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arXiv · 2007.09514

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

Michael Griffin·Ken Ono·Wei-Lun Tsai
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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 $$ \widehat{h}(P)> \frac{|E_{\mathrm{tor}}(\mathbb{Q})|^2}{\left( H(-D)+ |E_{\mathrm{tor}}(\mathbb{Q})|\right)^2}\cdot T_E(-D). $$

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Michael Griffin, Ken Ono, Wei-Lun Tsai. 2020-07-18. Heights of points on elliptic curves over $\mathbb Q$. https://arxiv.org/abs/2007.09514

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