Tate Cycles on Abelian Varieties with Complex Multiplication

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 $\leq C$. 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 N\'eron model of $A$ is isomorphic to the space of Tate cycles on $A$ itself.