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Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture. (English) Zbl 1294.11113
Assuming the Birch and Swinnerton-Dyer conjecture (the precise version with the explicit constant), this paper proves an explicit upper bound for the $$x$$-coordinates of $$S$$-integral points on an elliptic curve (Theorem 3.4). The bound is in terms of the discriminant and the degree of the number field. As a consequence, it follows from the Chevalley-Weil theorem that the radical of $$abc$$ has to be large when $$a+b=c$$. The inequality they obtain is worse than the published results (e.g. [K. Györy, Acta Arith. 133, No. 3, 281–295 (2008; Zbl 1188.11011)]), but this paper brings to light a connection between the BSD and the $$abc$$ conjectures.
To prove Theorem 3.4, the authors quote their earlier work [Ramanujan J. 32, No. 1, 125–141 (2013; Zbl 1362.11065)] bounding the $$S$$-integral points in terms of the regulator of the elliptic curve and the Néron-Tate heights of the generators of the Mordell-Weil group. This is where they use the known estimates of linear forms in elliptic logarithms ([S. David, Mém. Soc. Math. Fr., Nouv. Sér. 62, 143 p. (1995; Zbl 0859.11048)]. The second author [“On the Mordell-Weil group and the Tate-Shafarevich group of an abelian variety”, Preprint, arXiv:0801.1054] has previously obtained an upper bound for the regulator in terms of the discriminant assuming BSD, and this in turn bounds the heights of the generators.

##### MSC:
 11G50 Heights 11G05 Elliptic curves over global fields 11J86 Linear forms in logarithms; Baker’s method 14G05 Rational points 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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