Effective lower bound for the class number of a certain family of real quadratic fields

JOURNAL OF NUMBER THEORY

卷:132

Article

Lapkova, Kostadinka

关键词: Class number;    Real quadratic fields;    Elliptic curves;   

DOI  :  10.1016/j.jnt.2012.05.029

来源: Elsevier

【 摘 要 】

In this work we establish an effective lower bound for the class number of the family of real quadratic fields Q(root d), where d = n(2) + 4 is a square-free positive integer with n = m (m(2) - 306) for some odd m, with the extra condition (d/N) = -1 for N = 2(3) . 3(3.) 103 . 10303. This result can be regarded as a corollary of a theorem of Goldfeld and some calculations involving elliptic curves and local heights. The lower bound tending to infinity for a subfamily of the real quadratic fields with discriminant d = n(2) + 4 could be interesting having in mind that even the class number two problem for these discriminants is not yet solved unconditionally. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free