Split reductions of simple abelian varieties

Consider an absolutely simple abelian variety $X$ over a number field $K$. We show that if the absolute endomorphism ring of $X$ is commutative and satisfies certain parity conditions, then $X_\idp$ is absolutely simple for almost all primes $\idp$. Conversely, if the absolute endomorphism ring of $X$ is noncommutative, then $X_\idp$ is reducible for $\idp$ in a set of positive density.

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Published 1 January 2009