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On finiteness of endomorphism rings of Abel varieties. (English) Zbl 1231.14037
The endomorphism ring of an abelian variety over a field $$k$$ is an order in a semisimple $$Q$$-algebra. The paper bounds the $$p$$-power in the index of this order in a maximal order, provided $$k$$ has characteristic $$p$$. The proof reduces to $$k$$ algebraically closed and then the analogue for $$p$$-divisible groups (the analogue is false if $$k$$ is not algebraically closed).
Finally one applies Dieudonné theory. The author shows that the result is wrong for $$l$$-powers $$(l\neq p)$$, and determines for principally polarized abelian varieties of dimension two the stratification on the moduli space defined by the index.

##### MSC:
 14L05 Formal groups, $$p$$-divisible groups
##### Keywords:
abelian varieties; endomorphisms
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