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

Comments. 1) This paper was published as Proceedings of the American Mathematical Society 149 (2021), 5093-5100. 2) After the paper was published, we discovered a typographical error in the example after Theorem 1.1. The number 0.043 should be 0.035

References in corpus (1)

• Small rational points on elliptic curves over number fields

Cited by in corpus (1)

• Distribution of values of Gaussian hypergeometric functions