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

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.