Deformazioni di Rappresentazioni Galoisiane

The aim of this thesis is to study the basic elements of the theory of deformations of Galois representations, which has become an important number-theoretical tool and was originally developed by Barry Mazur, on whose famous paper most of this work has been based. After giving some preliminary notions on infinite Galois Theory, we will introduce the Galois representations and their deformations as particular equivalence classes of lifts. Then, fixed any absolutely irreducible residual Galois representation satisfying precise assumptions, we will prove the existence of a universal deformation ring and a universal deformation such that, for any coefficient ring R (that is, a complete noetherian local ring) and deformation r of the residual representation, there exists a morphism f from the universal deformation ring to R, such that r coincides with the composition of f with the universal representation, making all the involved diagrams commute. To do so, everything will be set up in the language of categories. This way, the existence of those universal objects will amount to the representability of a properly defined deformation functor D. Consequently, the next step will be to investigate the conditions under which D is representable. In particular, a detailed study of the work by Michael Schlessinger on a criterion for representability of functors of Artin rings will be presented and applied to the deformation functor. A final section will be dedicated to an explicit construction of the universal deformation ring when the residual representation is simply a character (which is the case n = 1).