 The (known) Tate conjecture for divisors on abelian varieties is equivalent to a powerful statement about homomorphisms between abelian varieties.Namely, for any abelian varieties "A" and "B" over a finitely generated field "k", the natural map

is an isomorphism.In particular, an abelian variety "A" is determined up to isogeny by the Galois representation on its Tate module "H"1("A""k"s, Zℓ).