On the problem of detecting linear dependence for products of abelian varieties and tori

Abstract

Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a point in G(K) and let L be a finitely generated subgroup of G(K). Suppose that for all but finitely many primes p of K the point (R mod p) belongs to (L mod p). Does it follow that R belongs to L? We answer this question affirmatively in three cases: if L is cyclic; if L is a free left End_K G-submodule of G(K); if L has a set of generators (as a group) which is a basis of a free left End_K G-submodule of G(K). In general we prove that there exists an integer m (depending only on G, K and the rank of L) such that mR belongs to the left End_K G-submodule of G(K) generated by L.