I am reading this paper on the complex multiplication of K3 surfaces. It seems that this is only defined for complex K3 surfaces, or K3 surfaces over number fields. Is there a more general defintion applying to general fields, or both number fields and local fields?
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1$\begingroup$ The definition of a CM K3 surface is Hodge theoretic (the Mumford-Tate group of $H^2$ is abelian). To work over a more general field, you would need another definition. $\endgroup$– Donu ArapuraCommented Jun 24 at 21:33
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$\begingroup$ @DonuArapura Sorry, I haven't made myself clear. My question is exactly whether there is a more general definition. If not, is there some reason that we cannot do this? $\endgroup$– Ja_1941Commented Jun 25 at 14:21
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1$\begingroup$ One can replace the Mumford-Tate group by (the identity component of) the motivic Galois group (defined, e.g., using Andre's category of motives) to get a definition over an arbitrary field. Over local fields I think it is unlikely that there is a definition in terms of the Galois representation on the (etale) $H^2$: is there such a definition for CM abelian surfaces? $\endgroup$– nafCommented Jun 26 at 2:21
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2$\begingroup$ Over a field of characteristic zero, you can embed the field in $\mathbb{C}$, and then use Deligne's theorem that Hodge classes on K3 surfaces are absolutely Hodge to see that the answer is independent of the embedding. $\endgroup$– anonCommented Jun 26 at 17:03
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