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If an abelian variety $A$ is not simple, it is isogenous to a product of simple lower dimensional abelian varieties. These simple abelian varieties $B_i$ are the isogeny factors of $A$, and we say that $A$ decomposes (up to isogeny) into the product of the $B_i$'s: $$A \sim B_1 \times \cdots \times B_n$$ Note that two elements of this product might be isogenous; in other words, elements of the decomposition may appear with multiplicity.

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• Last edited by Kiran S. Kedlaya on 2019-05-04 20:31:44
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