Universal deformations of dihedral representations

Abstract: 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 an R=T theorem (in the spirit
of Calegari-Geraghty) in the setting of Hilbert modular forms of parallel
weight one. We will begin with a brief introduction to the deformation
theory of Galois representations. This talk is based on joint work with
Gabor Wiese.