On torsion anomalous intersections

A deep conjecture on torsion anomalous varieties states that if V is a weak-transverse variety in an abelian variety, then the complement V^{ta} of all V-torsion anomalous varieties is open and dense in V. We prove some cases of this conjecture. We show that the V-torsion anomalous varieties of relative codimension one are non-dense in any weak-transverse variety V embedded in a product of elliptic curves with CM. We give explicit uniform bounds in the dependence on V. As an immediate consequence we prove the conjecture for V of codimension two in a product of CM elliptic curves. We also point out some implications on the effective Mordell-Lang Conjecture.