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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk


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First steps towards p-adic Langlands functoriality

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2007, Issue 610, Pages 149–180, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2007.070, December 2007

Publication History

Received:
2006-04-24
Revised:
2006-06-09
Published Online:
2007-12-07

Abstract

By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the corresponding locally algebraic representation carries an invariant norm. In the crystalline case, the Weil-Deligne group representation is unramified and the associated locally algebraic representation can be studied using the classical Satake isomorphism. By extending the latter to a specific norm completion of the Hecke algebra, we show that the existence of an invariant norm implies that our pair, indeed, comes from a crystalline representation. We also show, by using the formalism of Tannakian categories, that this latter fact is compatible with classical unramified Langlands functoriality and therefore generalizes to arbitrary split reductive groups.

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[2]
Claus Sorensen
Pacific Journal of Mathematics, 2015, Volume 275, Number 1, Page 191
[3]
Pierre Colmez and Gabriel Dospinescu
Algebra & Number Theory, 2014, Volume 8, Number 6, Page 1447
[4]
Claus Sorensen
Annals of Mathematics, 2013, Volume 177, Number 1, Page 367
[5]
Eknath Ghate and Narasimha Kumar
Pacific Journal of Mathematics, 2011, Volume 252, Number 2, Page 379

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