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∈ B(Gℚp,μh−1). For an individual Newton stratum, this paper constructs a stabilized formula for the alternating traces of Frobenius--Hecke correspondences on its compactly supported ℓ-adic cohomology, with coefficients in an ℓ-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.
Related
- Ekedahl-Oort strata with a superspecial factor in unitary Shimura varieties
- On the arithmetic Siegel--Weil formula for orthogonal Shimura varieties over a general totally real field
- Igusa stacks and the cohomology of Shimura varieties
- S = T S=T upper S equals upper T for Shimura varieties and moduli spaces of p -adic shtukas
- Heights of small CM cycles on Unitary Shimura Varieties
BibTeX
@misc{kelkar2026stable,
title = {{Stable Trace Formula for Newton strata of Shimura varieties}},
author = {Dhruva Kelkar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18665},
eprint = {2609.18665},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18665},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stable Trace Formula for Newton strata of Shimura varieties
AU - Dhruva Kelkar
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18665
UR - https://arxiv.org/abs/2609.18665
AB - 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∈ B(G_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 ℓ-adic cohomology, with coefficients in an ℓ-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.
ER -