Néron-tate Projection of Algebraic Points

Let X be a geometrically irreducible closed subvariety of an abelian variety A over a number field k such that X generates A. Let V be a finite-dimensional subspace of A(k) ⊗R, and let π : A(k) → V be the orthogonal projection relative to a Néron-Tate pairing 〈 , 〉 : A(k) × A(k) → R. For V = A(k) ⊗R, we prove that π(X(k)) = A(k) ⊗Q, and moreover, there exist c, c′ > 0 such that for any a ∈ A(k) ⊗ Q, {x ∈ X(k) : π(x) = a and h(x) < ch(a) + c′ } is Zariski dense in X.