 
Summary: SPLIT REDUCTIONS OF SIMPLE ABELIAN VARIETIES
JEFFREY D. ACHTER
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 Xp
is absolutely simple for almost all primes p. Conversely, if the absolute endomorphism ring of X is
noncommutative, then Xp is reducible for p in a set of positive density.
An absolutely simple abelian variety over a number field may or may not have absolutely sim
ple reduction almost everywhere. On one hand, let K = Q(5), and let X be the Jacobian of the
hyperelliptic curve with affine model
t2
= s(s  1)(s  1 5)(s  1 5 2
5 )(s  1 5 2
5 3
5 ),
considered as an abelian surface over K. Then X is absolutely simple [13, p.648] and has ordinary
reduction at a set of primes p of density one [12, Prop. 1.13]; at such primes Xp is absolutely simple.
On the other hand, let Y be the Jacobian of the hyperelliptic curve with affine model
t2
= s6
 12s5
