Compatibility of $F$-isocrystals on adjoint Shimura varieties

TL;DR

This work proves that on adjoint Shimura varieties in the superrigid regime, canonical $\ell$-adic local systems and canonical $F$-isocrystals are compatible in the sense that there exists an overconvergent $G$-$F$-isocrystal on mod-$p$ fibers whose Frobenius traces match the $\ell$-adic Frobenius data for all primes $\ell \neq p$, and for $p$ on the adjoint side. The approach combines Esnault–Groechenig crystallinity, Drinfeld–Kedlaya crystalline-to-étale companions, and Margulis superrigidity, and works through integral and toroidal-compactification techniques to produce a robust p-adic–l-adic compatibility framework. The results extend prior adjoint–local-system compatibilities to the overconvergent setting and connect to Langlands–Rapoport–Kottwitz-type descriptions by providing a precise matching of semisimple Frobenius data across realizations. This represents a significant step toward formulating Langlands-type descriptions (e.g., Kottwitz triples) for non-abelian type Shimura varieties, linking mod-$p$ geometry with global Galois representations via a unified motive-with-G-structure perspective.

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-étale companion construction of Drinfeld and Kedlaya.

Theorems & Definitions (37)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Corollary 1.4
• Remark 1.5
• Example 2.1
• Corollary 2.2
• proof
• Corollary 2.3
• proof