Lower bounds for the normalized height and non-dense subsets of varieties in an abelian variety

This work is the third part of a series of papers. In the first two we considered curves and varieties in a power of an elliptic curve. Here we deal with subvarieties of an abelian variety in general. Let V be an irreducible variety of dimension d embedded in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V . Then, for V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d + 1 translated by the points close to a subgroup Γ of finite rank, are non Zariski-dense in V . If Γ has rank zero, it is sufficient to assume that V is weak-transverse. The notion of closeness is defined using a height function.