Split reductions of simple abelian varieties

Abstract

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 simple reduction almost everywhere. On one hand, let K = Q(ζ 5 ), and let X be the Jacobian of the hyperelliptic curve with affine model Note that End K (X) ⊗ Q is the cyclotomic field Q(ζ 5 ), while End L (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 EndK(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) EndK(X) ⊗ Q ∼ = F a totally real field, and dim X/[F : Q] is odd; or

Explicit bounds for split reductions of simple abelian varieties

“…(Perhaps some remarks on the hypotheses, which are explained in greater detail in [1,Sec. 4], are in order here.…”

“…Finally, since the method of [1] relies crucially on the image of modGalois representations, it seems reasonable to explore the conjecture of Murty and Patankar in cases where the endomorphism ring alone does not determine these groups. Mumford described the simplest such situation in [12]; there are abelian fourfolds with trivial endomorphism ring whose Mumford-Tate group is smaller than the full symplectic group.…”

“…Using known instances of the Mumford-Tate conjecture and ideas of Chavdarov [4], the author made progress [1] towards affirming this conjecture. Moreover, a preprint of [1] elicited two further questions.…”

“…2(e)]. 1 On the other hand, if End(X) is trivial and dim(X) is odd [3], or if X has complex multiplication [13], then X p is almost always simple.…”

“…Moreover, a preprint of [1] elicited two further questions. W. Gajda shared a preprint of his work with Banaszak and Krasoń on the MumfordTate conjecture for abelian varieties of type III [2], and inquired whether it might be used to refine the results of [1] in that case. V.K.…”