Lang's conjecture and sharp height estimates for the elliptic curves \(y^2 = x^3 + ax\) (Q2840294)

Let \(E/\mathbb{Q}\) be an elliptic curve with minimal discriminant \(\mathfrak{D}_{E/\mathbb{Q}}\). It was conjectured by \textit{S. Lang} [Elliptic curves: Diophantine analysis. New York: Springer-Verlag (1978; Zbl 0388.10001)] that there exist constants \(C_1>0\) and \(C_2\) such that for any non-torsion point \(P\in E(\mathbb{Q})\), NEWLINE\[NEWLINE\hat{h}_E(P)\geq C_1\log\mathfrak{D}_{E/\mathbb{Q}}-C_2.NEWLINE\]NEWLINENEWLINENEWLINEThe authors establish an explicit bound of this form for the family of elliptic curves \(E_a:y^2=x^3+ax\), for \(a\in\mathbb{Z}\) fourth-power free. As noted in the paper, Lang's conjecture is known to hold for families of twists of a given curve (such as the family \(E_a\)), and explicit bounds can be deduced from a work of \textit{M. Hindry} and \textit{J. H. Silverman} [Invent. Math. 93, No. 2, 419--450 (1988; Zbl 0657.14018)] and subsequent improvements. The bounds so obtained, however, are far from the truth. The main result of this paper, extending a work of \textit{A. Bremner} et al. [J. Number Theory 80, No. 2, 187--208 (2000; Zbl 1009.11035)], is the following lower bound for non-torsion points \(P\in E_a(\mathbb{Q})\). NEWLINE\[NEWLINE\hat{h}_{E_a}(P)\geq \frac{1}{16}\log|a|-C_a,NEWLINE\]NEWLINE where \(C_a\) is given explicitly and depends on the sign of \(a\), and its residue class modulo 64. The authors also construct classes of examples to show that this lower bound is essentially the best possible.