Etale and crystalline companions, II
Kiran S. Kedlaya
TL;DR
Etale and crystalline companions, II establishes that every algebraic (or irreducible with finite determinant) $\ell$-adic coefficient on a smooth scheme $X$ over a finite field admits a crystalline companion in overconvergent $F$-isocrystals with matching Frobenius polynomials at all closed points. The paper merges Langlands correspondence for $\mathrm{GL}_n$ over function fields with Drinfeld–Abe–Kedlaya companion techniques, and develops a moduli-stack framework for truncated coefficient objects to globalize from curves to higher dimensions. It provides a robust transfer of properties between crystalline and étale theories, including Newton polygon stratifications and Wan’s $p$-adic meromorphy results for unit-root $L$-functions, and yields new tools such as skeleton-sheaves and divisor-based gluing to construct global companions. These results deepen our understanding of motivic-like behavior of coefficient objects and offer new avenues for relating $\ell$-adic and $p$-adic cohomological data across dimensions. The methods have potential implications for motives, independence of $\ell$, and the study of monodromy groups in a broader reductive setting.
Abstract
Let $X$ be a smooth scheme over a finite field of characteristic $p$. In answer to a conjecture of Deligne, we establish that for any prime $\ell \neq p$, an $\ell$-adic Weil sheaf on $X$ which is algebraic (or irreducible with finite determinant) admits a crystalline companion in the category of overconvergent $F$-isocrystals, for which the Frobenius characteristic polynomials agree at all closed points (with respect to some fixed identification of the algebraic closures of $\mathbb{Q}$ within fixed algebraic closures of $\mathbb{Q}_\ell$ and $\mathbb{Q}_p$). The argument depends heavily on the free passage between $\ell$-adic and $p$-adic coefficients for curves provided by the Langlands correspondence for $\mathrm{GL}_n$ over global function fields (work of L. Lafforgue and T. Abe), and on the construction of Drinfeld (plus adaptations by Abe-Esnault and Kedlaya) giving rise to étale companions of overconvergent $F$-isocrystals. As corollaries, we transfer a number of statements from crystalline to étale coefficient objects, including properties of the Newton polygon stratification (results of Grothendieck-Katz and de Jong-Oort-Yang) and Wan's theorem (previously Dwork's conjecture) on $p$-adic meromorphicity of unit-root $L$-functions.