 This is a theorem of Tate for abelian varieties over finite fields, and of Faltings for abelian varieties over number fields, part of Faltings's solution of the Mordell conjecture.Zarhin extended these results to any finitely generated base field.The Tate conjecture for divisors on abelian varieties implies the Tate conjecture for divisors on any product of curves "C"1 × ... × "C""n".