The multilinear support problem for products of abelian varieties and tori

Let G be the product of an abelian variety and a torus defined over a number field K. The aim of this paper is detecting the dependence among some given rational points of G by studying their reductions modulo all primes of K. We show that if some simple conditions on the order of the reductions of the points are satisfied then there must be a dependency relation over the ring of K-endomorphisms of G. We generalize Larsen's result on the support problem to several points on products of abelian varieties and tori.