An Explicit Manin-Dem'janenko Theorem in Elliptic Curves

Let e be a curve of genus at least 2 embedded in E-1 x . . . x E-N, where the E-i are elliptic curves for i = 1, . . . , N. In this article we give an explicit sharp bound for the Neron-Tate height of the points of C contained in the union of all algebraic subgroups of dimension < max(r(e) - t(e) , t(e)), where to (resp. r(e)) is the minimal dimension of a translate (resp. of a torsion variety) containing e. As a corollary, we give an explicit bound for the height of the rational points of special curves, proving new cases of the explicit Mordell Conjecture and in particular making explicit (and slightly more general in the CM case) the Manin-Dem'janenko method for curves in products of elliptic curves.