On direct summands of products of Jacobians over arbitrary fields

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 Q due to Petrov--Skorobogatov. We also give applications to the integral Tate conjecture for divisors and for 1-cycles on abelian varieties over finitely generated fields; our results also address a p-adic version of the integral Tate conjecture over finite fields of characteristic p, for the first time beyond the case of divisors.