Reference : Unramifiedness of Galois representations attached to weight one Hilbert modular eigen...
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Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p
Dimitrov, Mladen []
Wiese, Gabor mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
[en] The main result of this article states that the Galois representation attached to a Hilbert modular eigenform over Fp 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 Tp for p dividing p. The theorem is deduced from known local properties of the Galois representation attached to ordinary eigenforms in characteristic 0.

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