Duke Math. J. 165 (18), 3567-3595, (1 December 2016) DOI: 10.1215/00127094-3674441
KEYWORDS: $(\varphi,\Gamma)$-module, locally analytic vector, $p$-adic period, Lubin–Tate group, $p$-adic monodromy, 11S, 11F, 12H, 13J, 22E, 46S
Let be a finite extension of , and let . There is a very useful classification of -adic representations of in terms of cyclotomic -modules (cyclotomic means that where is the cyclotomic extension of ). One particularly convenient feature of the cyclotomic theory is the fact that the -module attached to any -adic representation is overconvergent.
Questions pertaining to the -adic local Langlands correspondence lead us to ask for a generalization of the theory of -modules, with the cyclotomic extension replaced by an infinitely ramified -adic Lie extension . 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 -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 inside some big modules defined using Fontaine’s rings of periods. We show that, in the cyclotomic case, we recover the usual overconvergent -modules. In the Lubin–Tate case, we can prove, as an application of our theory, a folklore conjecture in the field stating that -modules attached to -analytic representations are overconvergent.