On families of φ,Γ-modules

Kedlaya, Kiran; Liu, Ruochuan

doi:10.2140/ant.2010.4.943

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On families of $\varphi,\Gamma$-modules

Kiran Kedlaya and Ruochuan Liu

Vol. 4 (2010), No. 7, 943–967

DOI: 10.2140/ant.2010.4.943

Abstract

Berger and Colmez (2008) formulated a theory of families of overconvergent étale $(φ, Γ)$ -modules associated to families of $p$ -adic Galois representations over $p$ -adic Banach algebras. In contrast with the classical theory of $(φ, Γ)$ -modules, the functor they obtain is not an equivalence of categories. In this paper, we prove that when the base is an affinoid space, every family of (overconvergent) étale $(φ, Γ)$ -modules can locally be converted into a family of $p$ -adic representations in a unique manner, providing the “local” equivalence. There is a global mod $p$ obstruction related to the moduli of residual representations.

Keywords

$p$-adic Galois representations, $(\varphi,\Gamma)$-modules

Mathematical Subject Classification 2000

Primary: 11F80

Secondary: 11S20

Milestones

Received: 10 December 2009

Accepted: 10 January 2010

Published: 29 January 2011

Authors

Kiran Kedlaya
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 United States
http://www-math.mit.edu/~kedlaya/
Ruochuan Liu
Department of Mathematics University of Michigan 1844 East Hall 530 Church Street Ann Arbor, MI 48109-1043 United States
http://www-personal.umich.edu/~ruochuan

Vol. 4, No. 7, 2010