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.

中文翻译：

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带复数乘法的 Frobenius 提升和椭圆曲线

我们根据 Frobenius 电梯的交换族给出了 Shimura 型椭圆曲线的新表征，并加强了射线类场的旧主理想定理。这两个结果相结合，得出了此类曲线的全局最小模型的存在，概括了 Gross 的结果。在此过程中，我们还证明了一些关于具有复数乘法的椭圆曲线的小而新的结果。