Quadratic twists of elliptic curves and class numbers.

For positive rank r elliptic curves E (Q) , we employ ideal class pairings E (Q) × E − D (Q) → CL (− D) , for quadratic twists E − D (Q) with a suitable "small y -height" rational point, to obtain explicit class number lower bounds that improve on earlier work by the authors. For the curves E (a) : y 2 = x 3 − a , with rank r (a) , this gives h (− D) ≥ 1 10 ⋅ | E tor (Q) | R Q (E) ⋅ π r (a) 2 2 r (a) Γ (r (a) 2 + 1) ⋅ log ⁡ (D) r (a) 2 log ⁡ log ⁡ D , representing a general improvement to the classical lower bound of Goldfeld, Gross and Zagier when r (a) ≥ 3. We prove that the number of twists E − D (a) (Q) with such a suitable point (resp. with such a point and rank ≥2 under the Parity Conjecture) is ≫ a , ε X 1 2 − ε. We give infinitely many cases where r (a) ≥ 6. These results can be viewed as an analogue of the classical estimate of Gouvêa and Mazur for the number of rank ≥2 quadratic twists, where in addition we obtain "log-power" improvements to the Goldfeld-Gross-Zagier class number lower bound. [ABSTRACT FROM AUTHOR]