Algebraic cycles on an abelian variety without complex multiplication

We prove a theorem to the effect that if a natural number $d$ is not exceptional, then all $d$-dimensional abelian varieties without complex multiplication satisfy the Grothendieck version of the general Hodge conjecture. Exceptional numbers have density zero in the set of natural numbers. If $\operatorname{End}(J)=\mathbf Z$, $J$ is defined over a number field, and $\dim J=2p$, where $p$ is a prime number, $p\ne 2$ and $p\ne 5$, then the Mumford–Tate conjecture and the Tate conjecture on algebraic cycles hold for the variety $J$.