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I am currently learning some aspects of the theory of complex multiplication for elliptic curves, and the relationship with class field theory.

As I understand it, there is a very special class of elliptic curves $E$ whose automorphism rings are ideals $\mathfrak{a}$ of $\mathcal{O_K}$, the ring of integers of an imaginary quadratic field $K$. And conversely, for every ideal $\mathfrak{a}$ in an imaginary quadratic field $K$, there is an elliptic curve whose endomorphism ring is $\mathfrak{a}$. The correspondence goes by viewing $E$ as $\mathbb{C}/\Lambda$ for some lattice $\Lambda$, and the endomorphism ring as the set of complex numbers $z$ such that multiplication by $z$ maps $\Lambda$ into itself.

These special curves give an explicit version of class field theory for $K$: the Hilbert class field is generated by the $j$-invariant of a $E$, and the rest of the maximal abelian extension is generated by numbers determined by the $x$-coordinates of torsion points on $E$.

How does this generalize to the world of abelian varieties? I am dimly aware that the idea of complex multiplication is well-defined for an abelian variety $A$. Loosely, this corresponds to the case when the endomorphism ring of $A$ is larger than "expected", and this is related to some algebraic number field in a somewhat similar way to the above relationship.

What does this look like on the class field theory side? Does it give an explicit class field theory for more general types of fields than imaginary quadratic fields? Or does it correspond to something more exotic - some Langlands-y analogue of class field theory for non-abelian extensions?

I'm also vaguely aware that "explicit class field theory" in a form analogous to the case of complex multiplication has been developed for CM-fields, which are imaginary quadratic extensions of totally real number fields. Is this related to higher-dimensional abelian varieties?

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    $\begingroup$ It would be better to say that, loosely, CM for abelian varieties corresponds to the case when the endomorphism ring is "as big as possible". If an abelian variety $X$ of dimension $g$ is simple with $\mathrm{End}^{0}(X) = \mathrm{End}(X) \otimes \mathbb{Q}$, then $de$ divides $g$, where $d^{2}$ is the degree of the division algebra $\mathrm{End}^{0}(X)$ over its center, and $e$ is the degree of its center over $\mathbb{Q}$. The CM case is where $de = g$. $\endgroup$
    – Jeff Yelton
    Commented Jul 18, 2016 at 10:56
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    $\begingroup$ The endomorphism rings are orders $O$ in imaginary quadratic fields. For $E$ over $\mathbf{C}$ with ${\rm{End}}(E) = O$, ${\rm{H}}_1(E(\mathbf{C}),\mathbf{Z})$ is an invertible $O$-module and $K(j(E))/K$ is a ring class field. For a CM field $L$ and CM type $\Phi$ on $L$, $L$-linear isogeny classes of abelian varieties with CM type $\Phi$ over a finite extension $K$ of the reflex field correspond to certain algebraic Hecke characters $\mathbf{A}^{\times}_K\to L^{\times}$. See 2.5.1-2.5.2 and A.4.6 of amazon.com/… $\endgroup$
    – nfdc23
    Commented Jul 18, 2016 at 15:56

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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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    $\begingroup$ Although the Main Theorem of Complex Multiplication is due to Shimura and Taniyama, the optimal result on class field theory for CM fields in the spirit of the classical results in the imaginary quadratic case is due to Wafa Wei in her 1993 thesis; see the question and answer at mathoverflow.net/questions/128907/… $\endgroup$
    – nfdc23
    Commented Jul 18, 2016 at 15:32

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