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After the work of G. Frey, it is known that an appropriate bound for the Faltings height of elliptic curves in terms of the conductor (Frey's height conjecture) would give a version of the ABC conjecture. In this paper we prove a partial result towards Frey's height conjecture which applies to all elliptic curves over Q, not only Frey curves. Our bound is completely effective and the technique is based in the theory of modular forms. As a consequence, we prove effective explicit bounds towards the ABC conjecture of similar strength to what can be obtained by linear forms in logarithms, without using the latter technique. The main application is a new effective proof of the finiteness of solutions to the S-unit equation (that is, S-integral points of P 1 − {0, 1, ∞}), with a completely explicit and effective bound, without using any variant of Baker's theory or the Thue-Bombieri method.

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Abstract. We prove a strong form of the "n Squares Problem" over polynomial rings with characteristic zero constant field. In particular we prove : for all r ≥ 2 there exists an integer M = M (r) depending only on r such that, if z 1 , z 2 , ..., z M are M distinct elements of F and we have polynomials, with some x i non-constant, satisfiying the equations x r i = (z i + f ) r + g for each i, then g is the zero polynomial.