Capitulation for even K -groups in the cyclotomic ℤ p -extension.

Romain Validire[1]

  • [1] XLIM DMI - UMR CNRS 6172 123, avenue Albert Thomas 87060 LIMOGES CEDEX (France).

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 2, page 439-454
  • ISSN: 1246-7405

Abstract

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Let p be a prime number and F be a number field. Since Iwasawa’s works, the behaviour of the p -part of the ideal class group in the ℤ p -extensions of F has been well understood. Moreover, M. Grandet and J.-F. Jaulent gave a precise result about its abelian p -group structure.On the other hand, the ideal class group of a number field may be identified with the torsion part of the K 0 of its ring of integers. The even K -groups of rings of integers appear as higher versions of the class group. Many authors have already studied the behaviour of the higher even K -groups in a ℤ p -extension. Here, we prove that Grandet and Jaulent’s result on class group still holds for higher even K -groups in the cyclotomic ℤ p -extension.

How to cite

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Validire, Romain. "Capitulation for even $K$-groups in the cyclotomic $\mathbb{Z}_p$-extension.." Journal de Théorie des Nombres de Bordeaux 21.2 (2009): 439-454. <http://eudml.org/doc/10891>.

@article{Validire2009,
abstract = {Let $p$ be a prime number and $F$ be a number field. Since Iwasawa’s works, the behaviour of the $p$-part of the ideal class group in the $\mathbb\{Z\}_p$-extensions of $F$ has been well understood. Moreover, M. Grandet and J.-F. Jaulent gave a precise result about its abelian $p$-group structure.On the other hand, the ideal class group of a number field may be identified with the torsion part of the $K_0$ of its ring of integers. The even $K$-groups of rings of integers appear as higher versions of the class group. Many authors have already studied the behaviour of the higher even $K$-groups in a $\mathbb\{Z\}_p$-extension. Here, we prove that Grandet and Jaulent’s result on class group still holds for higher even $K$-groups in the cyclotomic $\mathbb\{Z\}_p$-extension.},
affiliation = {XLIM DMI - UMR CNRS 6172 123, avenue Albert Thomas 87060 LIMOGES CEDEX (France).},
author = {Validire, Romain},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {-groups; -extensions; étale cohomology},
language = {eng},
number = {2},
pages = {439-454},
publisher = {Université Bordeaux 1},
title = {Capitulation for even $K$-groups in the cyclotomic $\mathbb\{Z\}_p$-extension.},
url = {http://eudml.org/doc/10891},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Validire, Romain
TI - Capitulation for even $K$-groups in the cyclotomic $\mathbb{Z}_p$-extension.
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 2
SP - 439
EP - 454
AB - Let $p$ be a prime number and $F$ be a number field. Since Iwasawa’s works, the behaviour of the $p$-part of the ideal class group in the $\mathbb{Z}_p$-extensions of $F$ has been well understood. Moreover, M. Grandet and J.-F. Jaulent gave a precise result about its abelian $p$-group structure.On the other hand, the ideal class group of a number field may be identified with the torsion part of the $K_0$ of its ring of integers. The even $K$-groups of rings of integers appear as higher versions of the class group. Many authors have already studied the behaviour of the higher even $K$-groups in a $\mathbb{Z}_p$-extension. Here, we prove that Grandet and Jaulent’s result on class group still holds for higher even $K$-groups in the cyclotomic $\mathbb{Z}_p$-extension.
LA - eng
KW - -groups; -extensions; étale cohomology
UR - http://eudml.org/doc/10891
ER -

References

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