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This paper focuses on the proof of Serge Lang's Heights Conjecture in a formthat is completely effective. As a complementary result the author provides anew proof of Mazur-Merel theorem about a bound for the torsion of ellipticcurves in terms of the degree of the ground field and improves the known resultby providing a small polynomial bound in terms of this degree. (Submitted)

Proof of Serge Lang's Heights Conjecture and an almost optimal Bound for the Torsion of Elliptic Curves