Entanglement in the family of division fields of elliptic curves with complex multiplication

Francesco Campagna and Riccardo Pengo

Vol. 317 (2022), No. 1, 21–66

DOI: 10.2140/pjm.2022.317.21

Abstract

For every elliptic curve $E$ which has complex multiplication (CM) and is defined over a number field $F$ containing the CM field $K$, we prove that the family of $p^{\infty }$-division fields of $E$, with $p\in \mathbb{N}$ prime, becomes linearly disjoint over $F$ after removing an explicit finite subfamily of fields. We then give a necessary condition for this finite subfamily to be entangled over $F$, which is always met when $F=K$. In this case, and under the further assumption that the elliptic curve $E$ is obtained as a base-change from $\mathbb{Q}$, we describe in detail the entanglement in the family of division fields of $E$.

Keywords

elliptic curves, complex multiplication, division fields, entanglement

Mathematical Subject Classification

Primary: 11G05, 11G15, 14K22

Secondary: 11F80, 11S15

Milestones

Received: 25 July 2020

Revised: 10 March 2021

Accepted: 21 August 2021

Published: 19 June 2022