On the Langlands--Kottwitz--Scholze method
Abstract
Using recent advances in the integral canonical models of Shimura varieties using syntomic methods, we give an extension of the Langlands--Kottwitz--Scholze method from Scholze's paper on the Langlands--Kottwitz method for deformation spaces of p-divisible groups. Namely, we define analogues ϕ𝒢,μτ,h of the local test functions from op. cit. in full generality, show they satisfy reasonable harmonic-analytic properties, and give rise to trace formulae for the Galois-Hecke action on the cohomology of Shimura varieties of bad reduction. Additionally, we use these new local test functions and the work of Fargues--Scholze to state a more robust and unconditional version of the Scholze--Shin conjecture. Along the way we study the (𝒢,μ)-apertures of Drinfeld and Gardner--Madapusi, in particular showing that such (𝒢,μ)-apertures carry no more information than a Tannakian version of Fontaine--Laffaille theory, at least over nice bases, and give a very explicit description of the universal deformations of (𝒢,μ)-apertures.
Related
- Local Langlands functoriality for Yu's supercuspidals II: Fargues--Scholze's parametrization
- A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence
- Geometrization of Kottwitz-Shelstad transfer and Vogan's conjecture on local Arthur packets
- The Categorical Fargues-Scholze Correspondence at Generic Parameters
- Gross--Prasad conjecture in geometric Langlands
BibTeX
@misc{youcis2026langlandskottwitzscholze,
title = {{On the Langlands--Kottwitz--Scholze method}},
author = {Alex Youcis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30537},
eprint = {2609.30537},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30537},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On the Langlands--Kottwitz--Scholze method
AU - Alex Youcis
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30537
UR - https://arxiv.org/abs/2609.30537
AB - Using recent advances in the integral canonical models of Shimura varieties using syntomic methods, we give an extension of the Langlands--Kottwitz--Scholze method from Scholze's paper on the Langlands--Kottwitz method for deformation spaces of p-divisible groups. Namely, we define analogues ϕ^𝒢,μ_τ,h of the local test functions from op. cit. in full generality, show they satisfy reasonable harmonic-analytic properties, and give rise to trace formulae for the Galois-Hecke action on the cohomology of Shimura varieties of bad reduction. Additionally, we use these new local test functions and the work of Fargues--Scholze to state a more robust and unconditional version of the Scholze--Shin conjecture. Along the way we study the (𝒢,μ)-apertures of Drinfeld and Gardner--Madapusi, in particular showing that such (𝒢,μ)-apertures carry no more information than a Tannakian version of Fontaine--Laffaille theory, at least over nice bases, and give a very explicit description of the universal deformations of (𝒢,μ)-apertures.
ER -