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

Lapkova, Kostadinka (2012) Effective lower bound for the class number of a certain family of real quadratic fields. Journal of Number Theory, 132 (12). pp. 2736-2747. ISSN 0022-314X

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Abstract

In this work we establish an effective lower bound for the class number of the family of real quadratic fields Q(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 (dN)=-1 for N=2 3{dot operator}3 3{dot operator}103{dot operator}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. © 2012 Elsevier Inc.

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