Dihedral universal deformations (Q2660593)

A continuous absolutely irreducible representation \(\overline{\rho}: G \rightarrow \text{GL}_2(\mathbb{F})\) is called dihedral if it is induced from a character, where \(G\) is a profinite group and \(\mathbb{F}\) is a finite field of characteristic \(p\). Consider a deformation \(\rho: G \rightarrow \text{GL}_2(R)\) of \(\rho\) for any complete local Noetherian algebra \(R\) over \(W(\mathbb{F})\), the ring of Witt vectors of \(\mathbb{F}\), with residue field \(\mathbb{F}\). In the present paper, the authors give a characterisation of dihedral representations by Frattini quotients and show that the universal deformation of \(\overline{\rho}\) is dihedral if and only if all infinitesimal deformations are dihedral. They give new evidence towards Boston's strengthening of the unramified Fontaine-Mazur conjecture. In particular, they present some examples of irreducible dihedral representations \(\overline{\rho}: G \rightarrow \text{GL}_2(\mathbb{F})\) and determine whether their universal deformation relatively unramified outside a finite set \(S\) is dihedral or not.