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

Abstract: Let k be a number field, p$\ge$2 a prime and S a set of tame or wild finite places of k. We call K/k a totally S-ramified cyclic p-tower if Gal(K/k)=Z/p^NZ and if S non-empty is totally ramified. Using analogues of Chevalley's formula (Gras, Proc. Math. Sci. 127(1) (2017)),we give an elementary proof of a stability theorem (Theorem 3.1 for generalized p-class groups X_n of the layers k_n$\le$K:let $\lambda$=max(0, #S-1-$\rho$) given in Definition 1.1; then#X_n = #X_0 x p^{$\lambda$ n} for all n in [0,N], if and only if #X_1=#X_0 x p^$\lambda$. This improves the case $\lambda$ = 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 (e.g. Conjecture 4.1). Finally we apply our principles to the torsion groups T_n of abelian p-ramification theory. Numerical examples are given.