On some p‐adic Galois representations and form class groups

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