A CALCULATION OF POTENTIALLY BARSOTTI-TATE DEFORMATION RINGS

Let F be an unramified extension of Q(p). The first aim of this work is to develop a purely local method to compute the potentially Barsotti-Tate deformation rings with tame Galois type of level [F : Q-p] of irreducible two-dimensional representations of the absolute Galois group of F. We then apply our method in the particular case where F has degree 2 over Q-p and determine in this way almost all these deformation rings. In this particular case, we observe a close relationship between the structure of these deformation rings and the geometry of the associated Kisin variety. As a corollary and still assuming that F has degree 2 over Q(p), we prove, except in two very particular cases, a conjecture of Kisin which predicts that intrinsic Galois multiplicities are all equal to 0 or 1.