On the number of points on the reductions of abelian varieties

Let A,A' be either two elliptic curves or two abelian varieties fully of type GSp defined over a number field K. This includes for example principally polarized abelian varieties with geometric endomorphism ring Z and dimension 2 or odd. Let S be a density-one set of primes of K of good reduction for both A and A'. If for every p in S the prime numbers dividing the size of A(kp) also divide the size of A'(kp) (we thereby mean the number of points on the residue field) then A and A' are K-isogenous. We also generalize this statement to some extent for products of such varieties. This refines results of Hall and Perucca (2011) and of Ratazzi (2012).