Deformations of Pseudorepresentations

AbstractThe deformation theory of Galois representations has been the subject of much research in the past several years. This work is a study of deformations of pseudorepresentations, which were first introduced by Wiles. In general, representations might only admit a “versal” deformation and a versal deformation ring. On the other hand, pseudorepresentations always have a universal deformation. Let π be a residual pseudorepresentation and let ρ be an odd, 2-dimensional residual representation such that π comes from ρ. If this representation is absolutely irreducible, then it has a universal deformation, and the universal deformation rings Rπ and Rρ of π and ρ are (canonically) isomorphic. More generally, there is a natural map φ:Rπ→Rρ which can give some relationship between the two rings. Early on we define the kernel of a pseudorepresentation and its factorization through any quotient by a normal subgroup contained in this kernel. Later we use an appropriate factorization to find the universal deformation ring of a specific residual pseudorepresentation, for which the above map is not an isomorphism. This computation also relies on finding the tangent spaces of both the original and the factorized pseudorepresentations, which turn out to have the same dimension. Finally, we consider the deformation theory of pseudorepresentations in more generality, and we give a generalization of the above deformation problem by capturing its relevant properties