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

Published online by Cambridge University Press:  23 April 2018

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)

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.

Type
Research Article
Copyright
© Cambridge University Press 2018

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