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 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 simple reduction almost everywhere. On one hand, let K = Q(ζ5), and let X be the Jacobian of the hyperelliptic curve with affine model t = s(s− 1)(s− 1− ζ5)(s− 1− ζ5 − ζ 5 )(s− 1− ζ5 − ζ 5 − ζ 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 t = s − 12s + 9s − 32s + 3s + 18s+ 3, considered as an abelian surface over L = Q( √ 2, √ −3). Then Y is absolutely simple [3, Thm. 6.1], but Yq is reducible for each prime q of good reduction. (The conclusions about the simplicity of Xp and the reducibility of Yq follow from Tate’s description [25] of the endomorphism rings of abelian varieties over finite fields.) Note that EndK(X) ⊗ Q is the cyclotomic field Q(ζ5), while EndL(Y ) ⊗ Q is an indefinite quaternion algebra over Q. Murty and Patankar study the splitting behavior of abelian varieties over number fields, and advance the following conjecture: Conjecture. [20, Conj. 5.1] Let X/K be an absolutely simple abelian variety over a number field. The set of primes of K where X splits has positive density if and only if EndK̄(X) is noncommutative. (A similar question has been raised by Kowalski; see [14, Rem. 3.9].) The present paper proves this conjecture under certain parity and signature conditions on End(X). The first main result states that a member of a large class of abelian varieties with commutative endomorphism ring has absolutely simple reduction almost everywhere. (Throughout this paper, “almost everywhere” means for a set of primes of density one.) Theorem A. Let X/K be an absolutely simple abelian variety over a number field. Suppose that either (i) EndK̄(X)⊗Q ∼= F a totally real field, and dimX/[F : Q] is odd; or Received by the editors May 21, 2008.