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Capping groups and some cases of the Fontaine-Mazur conjecture

In this paper we will prove some cases of the Fontaine-Mazur conjecture. Let p be an odd prime and let G Q,{p} be the Galois group over (Q) of the maximal unramified-outside-p extension of Q. We show that under certain hypotheses, the universal deformation of the action of G Q,{p} on the 2-torsion of an elliptic curve defined over Q has finite image. We compute the associated universal deformation ring, and we show in the process that S 4 caps Q for the prime 2, where S 4 is the double cover of S 4 whose Sylow 2-subgroups are generalized quaternion groups.