Galois representations attached to elliptic curves with complex multiplication

TL;DR

This paper classifies the possible $p$-adic Galois representations attached to elliptic curves with complex multiplication over $Q(j(E))$, providing explicit descriptions of their image groups for various orders in imaginary quadratic fields.

Contribution

It offers an explicit classification of the images of $p$-adic Galois representations for CM elliptic curves over their field of definition, extending prior understanding to arbitrary orders in imaginary quadratic fields.

Findings

Explicit descriptions of Galois image groups for CM elliptic curves.

Classification applicable to all orders in imaginary quadratic fields.

Provides a comprehensive framework for understanding Galois representations in CM case.

Abstract

The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$. Let $p\geq 2$ be a prime, let $G_{\mathbb{Q}(j(E))}$ be the absolute Galois group of $\mathbb{Q}(j(E))$, and let $\rho_{E,p^\infty}\colon G_{\mathbb{Q}(j(E))}\to \operatorname{GL}(2,\mathbb{Z}_p)$ 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 $\operatorname{GL}(2,\mathbb{Z}_p)$ that can occur as images of $\rho_{E,p^\infty}$, up to conjugation, for an arbitrary order $\mathcal{O}_{K,f}$.