Étale and crystalline companions, I

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 `-adic Weil cohomology: these are lisse Weil sheaves in étale cohomology when ` , p, and overconvergent F-isocrystals in rigid cohomology when ` = p. Using the Langlands correspondence for global function fields in both the étale 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 étale coefficient object has étale companions. We adapt Drinfeld’s method to show that any crystalline coefficient object has étale companions; this has been shown independently by Abe–Esnault. We also prove some auxiliary results relevant for the construction of crystalline companions of étale coefficient objects; this subject will be pursued in a subsequent paper.