Relative Mordell–Lang conjecture for abelian schemes

Prove that, for a smooth quasi-projective variety S over a number field, an abelian scheme A/S of relative dimension g, an irreducible subvariety X contained in A that dominates S, and a finitely generated subgroup Lambda of A(S), the set of points in the fibers X_s that belong to the divisible hull Lambda_s^div is not Zariski dense in X whenever dim X is less than g and the geometric generic fiber X_bar-eta is not contained in any translate of a proper abelian subvariety by a point of Lambda_bar-eta^div.

Background

The paper defines the relative Mordell–Lang set M(X,Lambda) as the union over algebraic points s of the intersections X_s with the divisible hull of the specialization Lambda_s of a finitely generated group of sections. The stated transversality condition excludes the geometric generic fiber from being contained in a translate of a proper abelian subvariety by a point in the divisible hull of the generic specialized group.

The conjecture extends classical Mordell–Lang to varying abelian fibers. The paper proves that Pink’s conjecture would imply it and establishes it for section images in powers of abelian schemes whose nontrivial simple geometric generic factors have maximal variation, but leaves the conjecture unresolved in general.