Irreducible Components of Deformation Spaces: Wild 2-Adic Exercises

We prove that the irreducible components of the space of framed deformations of the trivial two-dimensional mod 2 representation of the absolute Galois group of Q(2) are in natural bijection with those of the trivial character, confirming a conjecture of Bockle. We deduce from this result that crystalline points are Zariski dense in that space: this provides the missing ingredient for the surjectivity of the p-adic local Langlands correspondence for GL(2)(Q(p)) in the case p=2 ( the result was already known for p >= 3).