Timeline for abelian varieties with the same CM type are isogenous
Current License: CC BY-SA 3.0
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| Feb 23, 2014 at 15:05 | comment | added | user76758 | This is proved in Mumford's book on abelian varieties in section 22 (see "First Example", and the Theorem there). As in Mumford's discussion, the notation should keep track of the choice of CM type $\Phi$ giving the identification of $K \otimes_{\mathbf{Q}} \mathbf{R}$ with $\mathbf{C}^d$ as $\mathbf{R}$-algebras; this is implicit to make sense of the quotient of $\mathbf{C}^d$ modulo orders in $\mathscr{O}_K$ or fractions ideals of $K$ (though since the CM order is not preserved by isogeny, it might be "better" to avoid formulating things in terms of orders). | |
| Feb 23, 2014 at 11:41 | history | answered | Venkataramana | CC BY-SA 3.0 |