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

Abstract

Let k be a number field, p ≥ 2 a prime and S a set of tame or wild finite places of k. We call K/k a totally S-ramified cyclic p-tower if Gal(K/k) ≃ Z/p N Z and if S = ∅ is totally ramified. Using analogues of Chevalley's formula (Gras, Proc. Math. Sci. 127(1) (2017)), we give an elementary proof of a stability theorem (Theorem 3.1) for generalized p-class groups Xn of the layers kn ⊆ K:This improves the case λ = 0 of Fukuda (1994), Li-Ouyang-Xu-Zhang (2020), Mizusawa-Yamamoto (2020), whose techniques are based on Iwasawa's theory or Galois theory of pro-p-groups. We deduce capitulation properties of X 0 in the tower. Finally we apply our principles to the torsion groups Tn of abelian p-ramification theory. Numerical examples are given. Contents 15 7.2. Case of real cyclic p-towers with tame ramification 16 References 17