Deformacoes de representacoes galoisianas ordinarias e de representacoes nao ramificadas

Abstract

In this thesis we treat two problema concerning deformations of a continuous residual Galois representation (*), p : cú(©/Q) --» c.L, (k) which are unrami6ed outside some ânite set S of primes containing the characteristic p of the rinite field k.Essentially, a deformation of P is a lift p of P, p : Gal(Q/Q) --' G.Z;2(.A), where.4 is a complete Noetherian local rins with residue âeld k. Mazur, IMalj, showed that the set of such liftings of P can be parametrized by a rins R(P) in the game category as .4 (provided P be absolutely irreducible). If we fix a certain behavior of an inertia subgroup .r at p in a representationnamely, the set of invariant elements in .A x .A âxed by -Z is a Á--submodule free of rank l and a direct sommand, we say that p is an ordinary representation. For an ordinary representation ('p) Mazur, IMall, showed that all ordinary liftings are likewise parametrized by a rins Ro(p).The ârst problem we treat is to ând ordinary deíormations of a residual representation (+), which is unramiBed at p. In this case, the rins R'(P) no longer exista, although there are ordinary liftings of ?. Wê do this in case P is a special dihedral representation (cf. Chapter 11) and the set S of primes is S = {Z,p,oo} and considering deíormation to the rins of l)--adie integers ZP' This is done in Chapter ll.The second problem is to consider an ordinary residual representation P an analyse the natural map d to R(p) --, n'(p).We prove that tais map is surjective and under mim hypothesis that its kemel can be uerated by two generators, thus generalizing a similar result of Mazur, IMa21.