Unramifiedness of weight 1 Hilbert Hecke algebras (Q6610063)

In their paper [Invent. Math. 211, No. 1, 297--433 (2018; Zbl 1476.11078)], \textit{F. Calegari} and \textit{D. Geraghty} showed that the Galois representations which are unramified at an odd prime \(p\) correspond to modular forms of weight one. More precisely, given an odd irreducible representation \(\overline{\rho}: G_{\mathbb{Q}} \rightarrow \mathrm{GL}(2, \overline{\mathbb{F}}_p)\) unramified outside a finite set of places \(S\) not containing \(p\), they show that \N\[\NR_{\mathbb{Q}, \overline{\rho}}^{S} \stackrel{\sim}{\longrightarrow} \mathbb{T}_{\rho}^{(1)},\N\]\Nwhere \(R_{\mathbb{Q}, \overline{\rho}}^{S}\) is the universal deformation ring parametrising deformations of \(\overline{\rho}\) which are unramified outside \(S\) and \(\mathbb{T}_{\rho}^{(1)}\) is the local component at \(\overline{\rho}\) of a weight one Hecke algebra of a certain level prime to \(p\).\N\NIn the present paper, the authors prove an analogous result for parallel weight one Hilbert modular forms over a totally real field \(F\) of degree \(d=[F: \mathbb{Q}] \geq 2\) and ring of integers \(\mathfrak{o}\). They give the construction of the Galois pseudo-representation with values in the parallel weight one Hecke algebra with \(p\)-power torsion coefficients and prove its local ramification properties. In particular, given a finite set \(S\) of places in \(F\) relatively prime to \(p\) and a totally odd irreducible representation \(\overline{\rho}: G_F \rightarrow \mathrm{GL}(2, \overline{\mathbb{F}}_p)\) unramified outside \(S\) they show that there exists a surjective homomorphism \N\[\NR_{F, \overline{\rho}}^{S} \twoheadrightarrow \mathbb{T}_{\rho}^{(1)},\N\]\Nwhere \(R_{F, \overline{\rho}}^{S}\) is the universal deformation ring parametrising deformations of \(\overline{\rho}\) which are unramified outside \(S\) and \(\mathbb{T}_{\rho}^{(1)}\) is the local component at \(\overline{\rho}\) of a weight one Hecke algebra of a certain level prime to \(p\).