Compatibility of canonical $\ell$-adic local systems on Shimura varieties, II

Stefan Patrikis

Compatibility of canonical $\ell$-adic local systems on Shimura varieties, II

Number Theory 2025-04-29 v2 Algebraic Geometry

Authors: Stefan Patrikis

Abstract

Let $(G, X)$ be a Shimura datum. In previous work with Christian Klevdal, we showed that the canonical $G(\mathbb{Q}_{\ell})$ -valued local systems on Shimura varieties for $G$ form compatible systems after projection to the adjoint group of $G$ . In this note, we strengthen this result to prove compatibility for the $G(\mathbb{Q}_{\ell})$ -local systems themselves. We also include the crystalline compatibility, extending the adjoint case established in our joint work Jake Huryn, Kiran Kedlaya, and Christian Klevdal.

Keywords

Shimura varieties $\ell$-adic local systems compatibility crystalline compatibility adjoint group

Cite

@article{arxiv.2504.00305,
  title  = {Compatibility of canonical $\ell$-adic local systems on Shimura varieties, II},
  author = {Stefan Patrikis},
  journal= {arXiv preprint arXiv:2504.00305},
  year   = {2025}
}

Comments

Comments welcome. This update: (1) fixes some cross-referencing with our joint work with Huryn, Kedlaya, and Klevdal; (2) gives more details on companions with reductive monodromy group; (3) changes the title to avoid confusion with our previous work with Klevdal

Compatibility of canonical $\ell$-adic local systems on Shimura varieties, II

Abstract

Keywords

Cite

Comments

Related papers