On Lang's conjecture on the lower bound of the Néron-Tate height for elliptic curves over \(\mathbb Q\) (Q2759141)

Let \(E/\mathbb{Q}\) be an elliptic curve and \(\widehat{h}:E(\mathbb{Q})\to\mathbb{R}\) its canonical height. Lang's conjecture asks for a real constant \(C>0\) such that, for every elliptic curve \(E/\mathbb{Q}\) with minimal discriminant \(\Delta_E\), every \(P\in E(\mathbb{Q})\) of infinite order satisfies \(\widehat{h}(P)>C\log|\Delta_E|\). For results in this direction see \textit{M. Hindry} and \textit{J. Silverman} [Invent. Math. 93, 419--450 (1988; Zbl 0657.14018)] and \textit{S. David} [J. Number Theory 64, 104--129 (1997; Zbl 0873.11035)]. NEWLINENEWLINENEWLINEThe author explicitly states the constant \(C\) for certain families of elliptic curves over \(\mathbb{Q}\). For instance, it is shown that one can take \(C\) to be \(10^{-3}\) if \(\Delta_E\) is square-free, \(10^{-5}\) if the conductor of \(E\) is a prime number, and \(10^{-6}\) if \(j_E\neq 0,1728\) is an integer. In the case where \(j_E=0\), respectively \(j_e=1728\), \(E/\mathbb{Q}\) has a Weierstrass equation of the form \(y^2=x^3+d\), respectively \(y^2=x^3+dx\), where \(d\) is an \(6\)-th power free, respectively \(4\)-th power free, integer, and in this case the lower bound obtained is \(\widehat{h}(P)>10^{-3}\log|d|+10^{-3}\), respectively \(\widehat{h}(P)>\frac 1{64}\log|d|\). In the case where \(d=-n^2\), \textit{A. Bremner, J. Silverman} and \textit{N. Tzanakis} [Integral points in arithmetic progression on \(y^2=x(x^2-n^2)\), J. Number Theory 80, 187--208 (2000; Zbl 1009.11035), Proposition 2.1] improved Proposition 4.1 of the paper under review.