A deep conjecture on torsion anomalous varieties states that if is a weak-transverse variety in an abelian variety, then the complement of all -torsion anomalous varieties is open and dense in . We prove some cases of this conjecture. We show that the -torsion anomalous varieties of relative codimension one are non-dense in any weak-transverse variety embedded in a product of elliptic curves with CM. We give explicit uniform bounds in the dependence on . As an immediate consequence we prove the conjecture for of codimension two in a product of CM elliptic curves. We also point out some implications on the effective Mordell-Lang Conjecture.
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Sara Checcoli, Francesco Veneziano, Evelina Viada, On torsion anomalous intersections. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 25 (2014), no. 1, pp. 1–36