On p-class groups of relative cyclic p-extensions

Mizusawa, Yasushi; Yamamoto, Kiyotatsu

doi:10.1007/s00013-021-01619-8

Cited by 3 publications

(2 citation statements)

References 12 publications

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“…For known results (all relative to λ = 0), one may cite Fukuda [2] using Iwasawa's theory, Li-Ouyang-Xu-Zhang [10, § 3] working in a non-abelian Galois context, in Kummer towers, via the use of the fixed points formulas [3,4], then Mizusawa [12] above Z 2 -extensions, and Mizusawa-Yamamoto [13] for generalizations, including ramification and splitting conditions, via the Galois theory of pro-p-groups.…”

Section: Introductionmentioning

confidence: 99%

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

Gras

2021

Preprint

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Let k be a number field and let p ≥ 2 be prime. We call K/k a cyclic p-tower if Gal(K/k) ≃ Z/p N Z, N ≥ 2. We give an elementary proof of a stability theorem for generalized p-class groups Xn of the layers kn (Theorem 3.1). Using generalizations of Chevalley's formula (Gras, J. Math. Soc. Japan 46(3) (1994), Proc. Math. Sci. 127(1) (2017)), we improve results by Fukuda (1994), Li-Ouyang-Xu-Zhang (2020), Mizusawa-Yamamoto (2020) and others, whose techniques are based on Iwasawa's theory, Galois theory of pro-p-groups or specific methods. The main difference, compared to these works, is that we introduce an explicit parameter λ ≥ 0 giving formulas of the form #X n = #X 0 • p λ•n , for all n ∈ [0, N ], as soon as this equality holds for n = 1.

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Section: Introductionmentioning

confidence: 99%

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

Gras

2021

Preprint

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“…(c) For k fixed, λ is unbounded (which only depends on the tower via S) and the condition #C 1 = #C of the literature (λ = t = 0, thus #S ≤ r 1 + r 2 ) is empty as soon as λ ≥ 1, whence the interest of the factor p λ to get examples whatever S. For known results (all relative to λ = 0), one may cite Fukuda [5] using Iwasawa's theory, Li-Ouyang-Xu-Zhang [25, § 3] working in a non-abelian Galois context, in Kummer towers, via the use of the fixed points formulas [6,9], then Mizusawa [27] above Z 2 -extensions, and Mizusawa-Yamamoto [28] for generalizations, including ramification and splitting conditions, via the Galois theory of pro-p-groups.…”

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confidence: 99%

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

Gras¹

2022

Int. J. Number Theory

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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 N Z and if S = ∅ 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 Xn of the layers kn ⊆ K: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. Finally we apply our principles to the torsion groups Tn of abelian p-ramification theory. Numerical examples are given. Contents 15 7.2. Case of real cyclic p-towers with tame ramification 16 References 17

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The Chevalley–Herbrand formula and the real abelian Main Conjecture (New criterion using capitulation of the class group)

Gras¹

2023

Journal of Number Theory

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On p-class groups of relative cyclic p-extensions

Cited by 3 publications

References 12 publications

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

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

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

The Chevalley–Herbrand formula and the real abelian Main Conjecture (New criterion using capitulation of the class group)

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