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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.

14L05 Formal groups, \(p\)-divisible groups
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