Vol. 317, No. 1, 2022

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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
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-division fields of E, with p 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 , 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
Authors
Francesco Campagna
University of Copenhagen
Copenhagen
Denmark
Riccardo Pengo
École normale supérieure de Lyon
Lyon
France