On the λ-stability of p-class groups along cyclic p-towers of a number field

Abstract

Let [Formula: see text] be a number field, [Formula: see text] a prime and [Formula: see text] a set of finite places of [Formula: see text]. We call [Formula: see text] a totally [Formula: see text]-ramified cyclic [Formula: see text]-tower if [Formula: see text] and if [Formula: see text] is totally ramified. Using analogues of Chevalley’s formula [G. Gras, Invariant generalized ideal classes—Structure theorems for p-class groups in p-extensions, Proc. Math. Sci. 127(1) (2017) 1–34], we give an elementary proof of a stability theorem (Theorem 3.1) for generalized [Formula: see text]-class groups [Formula: see text] of the layers [Formula: see text]: let [Formula: see text] given in Definition 1.1 ([Formula: see text] for ordinary class groups); then [Formula: see text] for all [Formula: see text], if and only if [Formula: see text]. This improves the case [Formula: see text] of [T. Fukuda, Remarks on [Formula: see text]-extensions of number fields, Proc. Japan Acad. Ser. A 70(8) (1994) 264–266; J. Li, Y. Ouyang, Y. Xu and S. Zhang, l-Class groups of fields in Kummer towers, Publ. Mat. 66(1) (2022) 235–267; Y. Mizusawa, K. Yamamoto, On [Formula: see text]-class groups of relative cyclic [Formula: see text]-extensions, Arch. Math. 117(3) (2021) 253–260] whose techniques are based on Iwasawa’s theory or Galois theory of pro-[Formula: see text]-groups. We deduce promising capitulation properties of [Formula: see text] in the tower giving Conjecture 4.1. Finally, we apply our principles to the torsion groups [Formula: see text] of abelian [Formula: see text]-ramification theory. Numerical examples are given with PARI programs.