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

TL;DR

This paper proves a stability theorem for p-class groups in cyclic p-towers of number fields, improving previous results by providing elementary proofs and applying to torsion groups in abelian p-ramification theory.

Contribution

It offers an elementary proof of a stability theorem for generalized p-class groups in cyclic p-towers, extending prior work that relied on Iwasawa and Galois theory.

Findings

The size of p-class groups grows predictably under certain conditions.

Capitulation properties of class groups are deduced in the tower.

Applications to torsion groups in abelian p-ramification theory are demonstrated.

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.