Galois representations attached to elliptic curves with complex multiplication

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 $f\ge 1$. Let $E$ be an elliptic curve with CM by ${\mathcal{O}}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$. Let $p\ge 2$ be a prime, let $G_{\mathbb{Q}(j(E))}$ be the absolute Galois group of $\mathbb{Q}(j(E))$, and let ${\rho }_{E,p^{\infty }}:G_{\mathbb{Q}(j(E))}\to \mathrm{GL}(2,{\mathbb{Z}}_p)$ 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 ${\mathcal{O}}_{K,f}$.

55 pages. Updated after referee reports. To appear in "Algebra and Number Theory"

References in corpus (3)

• Modular curves of prime-power level with infinitely many rational points
• Torsion points and Galois representations on CM elliptic curves
• Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9

Cited by in corpus (6)

• A classification of isogeny-torsion graphs of $\mathbb{Q}$-isogeny classes of elliptic curves
• Torsion points and isogenies on CM elliptic curves
• Explicit Kummer Theory for Elliptic Curves
• Odd degree isolated points on $X_1(N)$ with rational $j$-invariant
• Congruences of elliptic curves arising from non-surjective mod $N$ Galois representations
• Torsion groups of Mordell curves over number fields of higher degree