Unités cyclotomiques, unités semi-locales et ℤ ℓ -extensions. II

Roland Gillard

Annales de l'institut Fourier (1979)

  • Volume: 29, Issue: 4, page 1-15
  • ISSN: 0373-0956

Abstract

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Let K an abelian number field, ℓ a prime number, prime to [ K : Q ] , Y n the quotient of the group of semi-local units in K ( 1 ℓ n ) by the group of cyclotomic units. By giving the Galois structure of lim ← Y n , we generalise a theorem of Iwasawa and use this result for comparing classical conjectures about projective limits of class groups.

How to cite

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Gillard, Roland. "Unités cyclotomiques, unités semi-locales et ${\mathbb {Z}}_\ell $-extensions. II." Annales de l'institut Fourier 29.4 (1979): 1-15. <http://eudml.org/doc/74431>.

@article{Gillard1979,
abstract = {Soient $K$ un corps abélien réel, $\ell $ un nombre premier, premier à $[K:\{\bf Q\}]$ et $Y_n$ le quotient du groupe des unités semi-locales de $K(\@root \{\ell ^n\} \of \{1\})$ par celui des unités cyclotomiques : on donne la structure galoisienne de la limite projective des $Y_n$, généralisant un théorème d’Iwasawa, et on applique ceci à la comparaison de conjecture classique sur la limite projective des groupes de classes.},
author = {Gillard, Roland},
journal = {Annales de l'institut Fourier},
keywords = {abelian real field; group of units; cyclotomic unit; Galois group},
language = {fre},
number = {4},
pages = {1-15},
publisher = {Association des Annales de l'Institut Fourier},
title = {Unités cyclotomiques, unités semi-locales et $\{\mathbb \{Z\}\}_\ell $-extensions. II},
url = {http://eudml.org/doc/74431},
volume = {29},
year = {1979},
}

TY - JOUR
AU - Gillard, Roland
TI - Unités cyclotomiques, unités semi-locales et ${\mathbb {Z}}_\ell $-extensions. II
JO - Annales de l'institut Fourier
PY - 1979
PB - Association des Annales de l'Institut Fourier
VL - 29
IS - 4
SP - 1
EP - 15
AB - Soient $K$ un corps abélien réel, $\ell $ un nombre premier, premier à $[K:{\bf Q}]$ et $Y_n$ le quotient du groupe des unités semi-locales de $K(\@root {\ell ^n} \of {1})$ par celui des unités cyclotomiques : on donne la structure galoisienne de la limite projective des $Y_n$, généralisant un théorème d’Iwasawa, et on applique ceci à la comparaison de conjecture classique sur la limite projective des groupes de classes.
LA - fre
KW - abelian real field; group of units; cyclotomic unit; Galois group
UR - http://eudml.org/doc/74431
ER -

References

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  1. [1] N. BOURBAKI, Éléments de mathématique, Algèbre commutative, Chap. 7, Hermann, Paris, 1965. Zbl0141.03501
  2. [2] J. COATES, p-adic L-functions and Iwasawa's theory, Durham conference on Algebraic Number Theory, edited by A. Fröhlich, Academic Press, Londres, 1977. Zbl0393.12027MR57 #276
  3. [3] J. COATES et S. LICHTENBAUM, On l-adic zeta functions, Ann. of Math., 98 (1973), 498-550. Zbl0279.12005MR48 #8445
  4. [4] J. COATES et A. WILES, On p-adic L-functions and elliptic units, J. Austral. Math. Soc., series A 26 (1978), 1-25. Zbl0442.12007MR80a:12007
  5. [5] R. COLEMAN, Some modules attached to Lubin-Tate groups, à paraître. 
  6. [6] B. FERRERO, Iwasawa invariants of abelian number fields, Math. Ann., 234 (1978), 9-24. Zbl0347.12004MR58 #5584
  7. [7] B. FERRERO et L. WASHINGTON, The Iwasawa invariant µp vanishes for abelian number fields, Ann. of Math., à paraître. Zbl0443.12001
  8. [8] R. GILLARD, Unités cyclotomiques, unités semi-locales et Zl-extensions, Ann. Inst. Fourier, t. 29, fasc. 1 (1979), 49-79. Zbl0387.12002MR81e:12005a
  9. [9] R. GILLARD, Remarques sur les unités cyclotomiques et les unités elliptiques, J. of Numbers Theory, 11, 1 (1979), 21-48. Zbl0405.12008MR80j:12004
  10. [10] R. GREENBERG, On p-adic L-functions and cyclotomic fields, Nagoya Math. J., 56 (1974), 61-77. Zbl0315.12008MR50 #12984
  11. [11] R. GREENBERG, On p-adic L-functions and cyclotomic fields II, Nagoya Math. J., 67 (1977), 139-158. Zbl0373.12007MR56 #2964
  12. [12] R. GREENBERG, On 2-adic L-functions and cyclotomic invariants, Math. Zeit., 159 (1978), 37-45. Zbl0354.12014MR57 #16263
  13. [13] K. IWASAWA, On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan, vol. 16, n° 1, (1964), 42-82. Zbl0125.29207MR35 #6646
  14. [14] K. IWASAWA, On Zl-extensions of algebraic number fields, Ann. of Math., 98 (1973), 246-326. Zbl0285.12008MR50 #2120
  15. [15] K. IWASAWA, Lectures on p-adic L-functions, Ann. Math. Studies, 74, Princeton Univ. Press, 1972. Zbl0236.12001MR50 #12974
  16. [16] S. LANG, Cyclotomic fields, Springer Verlag, 1978. Zbl0395.12005MR58 #5578

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