Formules de classes pour les corps abéliens réels
Belliard, Jean-Robert ; Nguyen Quang Do, Thong
Annales de l'Institut Fourier, Tome 51 (2001), p. 903-937 / Harvested from Numdam

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.

Publié le : 2001-01-01
DOI : https://doi.org/10.5802/aif.1840
Classification:  11R23,  11R29,  11R18
Mots clés: groupes de classes, fonctions Lp-adiques, théorie d’Iwasawa
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     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},
     volume = {51},
     year = {2001},
     pages = {903-937},
     doi = {10.5802/aif.1840},
     mrnumber = {1849210},
     zbl = {1007.11063},
     language = {fr},
     url = {http://dml.mathdoc.fr/item/AIF_2001__51_4_903_0}
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Belliard, Jean-Robert; Nguyen Quang Do, Thong. Formules de classes pour les corps abéliens réels. Annales de l'Institut Fourier, Tome 51 (2001) pp. 903-937. doi : 10.5802/aif.1840. http://gdmltest.u-ga.fr/item/AIF_2001__51_4_903_0/

[BB] W. Bley; D. Burns Equivariant Tamagawa numbers, Fitting ideals and Iwasawa theory (2000) (A paraître dans Compositio Math.) | MR 1760494 | Zbl 0987.11069

[Be] J.-R. Belliard Sur la structure galoisienne des unités circulaires dans les p -extensions, J. of Number Theory, Tome 69 (1998), pp. 16-49 | Article | MR 1611081 | Zbl 0911.11051

[Coa] J. Coates p-adic L-functions and Iwasawa's theory, Proc. Sympos., Univ. Durham, Durham (1975), Academic Press, London (Algebraic number fields: L -functions and Galois properties) (1977), pp. 269-353 | Zbl 0393.12027

[Cor] P. Cornacchia Fitting ideals of class groups in a p -extension, Acta Arithm., Tome 87 (1998) no. 1, pp. 79-88 | MR 1659155 | Zbl 0926.11084

[FG] L. J. Federer; B. H. Gross Regulators and Iwasawa modules. With an appendix by W. Sinnott, Invent. Math., Tome 62 (1981) no. 3, pp. 443-457 | MR 604838 | Zbl 0468.12005

[Gi1] R. Gillard Unités cyclotomiques, unités semi-locales et l -extensions, Ann. Inst. Fourier, Grenoble, Tome 29 (1979) no. 1, pp. 49-79 | Article | Numdam | MR 526777 | Zbl 0387.12002

[Gi2] R. Gillard Unités cyclotomiques, unités semi-locales et l -extensions II, Ann. Inst. Fourier, Grenoble, Tome 29 (1979) no. 4, pp. 1-15 | Article | Numdam | MR 558585 | Zbl 0403.12006

[Gi3] R. Gillard Remarques sur les unités cyclotomiques et elliptiques, J. of Number Theory, Tome 11 (1979), pp. 21-48 | Article | MR 527759 | Zbl 0405.12008

[GJ] M. Grandet; J.-F. Jaulent Sur la capitulation dans une -extension, J. Reine Angew. Math., Tome 362 (1985), pp. 213-217 | Article | MR 809976 | Zbl 0564.12011

[Gra1] G. Gras Classes d'idéaux des corps abéliens et nombres de Bernoulli généralisés, Ann. Inst. Fourier, Tome 27 (1977) no. 1, pp. 1-66 | Article | Numdam | MR 450238 | Zbl 0336.12004

[Gra2] G. Gras Canonical divisibilities of values of p-adic L-functions, Journées Arithmétiques d'Exeter (1980) | Zbl 0494.12006

[Gree] R. Greenberg On the Iwasawa invariants of totally real number fields, Amer. J. Math., Tome 98 (1976), pp. 263-284 | Article | MR 401702 | Zbl 0334.12013

[Grei1] C. Greither Class groups of abelian fields and the Main Conjecture, Ann. Inst. Fourier, Tome 42 (1992) no. 3, pp. 449-499 | Article | Numdam | MR 1182638 | Zbl 0729.11053

