Tate Cycles on Abelian Varieties with Complex Multiplication

Abstract

Abstract. We consider Tate cycles on an Abelian variety A defined over a sufficiently large number field K and having complex multiplication. We show that there is an effective bound C = C(A, K) so that to check whether a given cohomology class is a Tate class on A, it suffices to check the action of Frobenius elements at primes v of norm ≤ C. We also show that for a set of primes v of K of density 1, the space of Tate cycles on the special fibre A v of the Néron model of A is isomorphic to the space of Tate cycles on A itself.