Abelian varieties isogenous to a Jacobian

(0.1) Question Given an abelian variety A; does there exist an algebraic curve C such that there is an isogeny between A and the Jacobian of C ? • If the dimension of A is at most three, such a curve exists; see (1.3). • For any g ≥ 4 there exists an abelian variety A of dim(A) = g over C such that there is no algebraic curve C which admits an isogeny A ∼ Jac(A), see (3.1). One of the arguments which proves this fact (uncountability of the ground field) does not hold over a countable field