Vol. 15, No. 8, 2021

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Frobenius lifts and elliptic curves with complex multiplication

Lance Gurney

Vol. 15 (2021), No. 8, 1921–1942
Abstract

We give a new characterisation of elliptic curves of Shimura type in terms of commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for such curves, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.

Keywords
complex multiplication, Frobenius lifts
Mathematical Subject Classification 2010
Primary: 11G05
Secondary: 11G15, 11R37
Milestones
Received: 22 November 2019
Revised: 30 November 2020
Accepted: 1 January 2021
Published: 10 November 2021
Authors
Lance Gurney
Mathematical Sciences Institute
Australian National University
Canberra
Australia