  We prove the integral Hodge conjecture for one-cycles on a principally
polarized complex abelian variety whose minimal class is algebraic. In
particular, any product of Jacobians of smooth projective curves over the
complex numbers satisfies the integral Hodge conjecture for one-cycles. The
main ingredient is a lift of the Fourier transform to integral Chow groups.
Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian
of a proper curve of compact type over the separable closure of a finitely
generated field. Furthermore, abelian varieties satisfying such a conjecture
are dense in their moduli space.
