Galois representations attached to elliptic curves with complex multiplication

Abstract

We give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}\left(j\right(E\left)\right)$. More precisely, let $K$ be an imaginary quadratic field, and let ${\mathcal{𝒪}}_{K,f}$ be an order in $K$ of conductor $f\ge 1$. Let $E$ be an elliptic curve with CM by ${\mathcal{O}}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}\left(j\right(E\left)\right)$. Let $p\ge 2$ be a prime, let $G_{\mathbb{Q}\left(j\right(E\left)\right)}$ be the absolute Galois group of $\mathbb{Q}\left(j\right(E\left)\right)$, and let ${\rho }_{E,p^{\infty }}:G_{\mathbb{Q}\left(j\right(E\left)\right)}\to \mathrm{GL}(2,{\mathbb{Z}}_p)$ be the Galois representation associated to the Galois action on the Tate module $T_p(E)$. The goal is then to describe, explicitly, the groups of $\mathrm{GL}(2,{\mathbb{Z}}_p)$ that can occur as images of ${\rho }_{E,p^{\infty }}$, up to conjugation, for an arbitrary order ${\mathcal{O}}_{K,f}$.