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Explicit bounds for split reductions of simple abelian varieties

Jeffrey D. Achter (2012)

Journal de Théorie des Nombres de Bordeaux

Let X / K be an absolutely simple abelian variety over a number field; we study whether the reductions X 𝔭 tend to be simple, too. We show that if End ( X ) is a definite quaternion algebra, then the reduction X 𝔭 is geometrically isogenous to the self-product of an absolutely simple abelian variety for 𝔭 in a set of positive density, while if X is of Mumford type, then X 𝔭 is simple for almost all 𝔭 . For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound...

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