On local Galois representations associated to ordinary Hilbert modular forms

On the eigenvariety of Hilbert modular forms at classical parallel weight one points with dihedral projective image

“…Suppose dim t D > n − 1. Then, from Proposition 2, it follows that dim t D = n. From the equation (2) in the proof of Proposition 2 above, we see that, after fixing a basis (e 1,i 0 , e 2,i 0 ), dim t D is equal to the dimension of the subspace of (Q p ) n consisting of n-tuples (b 1 , ..., b n ) such that if we substitute them in the RHS of (2), we get a matrix with entries in H 1 (H, Q p ). Hence, the equality dim t D = n implies that the matrix…”

“…Note that these operators coincide with the classical U (p i )'s on parallel weight Hilbert modular forms. See remark 4.7 of [1] and [2,Section 3] for more details. Let T be Res O F /Z G m and M be a finite extension of Q p which splits F .…”

Classical weight one forms in Hida families: Hilbert modular case

“…Further the number of such forms inside a non-CM family is bound by an explicit constant due to Dimitrov and Ghate ([5]). The former result in the specialization was generalized to the case of totally real fields by Balasubramanyam, Ghate and Vatsal ( [1]). Motivated by these works, in this paper, we pursue a generalization of the results of Dimitrov and Ghate.…”

“…This paper is organized as follows: In Section 2, we recall basics of ordinary Hida families of Hilbert cusp forms and Galois representations attached to them. In the first two subsections of Section 3, we discuss specializations (including in weight one) of CM and non-CM families and then state the finiteness result for non-CM families in [1]. The rest of the section is a summary of projective images of the residual representations of Hida families and of Artin representations attached to classical weight one forms.…”

“…They had to impose the assumptions (A1) and (A2) because they used modularity lifting theorem (Theorem 3; p. 999 of [13]) to observe that (C1) implies any weight one specializations of F is classical. More precisely, (A1) is a condition that is assumed in the modularity lifting theorem (see the remark in p. 518 of [1]), and (A2) is used to verify that they can apply modularity lifting theorem to the Galois representation attached to each weight one specialization of F . We do not need modularity lifting theorem to establish the equivalence in Theorem 3.3, and hence we assume neither (A1) nor (A2).…”