On the Iwasawa \(\lambda\)-invariant of the cyclotomic \(\mathbb Z_2\)-extension of a real quadratic field (Q2566553)

In [Manuscr. Math. 94, No. 4, 437--444 (1997; Zbl 0935.11040)] \textit{M. Ozaki} and \textit{H. Taya} study the \(\lambda\)--invariant of the cyclotomic \({\mathbb Z}_ 2\)-extension of a real quadratic field \(k\). In that paper \(k\) was taken of the form \(k= {\mathbb Q}(\sqrt{m})\) or \({\mathbb Q}(\sqrt{2m})\) and \(m=p\) or \(pq\) with \(p\) and \(q\) prime numbers for various cases of \(p, q \pmod 8\). In the paper under review, the authors study the \(\lambda\)-invariant of the cyclotomic \({\mathbb Z}_ 2\)-extension of \(k={\mathbb Q}(\sqrt{pq})\) with \(p\equiv 3\bmod 8\), \( q\equiv 1\bmod 8\) and the Legendre symbol \(({q\over p})\) equal to \(-1\). The results obtained are: 1) if \(q\equiv 1\bmod 2^ {m+2}\) and \(q\not\equiv 1\bmod 2^ {m+3}\), then \(\lambda\) is zero or \(2^ m\); 2) If \(2^ {q-1\over 4}\not\equiv 1\bmod q\), then \(\lambda\) is zero.