Local cyclicity of isogeny classes of abelian varieties defined over finite fields

For a prime number l, an isogeny class A of abelian varieties is called l-cyclic if every variety in A have a cyclic l-part of its group of rational points. More generally, for a finite set of prime numbers S, A is said to be S-cyclic if it is l-cyclic for every l is an element of S. We give lower and upper bounds on the fraction of S-cyclic g-dimensional isogeny classes of abelian varieties defined over the finite field F-q, when q tends to infinity. (C) 2019 Elsevier Inc. All rights reserved.