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 \'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.

Drinfeld's lemma for F-isocrystals, II: Tannakian approach

“…The map (4.2.3.1) has a canonical lift to a -equivariant map (cf. [Dri18, §§ 7.2.4, 7.3.5]) By the companion theorem [Abe18b, AE19, Ked22b, Ked23b], the above map factors as [Dri18, Corollary 7.4.2] For any non-Archimedean place of , we have the following diagram of sets. Via the isomorphism , we obtain the following diagram of sets.…”

“…we do not consider partial Frobenius structures), Drinfeld proved the above result for higher-dimensional smooth geometrically connected -varieties if do not divide and the result in the curve case when or divides (see [Dri18, Theorem 1.4.1, 5.2.1]). Now we can obtain the full generality of the theorem in the case using the recent breakthrough in the companion theorem for -adic coefficients [AE19, Ked22b, Ked23b] over smooth -varieties and the same argument of [Dri18].…”