The Prime-to-Adjoint Principle and Unobstructed Galois Deformations in the Borel Case Original Research Article

For a given odd two-dimensional representation ρ over Fp of the absolute Galois group GE of a totally real field E which is unramified outside a finite set of places S, Mazur defined a universal deformation ring RGS(ρ). By obstruction theory, the group -- EQUATION OMITTED -- measures to what extent RGS(ρ) is determined by local relations. Using devissage on ad ρ, we give criteria for the vanishing of -- EQUATION OMITTED -- in terms of vanishing of S-class groups, in terms of Iwasawa invariants, and in terms of special values of p-adic L-functions. If S is the set of places above p and ∞, the condition -- EQUATION OMITTED -- implies that RGS(ρ) is free of dimension 2[E : Q]+1. In this case, we obtain a reformulation of Vandiverʹs conjecture and asymptotic connections between Greenbergʹs conjecture and the freeness of RGS(ρ). For larger S, we relate the freeness of the universal deformation ring for minimal deformations to the vanishing of a modified obstruction group -- EQUATION OMITTED -- . Based on this, we can calculate non-free rings RGS(ρ) for some explicit reducible ρ coming from the action of GQ on p-torsion points of elliptic curves.