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


Author: Tomoyuki Abe
Journal: J. Amer. Math. Soc. 31 (2018), 921-1057
MSC (2010): Primary 14F30, 11R39; Secondary 11S37
DOI: https://doi.org/10.1090/jams/898
Published electronically: May 22, 2018
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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 $ \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.


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Additional Information

Tomoyuki Abe
Affiliation: Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583, Japan
Email: tomoyuki.abe@ipmu.jp

DOI: https://doi.org/10.1090/jams/898
Received by editor(s): January 27, 2015
Received by editor(s) in revised form: April 25, 2016, August 31, 2016, and November 14, 2017
Published electronically: May 22, 2018
Additional Notes: This work is supported by Grant-in-Aid for Research Activity Start-up 23840006, Grant-in-Aid for Young Scientists (B) 25800004, and Grant-in-Aid for Young Scientists (A) 16H05993.
Article copyright: © Copyright 2018 American Mathematical Society

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