Multivariable (phi,Gamma)-modules and locally analytic vectors

Laurent Berger (1, 2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 IUF - Institut universitaire de France

Let K_\infty/K be a Galois extension such that K_\infty contains the cyclotomic extension and such that Gamma=Gal(K_\infty/K) is a p-adic Lie group. We construct (phi,Gamma)-modules over the ring of locally analytic vectors (for the action of Gamma) of some of Fontaine's rings. When K_\infty is the cyclotomic extension, these locally analytic vectors are closely related to the Robba ring, and we recover the classical theory. We determine some of these locally analytic vectors in the Lubin-Tate setting. This allows us to prove that a certain monodromy conjecture implies that the Lubin-Tate (phi,Gamma)-modules of F-analytic representations are overconvergent.

Mots clés: $(\phi; \Gamma)$-module; locally analytic vector; $p$-adic period; Lubin-Tate group; $p$-adic monodromy

Domaines: Théorie des nombres [math.NT]; Théorie des représentations [math.RT]

Laurent Berger. Multivariable (phi,Gamma)-modules and locally analytic vectors. Duke Mathematical Journal, 2016, 165 (18), pp.3567-3595. ⟨10.1215/00127094-3674441⟩. ⟨hal-00920290⟩