Formules de classes pour les corps abéliens réels
Belliard, Jean-Robert1 ; Nguyen Quang Do, Thong1
1 Université de Franche-Comté, Laboratoire de Mathématiques, CNRS UMR 6623, 16 route de Gray, 25030 Besançon Cedex (France)
Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 903-937.

Nous montrons des raffinements p-adique et “caractères par caractères” de la formule d’indice de Sinnott pour un corps abélien totalement réel. De tels raffinements ont aussi été obtenus par Kuz’min avec des méthodes différentes (voir les commentaires en introduction). Nous donnons des applications à la théorie d’Iwasawa des unités semi- locales et cyclotomiques.

We show p-adic and “character by character” refinements of Sinnott’s index formula for a totally real abelian number field. Such refinements have also been obtained by Kuz’min by different methods (but see comments in the introduction). Applications are given to Iwasawa theory of semi-local units and cyclotomic units.

DOI : 10.5802/aif.1840
Classification : 11R23, 11R29, 11R18
Mot clés : groupes de classes, fonctions $L\,p$-adiques, théorie d’Iwasawa
Keywords: class groups, $p$-adic $L$ functions, Iwasawa’s theory
Belliard, Jean-Robert 1 ; Nguyen Quang Do, Thong 1

1 Université de Franche-Comté, Laboratoire de Mathématiques, CNRS UMR 6623, 16 route de Gray, 25030 Besançon Cedex (France) 
@article{AIF_2001__51_4_903_0,
     author = {Belliard, Jean-Robert and Nguyen Quang Do, Thong},
     title = {Formules de classes pour les corps ab\'eliens r\'eels},
     journal = {Annales de l'Institut Fourier},
     pages = {903--937},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {51},
     number = {4},
     year = {2001},
     doi = {10.5802/aif.1840},
     mrnumber = {1849210},
     zbl = {1007.11063},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.5802/aif.1840/}
}
TY  - JOUR
AU  - Belliard, Jean-Robert
AU  - Nguyen Quang Do, Thong
TI  - Formules de classes pour les corps abéliens réels
JO  - Annales de l'Institut Fourier
PY  - 2001
SP  - 903
EP  - 937
VL  - 51
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1840/
DO  - 10.5802/aif.1840
LA  - fr
ID  - AIF_2001__51_4_903_0
ER  - 
%0 Journal Article
%A Belliard, Jean-Robert
%A Nguyen Quang Do, Thong
%T Formules de classes pour les corps abéliens réels
%J Annales de l'Institut Fourier
%D 2001
%P 903-937
%V 51
%N 4
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.1840/
%R 10.5802/aif.1840
%G fr
%F AIF_2001__51_4_903_0
Belliard, Jean-Robert; Nguyen Quang Do, Thong. Formules de classes pour les corps abéliens réels. Annales de l'Institut Fourier, Tome 51 (2001) no. 4, pp. 903-937. doi : 10.5802/aif.1840. http://archive.numdam.org/articles/10.5802/aif.1840/

[BB] W. Bley; D. Burns Equivariant Tamagawa numbers, Fitting ideals and Iwasawa theory (2000) (A paraître dans Compositio Math.) | MR | Zbl

[Be] J.-R. Belliard Sur la structure galoisienne des unités circulaires dans les ℤ p -extensions, J. of Number Theory, Volume 69 (1998), pp. 16-49 | DOI | MR | Zbl

[Coa] J. Coates p-adic L-functions and Iwasawa's theory, Proc. Sympos., Univ. Durham, Durham (1975) (Algebraic number fields: $L$ L -functions and Galois properties) (1977), pp. 269-353 | Zbl

[Cor] P. Cornacchia Fitting ideals of class groups in a ℤ p -extension, Acta Arithm., Volume 87 (1998) no. 1, pp. 79-88 | MR | Zbl

[FG] L. J. Federer; B. H. Gross Regulators and Iwasawa modules. With an appendix by W. Sinnott, Invent. Math., Volume 62 (1981) no. 3, pp. 443-457 | MR | Zbl

[Gi1] R. Gillard Unités cyclotomiques, unités semi-locales et ℤ l -extensions, Ann. Inst. Fourier, Grenoble, Volume 29 (1979) no. 1, pp. 49-79 | DOI | Numdam | MR | Zbl

[Gi2] R. Gillard Unités cyclotomiques, unités semi-locales et ℤ l -extensions II, Ann. Inst. Fourier, Grenoble, Volume 29 (1979) no. 4, pp. 1-15 | DOI | Numdam | MR | Zbl

[Gi3] R. Gillard Remarques sur les unités cyclotomiques et elliptiques, J. of Number Theory, Volume 11 (1979), pp. 21-48 | DOI | MR | Zbl

[GJ] M. Grandet; J.-F. Jaulent Sur la capitulation dans une ℤ ℓ -extension, J. Reine Angew. Math., Volume 362 (1985), pp. 213-217 | DOI | MR | Zbl

