The prime-to-adjoint principle and unobstructed Galois deformations in theborel case

For a given odd two-dimensional representation rho over F-p of the absolute Galois group G(E) Of a totally real field E which is unramified outside a finite set of places S, Mazur defined a universal deformation ring R-GS<(rh o)over bar>) By obstruction theory, the group IIIS2(E, ad <(rho)over bar> m easures to what extent R-GS(<(rho)over bar>) is determined by local relatio ns. Using devissage on ad <(rho)over bar> we give criteria for the vanishin g of IIIS2(E, ad <(rho)over bar>) in terms of vanishing of S-class groups, in terms of Iwasawa invariants, and in terms of special values of rho-adic L-functions. If S is the set of places above rho and infinity, the conditio n IIIS2 (E, ad <(rho)over bar>) = 0 implies that R-GS (<(rho)over bar>) is free of dimension 2[E: Q] + 1. In this case, we obtain a reformulation of V andiver's conjecture and asymptotic connections between Greenberg's conject ure and the Freeness of R-GS(<(rho)over bar>). For larger S, we relate the freeness of the universal deformation ring for minimal deformations to the vanishing of a modified obstruction group IIIS2.(S rho)(E, ad <(rho)over ba r>). Based on this, we a can calculate non-fret rings R-GS(<(rho)over bar>) for some explicit reducible <(rho)over bar> coming from the action of G(Q) on rho-torsion points of elliptic curves. (C) 1999 academic Press.