Etale 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 $\ell$-adic Weil cohomology: these are lisse Weil sheaves in étale cohomology when $\ell \ne p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell =p$. 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.

30 pages; v6: published version

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Cited by in corpus (3)

• Maximal tori of monodromy groups of $F$-isocrystals and an application to abelian varieties
• Geometric Langlands for hypergeometric sheaves
• Finiteness of logarithmic crystalline representations II