Irreducibility of versal deformation rings in the (p,p)(p,p)-case for 2-dimensional representations

Let GKGK be the absolute Galois group of a finite extension K   of QpQp and ρ¯:GK→GLn(F) be a continuous residual representation for F a finite field of characteristic p  . We investigate whether the versal deformation space X(ρ¯) of ρ¯ is irreducible. For n=2n=2 and p>2p>2 we obtain a complete answer in the affirmative based on the results of [2] and [4]. As a consequence we deduce from recent results of Colmez, Kisin and Nakamura [10], [22] and [28] that for n=2n=2 and p>2p>2 crystalline points are Zariski dense in the versal deformation space X(ρ¯).