Heights of points on elliptic curves over $\mathbb {Q}$

Abstract:

In this note we obtain effective lower bounds for the canonical heights of non-torsion points on $E(\mathbb {Q})$ by making use of suitable elliptic curve ideal class pairings \begin{equation*} \Psi _{E,-D}: \ E(\mathbb {Q})\times E_{-D}(\mathbb {Q})\mapsto \mathrm {CL}(-D). \end{equation*} In terms of the class number $H(-D)$ and $T_E(-D)$, a logarithmic function in $D$, we prove \begin{equation*} \widehat {h}(P)> \frac {|E_{\mathrm {tor}}(\mathbb {Q})|^2}{\left ( H(-D)+ |E_{\mathrm {tor}}(\mathbb {Q})|\right )^2}\cdot T_E(-D). \end{equation*}