Compatibility of $F$-isocrystals on adjoint Shimura varieties

TL;DR

This paper proves the compatibility of canonical $F$-isocrystals and $ ext{l}$-adic local systems on adjoint Shimura varieties in the superrigid case, extending previous results using advanced $p$-adic and superrigidity techniques.

Contribution

It extends prior work by establishing compatibility of $F$-isocrystals and $ ext{l}$-adic systems on Shimura varieties in the superrigid setting, employing new methods involving crystallinity and superrigidity.

Findings

Compatibility of $F$-isocrystals and $ ext{l}$-adic local systems established.

Uses crystallinity results of Esnault--Groechenig.

Employs Margulis superrigidity and crystalline-to-étale correspondence.

Abstract

In this article, we extend past results of the last two authors to include compatibility of canonical $\ell$-adic local systems and canonical $F$-isocrystals on adjoint Shimura varieties in the superrigid regime. Our method relies on the crystallinity of canonical $p$-adic local systems due to Esnault--Groechenig as well as Margulis superrigidity and the crystalline-to-\'etale companion construction of Drinfeld and Kedlaya.