Rationality of Tate classes

Determine whether a given Tate class for a variety over a number field is rational with respect to the non-archimedean notion of rationality, namely whether its specialization at a place of good reduction is represented by an algebraic cycle on the special fiber.

Background

The paper explains that, for a variety over a number field, a Tate class may be regarded as rational at a non-archimedean place of good reduction when its restriction to the special fiber is represented by an actual algebraic cycle. Establishing rationality of an individual Tate class is presented as difficult. The paper subsequently avoids resolving this individual-class problem by proving, for homomorphisms between abelian varieties, that the entire space of Tate classes is rational in the sense that it admits a rational basis, which is sufficient for the Tate conjecture in that setting.