An analogue of a conjecture of Mazur: a question in Diophantine approximation on tori

B. Mazur has considered the question of density in the Euclidean topology of the set of Q-rational points on a variety X defined over Q, in particular for Abelian varieties. In this paper we consider the question of closures of the image of finitely generated subgroups of T (Q) in Γ\T (R) where T is a torus defined over Q, Γ an arithmetic subgroup such that Γ\T (R) is compact. Assuming Schanuel’s conjecture, we prove that the closures correspond to algebraic sub-tori of T . Let V be a smooth algebraic variety over Q. The set V (R) acquires a topological structure from the Euclidean topology of R. It is known that V (R) has finitely many connected components. If V (Q) is Zariski dense in V , it was conjectured by B. Mazur, cf. [M1] and [M2], that the closure of V (Q) in V (R) is a finite union of connected components of V (R). This conjecture was shown to be false in this generality by Colliot-Thélène, Skorobogatov, and SwinnertonDyer for an elliptic surface who have proposed a slightly reformulated conjecture, cf. [CSS]. However, the present evidence seems to suggest that the following special case of Mazur’s conjecture is true. Conjecture 1 (Mazur’s conjecture for Abelian varieties): Let A be an abelian variety over Q, and G a subgroup of A(Q). Then the closure of G in the Euclidean topology of A(R) contains B(R) as a subgroup of finite index for a certain abelian subvariety B defined over Q.