Langlands correspondence for isocrystals and the existence of crystalline companions for curves

Author:

Abe Tomoyuki

Abstract

In this paper, we show the Langlands correspondence for isocrystals on curves, which asserts the existence of crystalline companions in the case of curves. For the proof we generalize the theory of arithmetic D \mathscr {D} -modules to algebraic stacks whose diagonal morphisms are finite. Finally, combining with methods of Deligne and Drinfeld, we show the existence of an “ \ell -adic companion” for any isocrystal on a smooth scheme of any dimension under the assumption of a Bertini-type conjecture.

Funder

Japan Society for the Promotion of Science

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference73 articles.

1. Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic 𝒟-modules;Abe, Tomoyuki;Rend. Semin. Mat. Univ. Padova,2014

2. Langlands program for 𝑝-adic coefficients and the petits camarades conjecture;Abe, Tomoyuki;J. Reine Angew. Math.,2018

3. [AC1] T. Abe and D. Caro, Theory of weights in 𝑝-adic cohomology, Amer. J. Math. (to appear), available at \url{http://arxiv.org/abs/1303.0662}.

4. [AC2] T. Abe and D. Caro, On Beilinson’s equivalence for 𝑝-adic cohomology, Selecta Math. (to appear), available at \url{http://arxiv.org/abs/1309.4517}.

5. [AE] T. Abe and H. Esnault, A Lefschetz theorem for overconvergent isocrystals with Frobenius structure, Ann. ENS. (to appear), available at \url{http://arxiv.org/abs/1607.07112}.

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