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I think this is due to Shimura: "Abelian Varieties with Complex Multiplication and Modular Functions", http://press.princeton.edu/titles/6242.html Edit: See also the link mentioned by nfdc23 in the comments to this response, http://math.stanford.edu/~conrad/vigregroup/vigre04/mainthm.pdf and http://jmilne.org/math/CourseNotes/cm.html.


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A reasonably good answer to your question of finding a number field $L$ is given in the comments to Is there an excplicit number field of definition for an Abelian Variety $A/\mathbb{C}$ with CM? . However, the answer is roughly "no, there's no easy way to find $L$". But first you need to understand the difference between the field of moduli and fields of ...



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