The main result of this article states that the Galois representation attached to a Hilbert modular eigenform over F_p^bar of parallel weight one 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 one. The proof is based on the observation that parallel weight one forms in characteristic p embed into the ordinary part of parallel weight p in two different ways per place above p, namely via `partial' Frobenius operators. These are defined in the article along with and based on Hecke operators T_P for P dividing p. The theorem is deduced from known local properties of the Galois representation attached to ordinary eigenforms in characteristic 0.
This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes algorithms and a database of modular forms orbits and higher congruences.
Last modification: 6 January 2017.