On Iwasawa \(\lambda_3\)-invariants of cyclic cubic fields of prime conductor (Q2723538)

Let \(k\) be a cubic cyclic field of prime conductor \(p\), and assume that \(p\) splits in \(k\). In [Acta Arith. 97, 387--398 (2001; Zbl 1059.11064)], \textit{M. Ozaki} and \textit{G. Yamamoto} showed that \(\lambda_3(k) = \mu_3(k) = 0\) for certain \(k\), where \(\lambda_3\) and \(\mu_3\) denote the Iwasawa invariants of the cyclotomic \(\mathbb Z_3\)-extension of \(k\). In this paper, some of the cases left open in the article above are covered. NEWLINENEWLINENEWLINEChecking that the \(\lambda\)-invariant vanishes requires showing that certain ideal classes capitulate; the capitulation of ideals in the layers of the \(\mathbb Z_3\)-extension of \(k\) is governed by certain unit groups, and it is by computing these unit groups (involving abelian fields of degree \(27\)) that the authors succeed in showing that \(\lambda_3(p) = 0\) for primes \(p \equiv 10, 19 \bmod 27\) below \(10,000\).