Etale and crystalline companions, I

TL;DR

This paper demonstrates that crystalline coefficient objects over smooth schemes in characteristic p have étale companions, extending the Langlands correspondence framework and building on Drinfeld's and Abe–Esnault's work.

Contribution

It adapts Drinfeld's method to prove the existence of étale companions for crystalline coefficient objects, expanding the Langlands correspondence to a broader class of schemes.

Findings

Crystalline coefficient objects have étale companions.

Extension of Langlands correspondence to higher-dimensional schemes.

Auxiliary results for constructing crystalline companions.

Abstract

Let $X$ be a smooth scheme over a finite field of characteristic $p$. Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adic Weil cohomology: these are lisse Weil sheaves in \'etale cohomology when $\ell \neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell=p$. Using the Langlands correspondence for global function fields in both the \'etale and crystalline settings (work of Lafforgue and Abe, respectively), one sees that on a curve, any coefficient object in one category has "companions" in the other categories with matching characteristic polynomials of Frobenius at closed points. A similar statement is expected for general $X$; building on work of Deligne, Drinfeld showed that any \'etale coefficient object has \'etale companions. We adapt Drinfeld's method to show that any crystalline coefficient object has \'etale companions; this has been shown independently by Abe--Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of \'etale coefficient objects; this subject will be pursued in a subsequent paper.