Mathematical Research Letters

Volume 16 (2009)

Number 2

Split reductions of simple abelian varieties

Pages: 199 – 213

DOI: http://dx.doi.org/10.4310/MRL.2009.v16.n2.a1

Author

Jeffrey D. Achter (Colorado State University)

Abstract

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.

Full Text (PDF format)