Abstract
We prove a theorem to the effect that if a natural number is not exceptional, then all -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 , is defined over a number field, and , where is a prime number, and , then the Mumford-Tate conjecture and the Tate conjecture on algebraic cycles hold for the variety .
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