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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 and let p≥2 be a prime number. 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 X_n of the layers k_n (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, pro-p-group Galois theory or specific methods. The main difference, compared to these works, is that we introduce a parameter lambda≥0 giving formulas of the form #X_n=#X_0 p^(lambda n), for all n in [0,N] as soon as this equality holds for n = 1.
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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-03155190v3⟩
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