Galois representations attached to elliptic curves with complex multiplication

Lozano-Robledo, Álvaro

doi:10.2140/ant.2022.16.777

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Galois representations attached to elliptic curves with complex multiplication

Álvaro Lozano-Robledo

Vol. 16 (2022), No. 4, 777–837

DOI: 10.2140/ant.2022.16.777

Abstract

We give an explicit classification of the possible $p$ -adic Galois representations that are attached to elliptic curves $E$ with CM defined over $ℚ (j (E))$ . More precisely, let $K$ be an imaginary quadratic field, and let $𝒪_{K, f}$ be an order in $K$ of conductor $f \geq 1$ . Let $E$ be an elliptic curve with CM by $𝒪_{K, f}$ , such that $E$ is defined by a model over $ℚ (j (E))$ . Let $p \geq 2$ be a prime, let $G_{ℚ (j (E))}$ be the absolute Galois group of $ℚ (j (E))$ , and let $ρ_{E, p^{\infty}} : G_{ℚ (j (E))} \to GL (2, ℤ_{p})$ 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 $GL (2, ℤ_{p})$ that can occur as images of $ρ_{E, p^{\infty}}$ , up to conjugation, for an arbitrary order $𝒪_{K, f}$ .

Keywords

elliptic curve, complex multiplication, p-adic, Galois representation, Cartan

Mathematical Subject Classification 2010

Primary: 11F80

Secondary: 11G05, 11G15, 14H52

Milestones

Received: 25 April 2019

Revised: 12 July 2021

Accepted: 12 August 2021

Published: 5 August 2022

Authors

Álvaro Lozano-Robledo
Department of Mathematics University of Connecticut Storrs, CT United States

Vol. 16, No. 4, 2022