V. Kumar Murty,
Vijay M. Patankar,
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
Néron model of $A$ is isomorphic to the space of Tate cycles
on $A$ itself.
Abelian varieties, complex multiplication, Tate cycles
11G10 - Abelian varieties of dimension $> 1$ [See also 14Kxx]
14K22 - Complex multiplication [See also 11G15]