On some <i>p</i> ?adic Galois representations and form class groups

Let $K$ be an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, let $H_K$ its Hilbert class field. For each prime $p$ we examine two $p$-adic abelian objects in view of theory. First, investigate the image a Galois representation $\mathrm{Gal}(\overline{\mathbb{Q}}/H_K)$ attached to elliptic curve with complex multiplication. Second, construct form group that is isomorphic quotient $\mathrm{Gal}(K_{(p^\infty)}/H_K)$, where $K_{(p^\infty)}$ maximal extension unramified outside ideals lying above $p$.