Bleher, Frauke M., Ted Chinburg, and Jennifer B. Froelich. Capping Groups and Some Cases of the Fontaine-Mazur Conjecture. Proceedings of the American Mathematical Society 137, no. 5 (2009): 1551-1560.

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In this paper we will prove some cases of the Fontaine-Mazur conjecture. Let p be an odd prime and let $G_{{\Bbb Q},{p}}$ 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 $G_{{\Bbb Q},{p}}$ 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 Ŝ₄ caps ℚ for the prime 2, where Ŝ₄ is the double cover of S₄ whose Sylow 2-subgroups are generalized quaternion groups.

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