[Gra1] G. Gras Classes d'idéaux des corps abéliens et nombres de Bernoulli généralisés, Ann. Inst. Fourier, Volume 27 (1977) no. 1, pp. 1-66 | DOI | Numdam | MR | Zbl

[Gra2] G. Gras Canonical divisibilities of values of p-adic L-functions, Journées Arithmétiques d'Exeter (1980) | Zbl

[Gree] R. Greenberg On the Iwasawa invariants of totally real number fields, Amer. J. Math., Volume 98 (1976), pp. 263-284 | DOI | MR | Zbl

[Grei1] C. Greither Class groups of abelian fields and the Main Conjecture, Ann. Inst. Fourier, Volume 42 (1992) no. 3, pp. 449-499 | DOI | Numdam | MR | Zbl

[Grei2] C. Greither The structure of some minus class groups, and Chinburg's third conjecture for abelian fields, Math. Zeit., Volume 229 (1998), pp. 107-136 | DOI | MR | Zbl

[I] K. Iwasawa On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan, Volume 16 (1964) no. 1, pp. 42-82 | DOI | MR | Zbl

[J] J.-F. Jaulent Classes logarithmiques des corps de nombres, J. Théor. Nombres, Bordeaux, Volume 6 (1994) no. 2, pp. 301-325 | DOI | Numdam | MR | Zbl

[K1] L. V. Kuz'min The Tate module of algebraic number fields, Math. USSR-Izv, Volume 6 (1972), pp. 263-321 | DOI | MR | Zbl

[K2] L. V. Kuz'min On formulas for the class number of real abelian fields, Math. USSR-Izv, Volume 60 (1996) no. 4, pp. 695-761 | MR | Zbl

[KNF] M. Kolster; T. Nguyen Quang Do; V. Fleckinger Twisted S-units, p-adic class number formulas, and the Lichtenbaum conjectures, Duke Math. J., Volume 84 (1996), pp. 679-717 | DOI | MR | Zbl

[KS] J. Kraft; R. Schoof Computing Iwasawa modules of real quadratic number fields, Special issue in honour of Frans Oort (Compositio Math.), Volume 97 (1995), pp. 135-155 | Numdam | Zbl

[La] S. Lang Introduction to Cyclotomic Fields I and II, GTM, 121, Springer-Verlag, New-York, 1990 | MR | Zbl

[Le1] G. Lettl The ring of integers of an abelian number field, J. reine angew. Math., Volume 404 (1990), pp. 162-170 | DOI | MR | Zbl

[Le2] G. Lettl Relative Galois module structure of integers of local abelian fields, Acta Arithm., Volume 85 (1998) no. 3, pp. 235-248 | MR | Zbl

[MW] B. Mazur; A. Wiles Class fields of abelian extensions of ℚ, Invent. Math., Volume 76 (1984) no. 2, pp. 179-330 | DOI | MR | Zbl

[O1] M. Ozaki On the cyclotomic unit group and the ideal class group of a real abelian number field I, J. of Number Theory, Volume 64 (1997), pp. 211-222 | DOI | MR | Zbl

[O2] M. Ozaki On the cyclotomic unit group and the ideal class group of a real abelian number field II, J. of Number Theory, Volume 64 (1997), pp. 223-232 | DOI | MR | Zbl

[Si1] W. Sinnott On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math., Volume 108 (1978) no. 1, pp. 107-134 | DOI | MR | Zbl

[Si2] W. Sinnott On the Stickelberger ideal and the circular units of an abelian field, Invent. Math., Volume 62 (1981), pp. 181-234 | DOI | MR | Zbl

[So1] D. Solomon On a construction of p-units in abelian fields, Invent. Math., Volume 109 (1992), pp. 329-350 | DOI | MR | Zbl

[So2] D. Solomon Galois relations for cyclotomic numbers and p-units, J. Number Theory, Volume 78 (1994), pp. 1-26 | MR | Zbl

[T] T. Tsuji Semi-local units modulo cyclotomic units, J. Number Theory, Volume 46 (1999), pp. 158-178 | MR | Zbl

[V] L. Villemot Etude du quotient des unités semi-locales par les unités cyclotomiques dans les ℤ p -extensions des corps de nombres abéliens réels (1981) (thèse, Orsay) | MR | Zbl

[Wa] L. Washington Introduction to Cyclotomic Fields, GTM, 83, Springer-Verlag, 1982 | MR | Zbl

[Wil] A. Wiles The Iwasawa conjecture for totally real fields, Ann. of Math., Volume 131 (1990), pp. 493-540 | DOI | MR | Zbl

[Win] K. Wingberg Duality theorems for Γ -extensions of algebraic number fields, Compositio Math., Volume 55 (1985), pp. 333-381 | Numdam | MR | Zbl

Cité par Sources :