Langlands correspondence for isocrystals and the existence of crystalline companions for curves
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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.References
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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
- 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.
- © Copyright 2018 American Mathematical Society
- Journal: J. Amer. Math. Soc. 31 (2018), 921-1057
- MSC (2010): Primary 14F30, 11R39; Secondary 11S37
- DOI: https://doi.org/10.1090/jams/898
- MathSciNet review: 3836562