Spencer Hamblen

Abstract:We investigate the problem of deforming n-dimensional mod p Galois representations to characteristic zero. Taylor and Ramakrishna have proven the existence of 2-dimensional deformations under certain conditions by allowing ramification at additional primes in order to annihilate a dual Selmer group. We use the same general methods to prove the existence of n-dimensional deformations.

We then examine under which conditions we may place restrictions on the shape of our deformations at p, with the goal of showing that under the correct conditions, the deformations may have locally geometric shape. We also use the existence of these deformations to prove the existence as Galois groups over Q of certain infinite subgroups of p-adic general linear groups.