The narrow class groups of the \(\mathbb Z_{17}\)- and \(\mathbb Z_{19}\)-extensions over the rational field (Q977335)

Let \(p\) be a prime and \(\mathbb B_\infty\) the \(\mathbb Z_p\)-extension of \(\mathbb Q\). For brevity, let us denote the class-group of this field by \(C_p\), and the narrow class-group by \(C_p'\). Further, for any prime \(l\), \(C_p(l)\) and \(C_p'(l)\) will denote the \(l\)-parts of these groups. Iwasawa showed that \(C_p(p)\) vanishes; the article under review considers the vanishing of \(C_p(l)\) and \(C_p'(l)\) for \(l\neq p\) when \(p= 17\) or \(19\). In [[*] J. Lond. Math. Soc., II. Ser. 66, No. 2, 257--275 (2002; Zbl 1011.11072)], the first author showed that, given an arbitrary prime \(p\), if \(l\neq p\) is a prime that (a) is primitive mod \(p^2\) and (b) exceeds a certain explicit bound \(M_p\) depending on \(p\), then \(C_p(l)\) vanishes. The first author and both authors respectively built on this in [*] and [Acta Arith. 135, No. 2, 159--180 (2008; Zbl 1158.11046)] to remove the condition \(l> M_p\) when \(p\leq 13\). The main aim of the present article is to extend this to the primes \(17\) and \(19\). To summarize their results, they prove: Let \(p= 17\) or \(19\) and let \(l\) be a prime distinct from \(p\). If \(l\) is primitive mod \(p^2\) or \(l= 2\), then \(C_p'(l)\) vanishes. In particular, \(C_p(l)\) vanishes also. This is achieved by analysing the class number \(h_n\) of the \(n\)th field in the \(\mathbb Z_p\)-extension \(\mathbb B_\infty/\mathbb Q\), and in particular the integer \(h_n/h_{n-1}\). Indeed, they show that if \(l\) divides this integer, then \(l\) and \(n\) are bounded above by explicit integers. A part of their argument is to reduce to cases that can be readily checked on a computer, showing ultimately that \(l\) in fact cannot divide \(h_n/h_{n-1}\). Some remarks should be made: If \(l\) is odd, the vanishing of \(C_p'(l)\) is trivially equivalent to the vanishing of \(C_p(l)\), and the calculations are made on the latter. Otherwise, the authors instead use a result of \textit{J. V. Armitage} and \textit{A. Fröhlich} [Mathematika 14, 94--98 (1967; Zbl 0149.29501)] to reduce to proving the vanishing of \(C_p(2)\). Since \(2\) is not primitive mod \(17^2\), the vanishing of \(C_{17}(2)\) is treated separately.