Height of rational points on quadratic twists of a given elliptic curve (Q2788658)

Let \(E_{/\mathbb{Q}}\) be an elliptic curve with Weierstrass equation \(y^2=x^3+Ax+B\) and, for any squarefree integer \(d\geqslant 1\), let \(E_d\) be its quadratic twist \(dy^2=x^3+Ax+B\). Let \(\hat{h}_{E_d}\) be the canonical height on \(E_d\) and define NEWLINE\[NEWLINE \log \eta_d(A,B):= \min\{\hat{h}_{E_d}(P)\,:\,P\in E(\mathbb{Q})-E(\mathbb{Q})_{\mathrm{tors}}\} NEWLINE\]NEWLINE when rank\(_\mathbb{Z}E(\mathbb{Q})\geqslant 1\) and \(\eta_d(A,B)=\infty\) otherwise. The paper presents a nice (conjectural) analogy between \(\eta_d(A,B)\) and the fundamental unit \(e_D\) of a real quadratic field \(\mathbb{Q}(\sqrt{D})\): namely they are conjectured to verify \(\eta_d(A,B)>e^{d^{1/2-\varepsilon}}\) (for all \(\varepsilon>0\) and almost all squarefree \(d\geqslant 1\)) and \(e_D>e^{D^{1/2-\varepsilon}}\) (for all \(\varepsilon>0\) and almost all fundamental discriminant \(D\geqslant 1\)).NEWLINENEWLINEThe author proves some bounds for \(\eta_d(A,B)\) far from the conjectured one but analogous to the available ones for \(e_D\) (see, e.g., [\textit{É. Fouvry} and \textit{F. Jouve}, Math. Z. 273, No. 3--4, 869--882 (2013; Zbl 1320.11026)]). Restricting to rank one curves and using a parametrization for points in \(E_d\), the author shows that \(\eta_d(A,B)>d^{1/4-\varepsilon}\) and is able to improve to \(\eta_d(A,B)>d^{5/8-\varepsilon}\) whenever the curves \(E_d\) have rational 2-torsion (like, for example, \((A,B)=(-1,0)\,\)) using explicit formulas for a full 2-descent on \(E_d\).