We prove that any abelian variety with CM by O-L of characteristic p is defined over a finite field, where O-L is the ring of integers of the CM field L. This generalizes a theorem of Grothendieck on isogeny classes of CM abelian varieties. We also provide a direct proof of the Grothendieck theorem, which does not require several ingredients based on Weil's foundation as the original proof does. A description of the isomorphism classes is given. We analyze the reduction map modulo p for the abelian varieties concerned and solve the lifting and algebraization problem. (C) 2003 Elsevier B.V. All rights reserved. [References: 20]
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