# Complex multiplication - Ray class fields

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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Dear Christopher, Do you understand the construction of the Hecke character $\psi$? And do you understand the basic class field theory computations occuring in the proof? If not, perhaps you should analyze an earlier point at which your understanding breaks down and ask about that. But if you do, then perhaps be more precise about at exactly where your understanding of what is written in the proof breaks down. (This argument combines a lot of what has gone before, and summarizing all of that would be a bit much!) Regards, –  Matt E Dec 13 '12 at 20:12
@MattE I understand how $\psi$ was constructed and I think that (i) follows fairly quickly (for $P \in E[\mathfrak{af}]$ and $x \equiv 1 \bmod \mathfrak{af}$ we have $[x,K^{ab}/K]P = P$ so $P \in E(K(\mathfrak{af}))$, right?) but I don't see how to prove (ii)... more importantly, though, I think, I don't understand how the result is what I want (which is a description of the ray class fields). –  Christopher Poole Dec 13 '12 at 20:20
Is it possible to bump a question here? –  Christopher Poole Dec 14 '12 at 8:41
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). –  Álvaro Lozano-Robledo Dec 24 '12 at 2:16