LAUSR

We prove that an abelian variety whose endomorphism ring is a maximal order can be written as a direct product of simple factors with the same property, in which furthermore two isogenous factors have isomorphic nth powers for some n. Conversely every such product has a maximal order as endomorphism ring. We deduce from this some properties for arbitrary abelian varieties, in particular for almost complements of abelian subvarieties.