Heights of points on elliptic curves over ℚ

Abstract

In this note we obtain effective lower bounds for the canonical heights of non-torsion points on E ( Q ) E(\mathbb {Q}) by making use of suitable elliptic curve ideal class pairings Ψ E , − D : E ( Q ) × E − D ( Q ) ↦ C L ( − D ) . \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 ) H(-D) and T E ( − D ) T_E(-D) , a logarithmic function in D D , we prove h ^ ( P ) > | E t o r ( Q ) | 2 ( H ( − D ) + | E t o r ( Q ) | ) 2 ⋅ T E ( − D ) . \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*}