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Old and new conjectured Diophantine inequalities. (English) Zbl 0714.11034
This paper is a general survey of certain Diophantine conjectures of current interest, and relations between them.
In this case, the discussion revolves around the Szpiro conjecture relating the modular height and conductor of elliptic curves defined over a fixed number field. The author shows that this is equivalent to the “\(abc\)” conjecture (if \(a,b,c\in\mathbb Z\) are relatively prime with \(a+b+c=0\), then for all \(\varepsilon >0\), \[ \max (| a|,| b|,| c|)\ll_{\varepsilon}\prod_{p| abc}p^{1+\varepsilon}) \] and to a generalized Hall conjecture (giving lower bounds for \(| x^ m-y^ n|\), \(x,y,m,n\in\mathbb Z)\). He also shows that these conjectures would follow from Conjecture 5.5.0.1 of the reviewer [Diophantine approximations and value distribution theory, Lect. Notes Math. 1239 (1987; Zbl 0609.14011), see Appendix 5.ABC for another treatment of the above implications].
This paper concludes by discussing two possible consequences of the Szpiro conjecture. First, in certain cases it is possible to prove upper bounds on the number of torsion elements in \(E(K)\), where \(K\) is a fixed number field and \(E\) is an elliptic curve defined over \(K\). Also, the author shows that Szpiro’s conjecture implies a lower bound on the canonical height \(h(P)\) for a non-torsion point \(P\) on an elliptic curve over \(\mathbb Q\), in terms of the discriminant of that curve. For more on these two results, see M. Hindry and J. H. Silverman [Invent. Math. 93, No. 2, 419–450 (1988; Zbl 0657.14018) and Sémin. Théor. Nombres, Paris/Fr. 1987–1988, Prog. Math. 81, 119–129 (1990; Zbl 0714.14023)].

MSC:
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11D75 Diophantine inequalities
11D41 Higher degree equations; Fermat’s equation
11G05 Elliptic curves over global fields
14H25 Arithmetic ground fields for curves
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14G25 Global ground fields in algebraic geometry
11J25 Diophantine inequalities
14H52 Elliptic curves
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