On \({\mathbb{Z}}_ p\)-extensions of real quadratic fields (Q1073092)

The Greenberg conjecture in the case of a real quadratic field is investigated. Let p be an odd prime number and \(k={\mathbb{Q}}(\sqrt{m})\) a real quadratic field in which p splits. Two natural numbers \(n_ 1\) and \(n_ 2\) are defined from a prime ideal of k lying over p and the fundamental unit of k. It is shown that the Iwasawa invariants \(\mu_ p(k)\) and \(\lambda_ p(k)\) both vanish under certain conditions concerning \(n_ 1\), \(n_ 2\), the class number of k, and the ideal class group of \(k(e^{2\pi i/p})\). Examples are given in the table when \(p\leq 7\) and \(m<2000\).