2018/11/01 by Kiran S. Kedlaya, Kedlaya, Kiran S. · 1 citation
Mathematics · #14F20 #14F30 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1811.00204
openalex publication_date 2018/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a smooth scheme over a finite field of characteristic p. Consider the coefficient objects of locally constant rank on X in ℓ-adic Weil cohomology: these are lisse Weil sheaves in étale cohomology when ℓ ≠ p, and overconvergent F-isocrystals in rigid cohomology when ℓ=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.