The isomorphism classes of abelian varieties of CM-type

We prove that any abelian variety with CM by OL of characteristic p is defined over a finite field, where OL 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 Weilʹs 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.