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I'm pretty new to complex multiplication and am struggling with Corollary 5.20 in Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer by Rubin.

According to a comment here, we should be able to use this and know how the ray class groups are generated (torsion points). However I don't see how that is done.

I'm also not sure about the proof (which is mostly not given) and would be grateful if someone could explain it a little.

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    I suggested Rubin's paper in MO because the OP there wanted to know the significance of the main theorem of CM. If you want to know why the ray class fields are generated by torsion points, then Silverman's "Advanced topics in the arithmetic of elliptic curves" is a better, simpler reference (see Theorem 5.6 of Chapter II).2012-12-24

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