PreprintCC BY 4.0arXiv
A geometric Jacquet--Langlands correspondence for Shimura varieties
Abstract
We formulate a conjecture predicting the existence of exotic isomorphisms between Igusa stacks associated to different Shimura data whose underlying groups are pure inner forms of each other, and prove it in many cases of interest. In combination with the spectral action of Fargues--Scholze, this allows us to relate the cohomology groups of the two Shimura varieties, giving a geometric incarnation of the Jacquet--Langlands correspondence. An important ingredient in our proofs is the work of Xiao--Zhu on exotic Hecke correspondences between the perfect special fibers of different Shimura varieties, which we reinterpret in terms of exotic isomorphisms of perfect Igusa stacks.
Related
- Relative Hitchin fibrations and endoscopy for Unitary Friedberg--Jacquet varieties
- 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
- Oort's conjecture for split unitary Shimura varieties
BibTeX
@misc{hoften2026geometric,
title = {{A geometric Jacquet--Langlands correspondence for Shimura varieties}},
author = {Pol van Hoften and Jack Sempliner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08549},
eprint = {2609.08549},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08549},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A geometric Jacquet--Langlands correspondence for Shimura varieties
AU - Pol van Hoften
AU - Jack Sempliner
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08549
UR - https://arxiv.org/abs/2609.08549
AB - We formulate a conjecture predicting the existence of exotic isomorphisms between Igusa stacks associated to different Shimura data whose underlying groups are pure inner forms of each other, and prove it in many cases of interest. In combination with the spectral action of Fargues--Scholze, this allows us to relate the cohomology groups of the two Shimura varieties, giving a geometric incarnation of the Jacquet--Langlands correspondence. An important ingredient in our proofs is the work of Xiao--Zhu on exotic Hecke correspondences between the perfect special fibers of different Shimura varieties, which we reinterpret in terms of exotic isomorphisms of perfect Igusa stacks.
ER -