Abelian fibrations and rational points on symmetric products

Let X be an algebraic variety defined over a number field K and X(K) its set of K-rational points. We are interested in properties of X(K) imposed by the global geometry of X. We say that rational points on X are potentially dense if there exists a finite field extension L/K such that X(L) is Zariski dense. It is expected at least for surfaces that if there are no finite étale covers of X dominating a variety of general type then rational points on X are potentially dense. This expectation complements the conjectures of Bombieri, Lang and Vojta predicting that rational points on varieties of general type are always contained in Zariski closed subsets. This dichotomy holds for curves: the nondensity for curves of genus ≥ 2 is a deep theorem of Faltings and the potential density for curves of genus 0 and 1 is classical.