On some $p$-adic Galois representations and form class groups

Let $K$ be an imaginary quadratic field of discriminant $d_K$ with ring of integers $\mathcal{O}_K$. When $K$ is different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$, we consider a certain specific model for the elliptic curve $E_K$ with $j(E_K)=j(\mathcal{O}_K)$ which is defined over $\mathbb{Q}(j(E_K))$. In this paper, for each positive integer $N$ we compare the extension field of $\mathbb{Q}$ generated by the coordinates of $N$-torsion points on $E_K$ with the ray class field $K_{(N)}$ of $K$ modulo $N\mathcal{O}_K$. By using this result we investigate the image of a $p$-adic Galois representation attached to $E_K$ for a prime $p$, in terms of class field theory. Second, we construct the definite form class group of discriminant $d_K$ and level $N$ which is isomorphic to $\mathrm{Gal}(K_{(N)}/\mathbb{Q})$.