Timeline for abelian varieties with the same CM type are isogenous
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 23, 2014 at 21:30 | comment | added | Damian Rössler | @user76758: the descent of homomorphisms needed here can be obtained by noticing that the graph of a homomorphism has a dense subset of torsion points and thus must descend to $k$ (because Zariski closure commutes with field extensions). | |
| Feb 23, 2014 at 19:37 | answer | added | anon | timeline score: 4 | |
| Feb 23, 2014 at 14:28 | comment | added | user76758 | @KeerthiMadapusiPera: One needs a bit more: if a pair of CM abelian varieties (for a common CM field) over an algebraically closed field $k$ become isogenous (linearly over the CM field) over an extension $K/k$ then they're isogenous (linearly over the CM field) over $k$. As you know, this is a standard "specialization" argument, by descending from $K$ to a finitely generated $k$-subalgebra $R$ (so now working with abelian schemes over $R$) and then passing to fibers over a $k$-point of $R$. (The much stronger results on descent of homomorphism for abelian varieties are not needed.) | |
| Feb 23, 2014 at 12:48 | comment | added | Keerthi Madapusi | The main point is that the CM type completely determines the $\mathbb{Q}$-Hodge structure attached to $A$ as a $K$-module. | |
| Feb 23, 2014 at 12:46 | comment | added | Keerthi Madapusi | I think these notes should do it: math.stanford.edu/~conrad/vigregroup/vigre04/cm.pdf | |
| Feb 23, 2014 at 11:41 | answer | added | Venkataramana | timeline score: 2 | |
| Feb 23, 2014 at 11:29 | review | First posts | |||
| Feb 23, 2014 at 11:44 | |||||
| Feb 23, 2014 at 11:10 | history | asked | questioner | CC BY-SA 3.0 |