G. Böckle: Irreducibility of local versal deformation rings in the (p,p)-case

Abstract:

Let r be a mod p Galois representation of the absolute Galois group of a p-adic local field. The p-adic Galois representations that have r as a reduction are precisely the closed points of the generic fiber X_r of the local versal deformation space of r. The talk addresses the question whether the subspace X_r^a of X_r of fixed given determinant a is irreducible. For p>2 and any 2-dimensional representations we indicate a proof. In light of recent  work of Colmez, Kisin, Chenevier and Nakamura this implies that in this case benign crystalline points are dense in X_r^a. In the talk we also introduce a strategy to resolve the question for representations of arbitrary dimension. The work is joint with A.-K. Juschka.