Abstract: We show that a principally polarized abelian variety over a field k is, as an abelian variety, a direct summand of a product of Jacobians of curves which contain a k-point if and only if the polarization and the minimal class are both algebraic over k. This extends results of Beckmann-de Gaay Fortman and Voisin over the complex numbers to arbitrary fields, and refines an obstruction to the direct summand property over the rational numbers due to Petrov-Skorobogatov. We then give applications to the integral Tate conjecture for 1-cycles on abelian varieties over finite fields, including the case ell=p. This is joint work with Federico Scavia.