Modular forms and effective Diophantine approximation

Abstract

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.

Integral points on moduli schemes of elliptic curves

“…As already mentioned in the introduction, this corollary is an effective version of Frey's remark in [29, p. 544 The method used in [52] is similar to our proof of Corollary 7.2. To conclude the discussion, we point out that the results were obtained completely independently: We obtained the results of this paper without knowing anything of the related work of Hector Pasten and Ram Murty, and they obtained the results of [52] without knowing anything of our related work). for s = |S|.…”

“…It turns out that one can make Frey's remark in [29] effective and one obtains for example the following explicit result (see Corollary 7.2): Any solution (x, y) of the S-unit equation (1.1) satisfies h(x), h(y) 3 2 n S (log n S ) 2 + 65, n S = 2 7 N S . (After we submitted this paper, Hector Pasten informed us about his joint work with Murty [52] in which they independently obtain a (slightly) better version [52, Theorem 1.1] of the displayed height bound by using a similar method; see below Corollary 7.2 for more details. We would like to thank Hector Pasten for informing us.)…”

Integral points on curves: Siegel’s theorem after Siegel’s proof

“…However, to date such a method has not yet been extended to cover sensibly more general cases of Siegel's theorem. A further sophisticated method is in the paper by H. Pasten and M. Ram Murty [43], which uses results on modular forms, through an approach related to G. Frey's ideas concerning Fermat's Last Theorem.…”

“…4 For these equations and curves, an effective method alternative to Baker's was found by Bombieri; see [10]. (For a recent and completely different approach, working over Q, see also the already quoted paper [43]. )…”

On Some Applications of Diophantine Approximations

“…This allowed to prove effective bounds for the integral (or S-integral) points on certain classes of affine curves; these include curves defined by Thue's equations (cf. (For a recent and completely different approach, working over Q, see also the already quoted paper [43]. 4 For these equations and curves, an effective method alternative to Baker's was found by Bombieri; see [10].…”