Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p

We show here that in order to strip powers of p from the level of a Galois representation which is strongly modular, it is necessary in general to consider the Galois

The main objective, however, is to extend the representation ρ g to a 2-dimensional Galois representation with coefficients in the weight 1 Hecke algebra and to show, in most

For example, if G is any of the three groups A 4 , S 4 , A 5 , by using results on the existence or non-existence of generic or parametric polynomials/extensions with

We will consider from now on rational `-adic representations, so we will be able to compare different rational representations (even over different completions) just by comparing

As in the previous proof, for p > 2 by [Edixhoven 1992, Theorem 4.5], together with the remark at the end of the introduction to that article, the existence of a weight 1 form

Finally, in Chapter 4, we will prove that a partial weight one Hilbert modular form, with parallel weight one for the places above a prime p, has associated modulo $

Nevertheless, we show that when the local structure of ρ n “mimics” the (mod p n reduction of) the local structure of the p-adic Galois representation attached to a p-ordinary

It allows one to see such a Galois representation that is modular of level N, weight 2 and trivial Nebentypus as one that is modular of level N p, for a prime p coprime to N , when