Vol. 4, No. 7, 2010

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

Kiran Kedlaya and Ruochuan Liu

Vol. 4 (2010), No. 7, 943–967
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