On the λ-stability of p-class groups along cyclic p-towers of a number field
Résumé
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^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≤K:
let λ=max(0, #S-1-ρ) given in Definition 1.1; then
#X_n = #X_0 x p^{λ n} for all n in [0,N], if and only if #X_1=#X_0 x p^λ.
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 (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.
Origine : Fichiers produits par l'(les) auteur(s)