Lubin–Tate theory and overconvergent Hilbert modular forms of low weight

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Let K be a finite extension of ℚp and let Γ be the Galois group of the cyclotomic extension of K. Fontaine’s theory gives a classification of p-adic representations of Ga( $$\overline K /K$$ ) in terms of (φ, Γ)-modules. A useful aspect of this classification is Berger’s dictionary which expresses invariants coming from p-adic Hodge theory in terms of these (φ, Γ)-modules. In this article, we use the theory of locally analytic vectors to generalize this dictionary to the setting where Γ is the Galois group of a Lubin—Tate extension of K. As an application, we show that if F is a totally real number field and v is a place of F lying above p, then the p-adic representation of Gal( $${\overline F _v}/{F_v}$$ ) associated to a finite slope overconvergent Hilbert eigenform which is Fv-analytic up to a twist is Lubin—Tate trianguline. Furthermore, we determine a triangulation in terms of a Hecke eigenvalue at v. This generalizes results in the case F = ℚ obtained previously by Chenevier, Colmez and Kisin.