On the vanishing of Iwasawa invariants of certain cyclic extensions of \(\mathbb{Q}\) with prime degree. II
Let \(\ell\) be an odd prime number. For a prime number \(p\) congruent to \(1\) modulo \(\ell\), let \(k _p\) denote the unique subfield of \({{\mathbb Q}} (\zeta _p)\) of degree \(\ell\) over \({\mathbb Q}\), where \(\zeta _p\) is a primitive \(p\)-th root of unity. Let \(({a \over p}) _\ell\) be the \(\ell\)-th power residue symbol for an integer \(a\). In Part I [\textit{T. Fukuda}, Proc. Japan Acad., Ser. A 73, No. 6, 108-110 (1997; Zbl 0899.11052)] the author finds, for two different prime numbers \(p\) and \(q\) congruent to \(1\) modulo \(\ell\) satisfying \(({\ell \over p}) _\ell \neq 1\), \( ({p \over q}) _\ell\neq 1\) and \(q\not\equiv 1 \bmod \ell ^2\), conditions under which the Iwasawa invariants \(\lambda _\ell (k)\), \(\mu _\ell (k)\) are \(0\) for any subfield \(k\) of \(k _p k_q\) of degree \(\ell\) over \({\mathbb Q}\). In this paper the author investigates the case \(({p \over q}) _\ell = 1\). Two cases are considered separately: when \(({\ell \over q}) _\ell = 1\) and when \(({\ell \over q}) _\ell \neq 1\). It is assumed that \(({\ell \over p}) _\ell \neq 1\), \(({q \over p}) _\ell = ({p \over q}) _\ell = 1\), \(p \not\equiv 1 \bmod \ell ^2\) and \(q \not\equiv 1 \bmod \ell ^2\). When \(({\ell \over q}) _\ell =1 \) and \(k\) is such that \(p\not\in E _k k ^{\times ^\ell}\), \(E _k\) the unit group of \(k\), it is proved that \(\lambda _\ell (k) = \mu _\ell (k) = 0\). When \(({\ell \over q}) _\ell \neq 1\) conditions on \(k\) are found for \(\lambda _\ell (k)\) and \(\mu _\ell (k)\) to be \(0\). The author also provides a correction to a gap in his previous paper. Finally, for \(\ell = 3\), the conditions of the results proved here are tested for all \(p\) and \(q\) satisfying \(p q < 100 000\).
scientific article; zbMATH DE number 1275060
scientific article