I want to understand the meaning of the main theorem of complex multiplication (of elliptic curves) as given in Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, II.8, or Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, 5.3.
I want to derive the theorems about the ray class fields and the Hilbert class fields of imaginary quadratic fields from this main theorem (as in Shimura, but not as in Silverman).
What I understand is: The main theorem gives an algebraic description of something analytic. But is there an easy understanding why all great theorems on CM elliptic curves follow from this main theorem?
Sorry for this vague question.
Best Answer
I would suggest the following reference: Karl Rubin's "Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer". In particular, in Section 5 he states the main theorem of complex multiplication and in a succinct manner deduces many of the interesting theorems in the theory of CM elliptic curves (e.g., $K(j(E))$ is the Hilbert class field of $K$ [Cor. 5.12]; the existence of the Hecke character [Thm. 5.15]; etc). And then, of course, he goes on to discuss the Birch and Swinnerton-Dyer for CM elliptic curves.