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

Let k be a number field, p >= 2 a prime and S not equal empty set a set of finite places of k. We call K/k a totally S-ramified cyclic p-tower if Gal(K/k) similar or equal to Z/p(N)Z and if S 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 p-class groups X-n of the layers k(n) subset of K: let lambda = max(0, #S - 1 - rho) given in Definition 1.1 (rho = r(1)(k)+r(2)(k) - 1 for ordinary class groups); then #X-n = #X-0.p(lambda.n) for all n is an element of [0, N], if and only if #X-1 = #X-0 . p(lambda). This improves the case lambda = 0 of [T. Fukuda, Remarks on Z(p)-extensions of number fields, Proc. Japan Acad. Ser. A70(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 p-class groups of relative cyclic p-extensions, Arch. Math.; 117(3) (2021) 253-260] whose techniques are based on Iwasawa's theory or Galois theory of pro-p-groups. We deduce promising capitulation properties of X-0 in the tower giving Conjecture 4.1. Finally, we apply our principles to the torsion groups J(n) of abelian p-ramification theory. Numerical examples are given with PARI programs.