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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 \(0\).

14K22 Complex multiplication and abelian varieties
11G15 Complex multiplication and moduli of abelian varieties
Full Text: DOI
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