On the order of CM abelian varieties over finite prime fields

摘要

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 G(l) of the l-division field of A lies in a maximal torus of the general symplectic group of dimension 2d over F-l. Relying on a method of Weng, we explicitly write down this maximal torus as a matrix group. We restrict ourselves to the case that G(l) equals the maximal torus. If p is a prime ideal in F with p vertical bar p, let A(p) be the reduction of A modulo p. By counting matrices with eigenvalue 1 in G(l) we obtain a formula for the density of primes p such that the l-rank of A(p) (E-p) 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 A(p) (E-p) has prime order. We describe the involved heuristics, generalize these conjectures to arbitrary d and provide examples with d= 3. (C) 2017 Elsevier Inc. All rights reserved.