Mathematical Research Letters

Volume 16, Issue 2, March 2009 pp. 199-213.

Split reductions of simple abelian varieties

Authors: Jeffrey D. Achter
Author institution: Colorado State University

Summary: 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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