Given a 2-dimensional dihedral representation of a profinite group over a finite field, we will give necessary and sufficient conditions for its universal deformation to be dihedral.

We will then specialize to the case of absolute Galois group of a number field and give sufficient conditions for the universal deformation unramified outside a finite set of primes to be dihedral. We will also see its applications to unramfied Fontaine-Mazur conjecture and to R=T theorem (in the spirit of Calegari-Geraghty) in the setting of Hilbert modular forms of parallel weight one. This talk is based on joint work with Gabor Wiese.

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