# Split reductions of simple abelian varieties

```@article{Achter2008SplitRO,
title={Split reductions of simple abelian varieties},
author={Jeff Achter},
journal={Mathematical Research Letters},
year={2008},
volume={16},
pages={199-213}
}```
• Jeff Achter
• Published 27 June 2008
• Mathematics
• Mathematical Research Letters
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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