Class numbers and Iwasawa invariants of certain totally real number fields
The author studies the density of real quadratic fields \(\mathbb{Q}(\sqrt D)\) with discriminant \(D\) and class number coprime to a given prime \(p>3\). Let \(S(X)\) denote the set of all such fields in the range \(0<D<X\). \textit{K. Ono} [Compos. Math. 119, 1-11 (1999; Zbl 1002.11080)] found, under a mild condition, a family of fields in \(S(X)\) large enough to prove that \(\# S(X)\gg_p \sqrt X/\log X\). The present author shows that this estimate is unconditionally true. He follows Ono's proof, which uses a construction by \textit{H. Cohen} [Math. Ann. 217, 271-285 (1975; Zbl 0311.10030)] of a half-integral weight Eisenstein series whose Fourier coefficients are given by generalized Bernoulli numbers attached to quadratic characters. The fields in \(S(X)\) found by him are different, however: unlike Ono, he constructs fields in which \(p\) is unramified. As noticed by both authors, the conditions they impose on the fields \({\mathbb{Q}}(\sqrt D) \in S(X)\) imply that the Iwasawa \(\lambda_p\)-invariant of \({\mathbb{Q}}(\sqrt D)\) vanishes. So their results give some support to Greenberg's conjecture that \(\lambda_p=0\) for all totally real fields.
scientific article; zbMATH DE number 1420253
scientific article