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UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ATTACHED TO HILBERT MODULAR FORMS MOD $p$ OF WEIGHT 1

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  23 April 2018

Mladen Dimitrov  and
Gabor Wiese
Mladen Dimitrov
Affiliation:
University of Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, 59000 Lille, France (mladen.dimitrov@univ-lille.fr)
Gabor Wiese
Affiliation:
University of Luxembourg, Mathematics Research Unit, 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Luxembourg (gabor.wiese@uni.lu)
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Abstract

The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over $\overline{\mathbb{F}}_{p}$ of parallel weight 1 and level prime to $p$ is unramified above $p$. This includes the important case of eigenforms that do not lift to Hilbert modular forms in characteristic 0 of parallel weight 1. The proof is based on the observation that parallel weight 1 forms in characteristic $p$ embed into the ordinary part of parallel weight $p$ forms in two different ways per prime dividing $p$, namely via ‘partial’ Frobenius operators.

Keywords

Hilbert modular forms modulo pweight oneGalois representations

MSC classification

Primary: 11F80: Galois representations
Secondary: 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F33: Congruences for modular and $p$-adic modular forms
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 2 , March 2020 , pp. 281 - 306
Copyright
© Cambridge University Press 2018

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