On universal norms in \(\mathbb{Z}_ p\)-extensions. (Sur les normes universelles dans les \(\mathbb{Z}_ p\)-extensions.) (French) Zbl 0833.11051

The author axiomatizes certain properties of the \(p\)-completion of the group of \(p\)-units of a number field \(F\), relating to: 1) existence of a norm map, 2) Galois descent, 3) rank on \(\mathbb{Z}_p\)-extensions, and 4) Galois descent modulo torsion.
He considers the functor of universal norms associated to a \(\mathbb{Z}_p\)- extension and computes its rank modulo torsion over the Iwasawa algebra. The level of generality allows application to obtain results for units of L. V. Kuz’min [Math. USSR, Izv. 6, No. 2, 263-321 (1973); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 36, 267-327 (1972; Zbl 0231.12013)]and E. de Shalit [Algebraic number theory in honor of K. Iwasawa, Adv. Stud. Pure Math. 17, 83-88 (1989; Zbl 0731.11060)]and extends also to the \(p\)-completion of \(K_3\). Several complements conclude the article, including a connection with the \(p\)-adic conjecture of Gross [cf. J.-F. Jaulent, Journées arithmétiques Luminy, Astérisque 198-200, 187-208 (1991; Zbl 0756.11033)].


11R18 Cyclotomic extensions
11R70 \(K\)-theory of global fields
Full Text: DOI Numdam EuDML EMIS


[1] Coates, J., On K2 and some classical conjectures in algebraic number theory, Ann. Math.95 (1972), 99-116. · Zbl 0245.12005
[2] Iwasawa, K., On Zl-extensions of algebraic number fields, Ann. Math.98 (1973), 246-326. · Zbl 0285.12008
[3] Jaulent, J.-F., Noyau universel et valeurs absolues, Journées arithmétiques de Luminy, Astérisque198-200 (1990), 187-209. · Zbl 0756.11033
[4] Kahn, B., Descente galoisienne et K2 des corps de nombres, K-theory7 (1993), 55-100. · Zbl 0780.12007
[5] Keune, F., On the structure of the K2 of the rings of integers in a number field, K-theory2 (1989), 625-645. · Zbl 0705.19007
[6] Kim, J.M., Bae, S. et Lee, I., Cyclotomic units in Zp-extensions, Israel J. Math.75 (1991), 161-165. · Zbl 0765.11042
[7] Kolster, M., An idelic approach to the wild kernel, Invent. Math.103 (1991), 9-24. · Zbl 0724.11056
[8] Kuz’min, L.V., The Tate module for algebraic number fields, Math. USSR Izv.6(2) (1972), 263-321. · Zbl 0257.12003
[9] Levine, M., The indecomposable K3 of a field, Ann. Sci. Ecole Norm. Sup22 (1989), 255-344. · Zbl 0705.19001
[10] Merkur’ev, A.S. et Suslin, A.A., The group K3 of a field, Math. USSR Izv.36 (1990), 541-565. · Zbl 0725.19003
[11] Shalit, E. de, A note on norm-coherent units in certain Zp-extensions, Algebraic number theory in honor of K. Iwasawa, Advanced Studies in Pure Math.17 (1989), 83-87. · Zbl 0731.11060
[12] Sinnott, W., On the Stilckelberger ideal and the circular units of an abelian field, Invent. Math62 (1980), 181-234. · Zbl 0465.12001
[13] Sinnott, W., Appendice à :Regulators and Iwasawa modules, par L. J. Federer et B. H. Gross, Invent. Math.62 (1981), 443-457. · Zbl 0468.12005
[14] Solomon, D., On a construction of p-units in abelian fields, Invent. Math.109 (1992), 329-350. · Zbl 0772.11043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.