On zeta-functions and cyclotomic \({\mathbb{Z}}_ p\)-extensions of algebraic number fields (Q762210)

Let \(\zeta_ k\) and \(\zeta_{k'}\), be Dedekind zeta functions of function fields k and k' respectively. Then Tate and Turner gave a necessary and sufficient condition for \(\zeta_ k=\zeta_{k'}\), by using Jacobian varieties J(\({\mathcal C})\) and J(\({\mathcal C}')\) of complete non-singular curves \({\mathcal C}\) and \({\mathcal C}'\) with function fields isomorphic to k and k' respectively [cf. \textit{J. Tate}, Invent. Math. 2, 134-144 (1966; Zbl 0147.203); and \textit{S. Turner}, Bol. Soc. Bras. Mat. 9, 89-95 (1978; Zbl 0427.12011)]. In this paper, the author intends to get an analogue of this result in algebraic number fields. Namely, for Dedekind zeta functions \(\zeta_ k\) and \(\zeta_{k'}\), of algebraic number fields k and k' respectively, he gives some necessary conditions for \(\zeta_ k=\zeta_{k'}\) by using the Galois group of the maximal (unramified) abelian pro-p extension over \(k_{\infty}\) \((k'_{\infty})\) or the Pontryagin dual of the p-primary subgroup of the ideal class group of \(k_{\infty}\) \((k'_{\infty})\), where \(k_{\infty}\) is the cyclotomic \({\mathbb{Z}}_ p\)-extension of k.