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

TL;DR

This paper extends the Berger dictionary to Lubin-Tate extensions using locally analytic vectors, and applies it to classify certain overconvergent Hilbert modular forms as Lubin-Tate trianguline, generalizing prior results.

Contribution

It generalizes Berger's dictionary to Lubin-Tate Galois groups and characterizes overconvergent Hilbert eigenforms as Lubin-Tate trianguline, with explicit triangulation descriptions.

Findings

Classification of $p$-adic Galois representations as Lubin-Tate trianguline.

Explicit triangulation in terms of Hecke eigenvalues.

Generalization of previous results from $Q$ to totally real fields.

Abstract

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\Gamma$ be the Galois group of the cyclotomic extension of $K$. Fontaine's theory gives a classification of $p$-adic representations of $\mathrm{Gal}\left(\overline{K}/K\right)$ in terms of $(\varphi,\Gamma)$-modules. A useful aspect of this classification is Berger's dictionary which expresses invariants coming from $p$-adic Hodge theory in terms of these $\left(\varphi,\Gamma\right)$-modules. In this paper, we use the theory of locally analytic vectors to generalize this dictionary to the setting where $\Gamma$ 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 $\mathrm{Gal}\left(\overline{F}_{v}/F_{v}\right)$ associated to a finite slope overconvergent Hilbert eigenform which is $F_{v}$-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 $F=\mathbb{Q}$ obtained previously by Chenevier, Colmez and Kisin.