On the order of CM abelian varieties over finite prime fields

Abstract

Let A be a principally polarized CM abelian variety of dimension d defined over a number field F containing the CM-field K. Let ℓ be a prime number unramified in K/Q. The Galois group Gℓ of the ℓ-division field of A lies in a maximal torus of the general symplectic group of dimension 2d over Fℓ. Relying on a method of Weng, we explicitly write down this maximal torus as a matrix group. We restrict ourselves to the case that Gℓ equals the maximal torus. If p is a prime ideal in F with p|p, let Ap be the reduction of A modulo p. By counting matrices with eigenvalue 1 in Gℓ we obtain a formula for the density of primes p such that the ℓ-rank of Ap(Fp) is at least 1. Thereby we generalize results of Koblitz and Weng who computed this density for d=1 and 2. Both authors gave conjectural formulae for the number of primes p with norm less than n such that Ap(Fp) has prime order. We describe the involved heuristics, generalize these conjectures to arbitrary d and provide examples with d=3.