Vol. 16, No. 4, 2022

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Galois representations attached to elliptic curves with complex multiplication

Álvaro Lozano-Robledo

Vol. 16 (2022), No. 4, 777–837
Abstract

We give an explicit classification of the possible p-adic Galois representations that are attached to elliptic curves E with CM defined over (j(E)). More precisely, let K be an imaginary quadratic field, and let 𝒪K,f be an order in K of conductor f 1. Let E be an elliptic curve with CM by 𝒪K,f, such that E is defined by a model over (j(E)). Let p 2 be a prime, let G(j(E)) be the absolute Galois group of (j(E)), and let ρE,p: G(j(E)) GL (2, p) be the Galois representation associated to the Galois action on the Tate module Tp(E). The goal is then to describe, explicitly, the groups of GL (2, p) that can occur as images of ρE,p, up to conjugation, for an arbitrary order 𝒪K,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