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Pré-Publication, Document De Travail Année : 2021

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.
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Dates et versions

hal-03155190 , version 1 (01-03-2021)
hal-03155190 , version 2 (21-03-2021)
hal-03155190 , version 3 (31-03-2021)
hal-03155190 , version 4 (19-05-2021)
hal-03155190 , version 5 (12-08-2021)

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Georges Gras. ON THE p-CLASS GROUP STABILITY ALONG CYCLIC p-TOWERS OF A NUMBER FIELD. 2021. ⟨hal-03155190v4⟩
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