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1 December 2016 Multivariable (φ,Γ)-modules and locally analytic vectors
Laurent Berger
Duke Math. J. 165(18): 3567-3595 (1 December 2016). DOI: 10.1215/00127094-3674441


Let K be a finite extension of Qp, and let GK=Gal(Q¯p/K). There is a very useful classification of p-adic representations of GK in terms of cyclotomic (φ,Γ)-modules (cyclotomic means that Γ=Gal(K/K) where K is the cyclotomic extension of K). One particularly convenient feature of the cyclotomic theory is the fact that the (φ,Γ)-module attached to any p-adic representation is overconvergent.

Questions pertaining to the p-adic local Langlands correspondence lead us to ask for a generalization of the theory of (φ,Γ)-modules, with the cyclotomic extension replaced by an infinitely ramified p-adic Lie extension K/K. It is not clear what shape such a generalization should have in general. Even in the case where we have such a generalization, namely, the case of a Lubin–Tate extension, most (φ,Γ)-modules fail to be overconvergent.

In this article, we develop an approach that gives a solution to both problems at the same time, by considering the locally analytic vectors for the action of Γ inside some big modules defined using Fontaine’s rings of periods. We show that, in the cyclotomic case, we recover the usual overconvergent (φ,Γ)-modules. In the Lubin–Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that (φ,Γ)-modules attached to F-analytic representations are overconvergent.


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Laurent Berger. "Multivariable (φ,Γ)-modules and locally analytic vectors." Duke Math. J. 165 (18) 3567 - 3595, 1 December 2016.


Received: 14 January 2015; Revised: 1 March 2016; Published: 1 December 2016
First available in Project Euclid: 12 September 2016

zbMATH: 06677445
MathSciNet: MR3577371
Digital Object Identifier: 10.1215/00127094-3674441

Primary: 11S
Secondary: 11F, 12H, 13J, 22E, 46S

Rights: Copyright © 2016 Duke University Press


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Vol.165 • No. 18 • 1 December 2016
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