Quadratic twists of elliptic curves and class numbers

JOURNAL OF NUMBER THEORY

卷:227

Article

Griffin, Michael¹  Ono, Ken²  Tsai, Wei-Lun² 

[1] Brigham Young Univ, Dept Math, 275 TMCB, Provo, UT 84602 USA

[2] Univ Virginia, Dept Math, Charlottesville, VA 22904 USA

关键词: Class numbers;    Elliptic curves;   

DOI  :  10.1016/j.jnt.2021.03.002

来源: Elsevier

【 摘 要 】

For positive rank r elliptic curves E(Q), we employ ideal class pairings E(Q) x 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 . vertical bar E-tor(Q)vertical bar/root R-Q(E) . pi(r(a)/2)/2(r(a))Gamma(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,epsilon) X1/2-epsilon. We give infinitely many cases where r(a) >= 6. These results can be viewed as an analogue of the classical estimate of Gouvea 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. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free