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We consider Tate cycles on an Abelian variety
defined over a sufficiently large number field
and having complexmultiplication. We show that there is an effective bound
so that to check whether a given cohomology class is a Tate class on
, it suffices to check the action of Frobenius elements at primes
. We also show that for a set of primes
of density 1, the space of Tate cycles on the special fibre
of the Néron model of
is isomorphic to the space of Tate cycles on
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