Frobenius lifts and elliptic curves with complex multiplication

TL;DR

This paper characterizes elliptic curves of Shimura type using Frobenius lifts, proves a principal ideal theorem for ray class fields, and establishes the existence of global minimal models for these curves, advancing understanding of elliptic curves with complex multiplication.

Contribution

It introduces a new characterization of Shimura type elliptic curves via Frobenius lifts and extends the existence of global minimal models, generalizing Gross's results.

Findings

Characterization of Shimura type elliptic curves through Frobenius lifts

Strengthening of principal ideal theorem for ray class fields

Existence of global minimal models for elliptic curves of Shimura type

Abstract

We give a new characterisation of elliptic curves of Shimura type in terms 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 elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.