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

DOI10.1007/S11856-022-2317-3zbMATH Open1509.11034arXiv2010.14574MaRDI QIDQ6352391

Gal Porat

Publication date: 27 October 2020

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 $mathrmGalleft(overlineK/Kight)$ 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 $left(varphi,Gammaight)$-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 $mathrmGalleft(overline{F}_v/F_{v}ight)$ 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 $F=mathbbQ$ obtained previously by Chenevier, Colmez and Kisin.

Mathematics Subject Classification ID

Galois theory (11S20) Congruences for modular and (p)-adic modular forms (11F33) Galois representations (11F80) Automorphic forms on (mbox{GL}(2)); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces (11F41)

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