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The isomorphism classes of abelian varieties of CM-type. (English) Zbl 1087.14030
It is well known (Grothendieck, Oort) that an abelian variety of CM-type over a field $$k$$ of positive characteristic is isogenous (over a finite extension) to an abelian variety defined over a finite field. A more precise criterion is obtained in this paper. The author shows that in positive characteristic, a CM abelian variety with CM field $$L$$ is defined over a finite field if its endomorphism ring contains $$O_L$$, the ring of integers of $$L$$. Based on this, a classification of the isomorphism classes of CM abelian varieties is given, and a criterion is obtained for a polarized abelian variety with CM by $$O_L$$ to be liftable to characteristic $$0$$.

##### MSC:
 14K22 Complex multiplication and abelian varieties 11G15 Complex multiplication and moduli of abelian varieties
##### Keywords:
abelian varieties; complex multiplication
Full Text:
##### References:
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