Existence of certain fundamental discriminants and class numbers of real quadratic fields

Abstract

Let D be the fundamental discriminant of the quadratic field Q( √ D), h(D) its class number, and χD := ( D · ) the usual Kronecker character. Let p be prime, Zp the ring of p-adic integers, and λp(Q( √ D)) the Iwasawa λ-invariant of the cyclotomic Zp-extension of Q( √ D). Let Rp(D) denote the p-adic regulator of Q( √ D), and | · |p denote the usual multiplicative p-adic valuation normalized so that |p|p = 1 p . In [9], by applying Sturm’s theorem on the congruence of modular forms to Cohen’s half integral weight modular forms, Ono proved the following theorem.