The generic fiber of the universal deformation space associated to a tame Galois representation

Abstract

We will study the generic fiber over of the universal deformation ring R Q , as defined by Mazur, for deformations unramified outside a finite set of primes Q of a given Galois representation , E a number field, k a finite field of characteristic l. The main result will be that, if ¯ρ is tame and absolutely irreducible, and if one assumes the Leopoldt conjecture for the splitting field E 0 of , then defines a smooth l-adic analytic variety, near the trivial lift ρ0 of ¯ρ, whose dimension is given by cohomological constraints and as predicted by Mazur. As a corollary it follows that, in the cases considered here, R Q is a quotient of by an ideal I generated by exactly m equations, where and . Under the above assumptions for and ¯ρ odd, using ideas of Coleman, Gouvea and Mazur it should now be possible to show that modular points are Zariski-dense in the component of , that contains the trivial lift ρ0, provided this lift satisfies the Artin conjecture and E 0 satisfies the Leopoldt conjecture. Furthermore, in the Borel case, we show that the Krull dimension of R Q can exceed any given number, provided Q is chosen appropriately. At the same time, we present some evidence that despite this fact, one might however expect that the dimension of the generic fiber is given by the same cohomological formula as in the tame case.