On \(p\)-adic zeta functions and \(\mathbb{Z}_p\)-extensions of certain totally real number fields
Let \(k\) be a totally real number field. For an odd prime \(p\) and for \(n \geq 0\), let \(A_n\) denote the Sylow \(p\)-group of the \(n\)th layer in the cyclotomic \(\mathbb{Z}_p\)-extension of \(k\). Denote by \(A_n^\Gamma\) the Galois-invariant part of \(A_n\). The author proves that if \(p\) splits completely in \(k\) and Leopoldt's conjecture is true for \(k\) and \(p\), then card\((A_n ^\Gamma)\) equals the \(p\)-part of \(\zeta^*(1,k)\), for every sufficiently large \(n\), where \(\zeta^*(s,k)=\zeta (s,k)/ \zeta (s,\mathbb{Q})\), the quotient of the \(p\)-adic zeta functions of \(k\) and \(\mathbb{Q}\). Moreover, this \(p\)-part is equal to card\((A_0)\) times the \(p\)-part of \(R_p(k) - [k:\mathbb{Q}] + 1\), where \(R_p(k)\) denotes the \(p\)-adic regulator and \([k:\mathbb{Q}]\) the absolute degree of \(k\). This generalizes a result proved by \textit{T. Fukuda} and \textit{K. Komatsu} [J. Number Theory 23, 238-242 (1986; Zbl 0593.12003)] for real quadratic fields \(k\). As a consequence the author provides an alternative formulation for Greenberg's theorem on the nonvanishing of the Iwasawa invariants \(\lambda\) and \(\mu\) under the above assumptions. He also gives some computational data in the case \([k:\mathbb{Q}]=3\) and \(p=3\).
scientific article; zbMATH DE number 1321871