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 $G_{\mathbb {Q},\{p\}}$ be the Galois group over $\mathbb {Q}$ of the maximal unramified-outside-$p$ extension of $\mathbb {Q}$. We show that under certain hypotheses, the universal deformation of the action of $G_{\mathbb {Q},\{p\}}$ on the $2$-torsion of an elliptic curve defined over $\mathbb {Q}$ has finite image. We compute the associated universal deformation ring, and we show in the process that $\hat {S}_4$ caps $\mathbb {Q}$ for the prime $2$, where $\hat {S}_4$ is the double cover of $S_4$ whose Sylow $2$-subgroups are generalized quaternion groups.