The prime divisors of the number of points on abelian varieties

Let A , A ' be elliptic curves or abelian varieties fully of type GSp defined over a number field K . This includes principally polarized abelian varieties with geometric endomorphism ring ℤ and dimension 2 or odd. We compare the number of points on the reductions of the two varieties. We prove that A and A ' are K -isogenous if the following condition holds for a density-one set of primes 𝔭 of K : the prime numbers dividing # A ( k 𝔭 ) also divide # A ' ( k 𝔭 ) . We generalize this statement to some extent for products of such varieties. This refines results of Hall and Perucca (2011) and of Ratazzi (2012).