The Hodge Conjecture as a Conjunction of Arithmetic and Geometric Problems: A Precise Reduction

We establish an exact reduction of the Hodge conjecture for complex projective varieties to the conjunction of two independent problems: (A) the algebraicity of reductions of Hodge classes modulo almost all primes, and (B) the possibility of lifting local algebraicity to global algebraicity. We prove that these two statements are together equivalent to the original Hodge conjecture. Statement (A) is shown to be closely related to the Tate conjecture, while Statement (B) is equivalent to a specialization problem for higher Chow groups.