Quadratic twists of elliptic curves and class numbers

Abstract

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

We correct minor typographical errors, including the formula in the abstract and equation (1.3)