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

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.

$p$-adic Galois representations, $(\varphi,\Gamma)$-modules
Mathematical Subject Classification 2000
Primary: 11F80
Secondary: 11S20
Received: 10 December 2009
Accepted: 10 January 2010
Published: 29 January 2011
Kiran Kedlaya
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139
United States
Ruochuan Liu
Department of Mathematics
University of Michigan
1844 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States