On the Zp-extensions of a totally p-adic imaginary quadratic field -- With an appendix by Jean-François Jaulent - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2024

On the Zp-extensions of a totally p-adic imaginary quadratic field -- With an appendix by Jean-François Jaulent

Résumé

Let k be an imaginary quadratic field, and let p be an odd prime number split in k. We analyze some properties of arbitrary Zp-extensions K/k. These properties are governed by the Hase norm residue symbol of the fundamental p-unit of k, in terms of the valuation δp(k) of a Fermat quotient, which determines the order of the logarithmic class group Clogk (Theorem 2.2, Appendix 1) and leads, under some conditions, to generalizations of Gold's criterion characterizing λp(K/k) = 1 (Theorems 5.1, 5.3, 5.5). This uses the higher rank Chevalley--Herbrand formulas, for the filtrations of the p-class groups in K, that we gave in the 1994's, and the theorem of λ-stability (2022). This study is in connection with articles of Gold, Sands, Dummit--Ford--Kisilevsky--Sands, Ozaki, Hubbard--Washington,Kataoka, Ray, Jaulent, Fujii. In Appendix A, is given a general proof, by Jaulent, of the link between Clogk and δp(k) in a broader context. Numerical illustrations (with pari/gp programs) are given.
Fichier principal
Vignette du fichier
Zp-extensions-arXiv.pdf (495.19 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04518815 , version 1 (24-03-2024)
hal-04518815 , version 2 (03-04-2024)
hal-04518815 , version 3 (05-05-2024)

Identifiants

  • HAL Id : hal-04518815 , version 3

Citer

Georges Gras. On the Zp-extensions of a totally p-adic imaginary quadratic field -- With an appendix by Jean-François Jaulent. 2024. ⟨hal-04518815v3⟩
17 Consultations
16 Téléchargements

Partager

Gmail Facebook X LinkedIn More