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

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 ( d N ) = −1 for N = 2 3 · 3 3 · 103 · 10 303. 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.

Class number problem for a family of real quadratic fields

“…for any integer m. In [La] and [BK1], there are similar results to Theorem 1.2 for subfamilies of narrow Richaud-Degert type. However, [La,Theorem 1.2] does not get an explicit lower bound and [BK1, Theorem 1.6] is less effective.…”