Towards the end of his official description of the Hodge conjecture, Deligne asked the following question:
Let $A$ be an abelian variety over the algebraic closure $\mathbb{F}$ of a finite field $\mathbb{F}_q$. Lift $A$ in two different ways to characteristic $0$, to complex abelian varieties $A_1$ and $A_2$ defined over $\bar{\mathbb{Q}}$. Pick Hodge classes $\mathfrak{z}_1$ and $\mathfrak{z}_2$ on $A_1$ and $A_2$, of complementary dimension. Interpreting $\mathfrak{z}_1$ and $\mathfrak{z}_2$ as $\ell$-adic cohomology classes, one can define the intersection number $\kappa$ of the reduction of $\mathfrak{z}_1$ and $\mathfrak{z}_2$ over $\mathbb{F}$. Is $\kappa$ a rational number?
Has this problem been solved?