The prime-to-adjoint principle and unobstructed Galois deformations in the Borel case, by Gebhard Boeckle, Ariane Mezard

Abstract: For a given odd two-dimensional representation $\bar\rho$ 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. By obstruction theory, the group $\Sha_S^2(E,ad\bar\rho)$ measures to what extend R is determined by local relations. Using devissage on $ad\bar\rho$, we give criteria for the vanishing of $\Sha_S^2(E,ad\bar\rho)$ 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 infinity, the condition $\Sha_S^2(E,ad\bar\rho)=0$ implies that R is free of dimension 2[E:Q]+1. In this case, we obtain a reformulation of Vandiver's conjecture and asymptotic connections between Greenberg's conjecture and the freeness of R. For larger S, we relate the freeness of the universal deformation ring for minimal deformations to the vanishing of a modified obstruction group $\Sha_{S,S_p}^2(E,ad\bar\rho)$. Based on this, we can calculate non-free rings R for some explicit reducible $\bar\rho$ coming from the action of $G_Q$ on p-torsion points of elliptic curves.

Gebhard Boeckle, Ariane Mezard <,>