Cyclic components of quotients of abelian varieties mod p

Let A an abelian variety of dimension r, defined over Q. For p a rational prime, we denote by F_p the finite field of cardinality p. If A has good reduction at p, let A¯_p be the reduction of A at p. Let Γ be a free subgroup of the Mordell–Weil group A(Q), and let Γ_p be the reduction of Γ at p. In this paper for abelian varieties of type I, II, III, and IV, under Generalized Riemann Hypothesis, Artin's Holomorphy Conjecture, and Pair Correlation Conjecture, we obtain asymptotic formulas for the number of primes p, with p≤x, for which the quotient [Formula presented] has at most 2r−1 cyclic components.