Frobenius lifts and elliptic curves with complex multiplication

Gurney, Lance

doi:10.2140/ant.2021.15.1921

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Frobenius lifts and elliptic curves with complex multiplication

Lance Gurney

Vol. 15 (2021), No. 8, 1921–1942

DOI: 10.2140/ant.2021.15.1921

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.

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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

Vol. 15, No. 8, 2021