Kiran S. Kedlaya - Etale and crystalline companions, I

epiga:6820 - Épijournal de Géométrie Algébrique, December 2, 2022, Volume 6 - https://doi.org/10.46298/epiga.2022.6820
Etale and crystalline companions, IArticle

Authors: Kiran S. Kedlaya ORCID

    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 $\ell \neq p$, 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.


    Volume: Volume 6
    Published on: December 2, 2022
    Accepted on: December 2, 2022
    Submitted on: October 4, 2020
    Keywords: Mathematics - Number Theory,Mathematics - Algebraic Geometry,14F30, 14F20
    Funding:
      Source : OpenAIRE Graph
    • Nonarchimedean Analysis, Geometry, and Computation; Funder: National Science Foundation; Code: 1802161
    • p-Adic Computation of L-Functions at Scale; Funder: National Science Foundation; Code: 2053473
    • Applications and extensions of p-adic Hodge theory; Funder: National Science Foundation; Code: 1501214

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