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

Stefan Patrikis

TL;DR

This work upgrades the canonical $G(Q_\\ell)$-local systems on Shimura varieties from adjoint-compatibility to full $G(Q_\\ell)$-compatibility, including crystalline compatibility. The strategy blends Drinfeld’s companion theory for reductive monodromy with recent integral-model results and prior adjoint-case analyses, enabling a robust $oldsymbol{ u}$-independence across all places. Under mild hypotheses on the center and real ranks of adjoint factors, there exists an integer $N$ and an integral model such that all $ ho_oldsymbol{ u}$ factor through the arithmetic fundamental group and their Frobenius conjugacy classes in $[G\sslash G](Q)$ are independent of $oldsymbol{ u}$, with a $G$-valued overconvergent $F$-isocrystal correspondence at primes dividing $p$. This represents a significant advance toward the Langlands–Rapoport program for non-abelian-type Shimura varieties by achieving strong compatibility and crystalline correspondence for full $G$-valued local systems.

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.

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

TL;DR

This work upgrades the canonical -local systems on Shimura varieties from adjoint-compatibility to full -compatibility, including crystalline compatibility. The strategy blends Drinfeld’s companion theory for reductive monodromy with recent integral-model results and prior adjoint-case analyses, enabling a robust -independence across all places. Under mild hypotheses on the center and real ranks of adjoint factors, there exists an integer and an integral model such that all factor through the arithmetic fundamental group and their Frobenius conjugacy classes in are independent of , with a -valued overconvergent -isocrystal correspondence at primes dividing . This represents a significant advance toward the Langlands–Rapoport program for non-abelian-type Shimura varieties by achieving strong compatibility and crystalline correspondence for full -valued local systems.

Abstract

Let be a Shimura datum. In previous work with Christian Klevdal, we showed that the canonical -valued local systems on Shimura varieties for form compatible systems after projection to the adjoint group of . In this note, we strengthen this result to prove compatibility for the -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.

Paper Structure

This paper contains 4 sections, 17 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Recollection on companions
  3. Ramification
  4. Shimura Varieties

Key Result

Theorem 1.1

Let $(G, X)$ be a Shimura datum such that $Z_G(\mathbb{Q})$ is a discrete subgroup of $Z_G(\mathbb{A}_f)$, let $K_0 \subset G(\mathbb{A}_f)$ be a neat compact open subgroup, and let $s \in \mathrm{Sh}(\mathbb{C})$, $S_{K_0, s}$ be as above. Assume that for all $\mathbb{Q}$-simple factors $H$ of $G^{

Theorems & Definitions (30)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • ...and 20 more