  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.
