This question was inspired by [ https://mathoverflow.net/questions/245364/has-the-following-problem-posed-by-deligne-in-the-official-description-of-the-ho ] (which did not get any reply). I am curious if testing (not proving) this conjecture stated by Deligne is currently possible. Assuming that we have two explicitly given abelian varieties over $\overline{\mathbb{Q}}$ possessing Hodge classes and with the same reduction to $\mathbb{F}_q$, is computing the intersection number of the reduced l-adic cohomology classes with sufficient precision feasible using existing methods? (A priory this intersection number belongs to $\mathbb{Q}_l$, and according to the conjecture it must be rational.)

This question was inspired by [ https://mathoverflow.net/questions/245364/has-the-following-problem-posed-by-deligne-in-the-official-description-of-the-ho ] (which did not get any reply). I am curious if testing (not proving) this conjecture stated by Deligne is currently possible. Assuming that we have two explicitly given abelian varieties over $\overline{\mathbb{Q}}$ possessing Hodge classes and with the same reduction to $\mathbb{F}_q$, is computing the intersection number of the reduced l-adic cohomology classes with sufficient precision feasible using existing methods? (A priori this intersection number belongs to $\mathbb{Q}_{\ell}$, and according to the conjecture it must be rational.)