This question was inspired by [ Has the following problem posed by Deligne in the official description of the Hodge conjecture been solved? ] (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.)
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$\begingroup$ What does ``sufficient precision" mean? Say we can compute them mod $\ell^n$. We have evidence for the conjecture if the residue class contains a rational number of height much less than $\ell^{n/2}$, but since every residue class contains a rational number of height at most $\ell^{n/2}$, we can never disprove the conjecutre Unless, I guess, we have a bound on what the numerator and denominator of the rational number should be? $\endgroup$– Will SawinCommented Apr 23, 2019 at 18:56
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$\begingroup$ My (not very educated) opinion is that the conjecture is more likely to be true then false. And, my wild guess is that $l^{100}$ or at least $l^{1000}$ should be enough to see the number unless the variety is not very nice. Of course, a much more interesting possibility is disproving the conjecture (and collecting $1M). But yes, one cannot do it this way having no rigorous bound on the height, and obtaining the latter may be rather difficult. $\endgroup$– Alex GavrilovCommented Apr 24, 2019 at 13:42
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