Mordell-Lang Plus Bogomolov

Let k be a number field. Let A be an almost split semiabelian variety over k; by this we mean that A is isogenous to the product of an abelian variety A0 and a torus T. We enlarge k if necessary to assume that T ∼ = Gn m. Let φ = (φ1, φ2) : A → A0 × Gn m be the isogeny. Let h1: A0(k) → R be a Néron-Tate canonical height associated to a symmetric ample line bundle on A0, and let h2: Gn m(k) → R be the sum of the naive heights of the coordinates. For x ∈ A(k), let h(x) = h1(φ1(x)) + h2(φ2(x)). For ǫ ≥ 0, let Bǫ = { z ∈ A(k) | h(z) ≤ ǫ}. Let Γ be a finitely generated subgroup of A(k), and define Γǫ: = Γ + Bǫ = { γ + z | γ ∈ Γ, h(z) ≤ ǫ}. Note that Γ0 = Γ + A(k)tors. Let X be a geometrically integral closed subvariety of A. Our main result is the existence of ǫ> 0 such that X(k) ∩ Γǫ is contained in a finite union ⋃ Zj where each Zj is a translate of a sub-semiabelian variety of A k = A ⊗k k by a point in Γ0 and Zj ⊆ X