Quadratic twists of elliptic curves and class numbers

For positive rank $r$ elliptic curves $E(\mathbb{Q})$, we employ ideal class pairings $E(\mathbb{Q})\times E_{-D}(\mathbb{Q}) \rightarrow \mathrm{CL}(-D),$ for quadratic twists $E_{-D}(\mathbb{Q})$ with a suitable ``small $y$-height'' rational point, to obtain effective class number lower bounds. For the curves $E^{(a)}: \ y^2=x^3-a,$ with rank $r(a),$ this gives $h(-D) \geq \frac{1}{10}\cdot \frac{|E_{\mathrm{tor}}(\mathbb{Q})|}{\sqrt{R_{\mathbb{Q}}(E)}}\cdot \frac{Γ\left (\frac{r(a)}{2}+1\right)}{(4π)^{\frac{r(a)}{2}}} \cdot \frac{\log(D)^{\frac{r(a)}{2}}}{\log \log D},$ representing an improvement to the classical lower bound of Goldfeld, Gross and Zagier when $r(a)\geq 3$. We prove that the number of twists $E_{-D}^{(a)}(\mathbb{Q})$ with such a point (resp. with such a point and rank $\geq 2$ under the Parity Conjecture) is $\gg_{a,\varepsilon} X^{\frac{1}{2}-\varepsilon}.$ We give infinitely many cases where $r(a)\geq 6$. These results can be viewed as an analogue of the classical estimate of Gouvêa and Mazur for the number of rank $\geq 2$ quadratic twists, where in addition we obtain ``log-power'' improvements to the Goldfeld-Gross-Zagier class number lower bound.

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