On the Iwasawa invariants of totally real number fields
The author finds conditions guaranteeing the vanishing of the Iwasawa invariants \(\lambda\) and \(\mu\), for an odd prime \(\ell\), of a totally real number field \(k\). Let \(K= k(e^{2\pi i/\ell})\) and denote by \(k_ n\) and \(K_ n\) the \(n\)th layers of the cyclotomic \(\mathbb{Z}_ \ell\)- extensions of \(k\) and \(K\), respectively \((n\geq 1)\). For \(K_ n\), let \(C_ n\) denote the \(\ell\)-class group factored by the subgroup generated by all ideals whose prime factors divide \(\ell\). In the group ring \(\mathbb{Z}_ \ell [\text{Gal}(K/k)]\) let \(\varepsilon\) be the idempotent determined by the Teichmüller character. The main result is that \(\lambda= \mu=0\) if either of the following conditions (A) and (B) is satisfied; (A) \(\varepsilon C_ 1= \{1\}\) and no prime dividing \(\ell\) decomposes completely in \(K/k\); (B) there is \(m\geq 1\) such that \(\varepsilon C_{m+1}= \{1\}\) and no prime dividing \(\ell\) decomposes in \(K_{m+1}/ K_ m\). Under the condition (B) it is also proved that the Leopoldt conjecture holds for \(\ell\) in \(k_ n\) for every \(n\). The proofs are based on results by G. Gras and J.-F. Jaulent; see, in particular, \textit{J.-F. Jaulent's} thesis [Publ. Math. Fac. Sci. Besançon, Théor. Nombres Années 1984/1985-1985/1986, No. 1, 349 p. (1986; Zbl 0601.12002)].
scientific article; zbMATH DE number 492773