ON THE p-CLASS GROUP STABILITY ALONG CYCLIC p-TOWERS OF A NUMBER FIELD
Résumé
Let k be a number field, p≥2 a prime and S any set of 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 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 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. 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) |
---|