THE TATE-CONJECTURE FOR GENERIC ABELIAN-VARIETIES

Let A --> V be a Kuga fibre variety of Mumford's Hodge type, defined over a finitely generated subfield of C, and let <(eta)under bar> be the generic point of V. We show that any element of H-et(2r)(A(eta), Q(l))(r) which is invariant under Gal(k(eta)/E), for some finite extension E of k(eta), is fixed by the semisimple part of the Hodge group of A(eta). If A --> V satisfies the H-2-condition, then the Hodge and Tate conjectures are equivalent for A(eta), and the Mumford-Tate conjecture is true.