论文标题
椭圆曲线上的点高度高于$ \ mathbb q $
Heights of points on elliptic curves over $\mathbb Q$
论文作者
论文摘要
在本说明中，我们通过使用合适的椭圆曲线理想的类配对$ ch _ { \ Mathrm {cl}（-d）。$$在类号码$ h（-d）$和$ t_e（-d）$，$ d $中的对数函数中\ frac {| e _ {\ mathrm {tor}}（\ Mathbb {q}）|^2} {\ left（h（h（-d）+ | e _ {\ mathrm {tor}}}}}}}}（\ mathbb {q}） $$
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 $$Ψ_{E,-D}: E(\mathbb{Q})\times E_{-D}(\mathbb{Q})\mapsto \mathrm{CL}(-D).$$ In terms of the class number $H(-D)$ and $T_E(-D)$, a logarithmic function in $D$, we prove $$ \widehat{h}(P)> \frac{|E_{\mathrm{tor}}(\mathbb{Q})|^2}{\left( H(-D)+ |E_{\mathrm{tor}}(\mathbb{Q})|\right)^2}\cdot T_E(-D). $$
