Quadratic twists of elliptic curves and class numbers

Abstract

For positive rank r elliptic curves E(Q), we employ ideal class pairingsE(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):y2=x3−a, with rank r(a), this givesh(−D)≥110⋅|Etor(Q)|RQ(E)⋅πr(a)22r(a)Γ(r(a)2+1)⋅log⁡(D)r(a)2log⁡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,εX12−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.