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

TL;DR

This paper establishes effective lower bounds for the canonical heights of non-torsion points on elliptic curves over rationals using elliptic curve ideal class pairings and class number estimates.

Contribution

It introduces a method to bound heights of points on elliptic curves via ideal class pairings and class number functions, providing explicit inequalities.

Findings

Derived explicit lower bounds for canonical heights.

Connected elliptic curve points with class number and ideal class pairings.

Provided a logarithmic bound involving class number and torsion subgroup size.

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 $$\Psi_{E,-D}: E(\mathbb{Q})\times E_{-D}(\mathbb{Q})\mapsto \mathrm{CL}(-D).$$ In terms of the class number $H(-D)$ and $T_E(-D)$, a logarithmic function in $D$, we prove $$ \widehat{h}(P)> \frac{|E_{\mathrm{tor}}(\mathbb{Q})|^2}{\left( H(-D)+ |E_{\mathrm{tor}}(\mathbb{Q})|\right)^2}\cdot T_E(-D). $$