Timeline for Does complex multiplication for higher dimensional abelian varieties give some generalization of class field theory?
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| Jul 18, 2016 at 15:56 | comment | added | nfdc23 | 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/… | |
| Jul 18, 2016 at 10:56 | comment | added | Jeff Yelton | 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 $de = g$. | |
| Jul 18, 2016 at 8:15 | answer | added | user19475 | timeline score: 2 | |
| Jul 18, 2016 at 6:16 | history | asked | dorebell | CC BY-SA 3.0 |