[Grei2] C. Greither The structure of some minus class groups, and Chinburg's third conjecture for abelian fields, Math. Zeit., Tome 229 (1998), pp. 107-136 | Article | MR 1649330 | Zbl 0919.11072

[I] K. Iwasawa On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan, Tome 16 (1964) no. 1, pp. 42-82 | Article | MR 215811 | Zbl 0125.29207

[J] J.-F. Jaulent Classes logarithmiques des corps de nombres, J. Théor. Nombres, Bordeaux, Tome 6 (1994) no. 2, pp. 301-325 | Article | Numdam | MR 1360648 | Zbl 0827.11064

[K1] L. V. Kuz'Min The Tate module of algebraic number fields, Math. USSR-Izv, Tome 6 (1972), pp. 263-321 | Article | MR 304353 | Zbl 0257.12003

[K2] L. V. Kuz'Min On formulas for the class number of real abelian fields, Math. USSR-Izv, Tome 60 (1996) no. 4, pp. 695-761 | MR 1416925 | Zbl 1007.11065

[KNF] M. Kolster; T. Nguyen Quang Do; V. Fleckinger Twisted S-units, p-adic class number formulas, and the Lichtenbaum conjectures, Duke Math. J., Tome 84 (1996), pp. 679-717 | Article | MR 1408541 | Zbl 0863.19003

[KS] J. Kraft; R. Schoof Computing Iwasawa modules of real quadratic number fields, Special issue in honour of Frans Oort (Compositio Math.) Tome 97 (1995), pp. 135-155 | Numdam | Zbl 0840.11043

[La] S. Lang Introduction to Cyclotomic Fields I and II, Springer-Verlag, New-York, GTM, Tome 121 (1990) | MR 1029028 | Zbl 0704.11038

[Le1] G. Lettl The ring of integers of an abelian number field, J. reine angew. Math., Tome 404 (1990), pp. 162-170 | Article | MR 1037435 | Zbl 0703.11060

[Le2] G. Lettl Relative Galois module structure of integers of local abelian fields, Acta Arithm., Tome 85 (1998) no. 3, pp. 235-248 | MR 1627831 | Zbl 0910.11050

[MW] B. Mazur; A. Wiles Class fields of abelian extensions of , Invent. Math., Tome 76 (1984) no. 2, pp. 179-330 | Article | MR 742853 | Zbl 0545.12005

[O1] M. Ozaki On the cyclotomic unit group and the ideal class group of a real abelian number field I, J. of Number Theory, Tome 64 (1997), pp. 211-222 | Article | MR 1453211 | Zbl 0879.11058

[O2] M. Ozaki On the cyclotomic unit group and the ideal class group of a real abelian number field II, J. of Number Theory, Tome 64 (1997), pp. 223-232 | Article | MR 1453211 | Zbl 0879.11059

[Si1] W. Sinnott On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math., Tome 108 (1978) no. 1, pp. 107-134 | Article | MR 485778 | Zbl 0395.12014

[Si2] W. Sinnott On the Stickelberger ideal and the circular units of an abelian field, Invent. Math., Tome 62 (1981), pp. 181-234 | Article | MR 595586 | Zbl 0465.12001

[So1] D. Solomon On a construction of p-units in abelian fields, Invent. Math., Tome 109 (1992), pp. 329-350 | Article | MR 1172694 | Zbl 0772.11043

[So2] D. Solomon Galois relations for cyclotomic numbers and p-units, J. Number Theory, Tome 78 (1994), pp. 1-26 | MR 1269250 | Zbl 0807.11054

[T] T. Tsuji Semi-local units modulo cyclotomic units, J. Number Theory, Tome 46 (1999), pp. 158-178 | MR 1706941 | Zbl 0948.11042

[V] L. Villemot Etude du quotient des unités semi-locales par les unités cyclotomiques dans les p -extensions des corps de nombres abéliens réels (1981) (thèse, Orsay) | MR 627614 | Zbl 0473.12003

[Wa] L. Washington Introduction to Cyclotomic Fields, Springer-Verlag, GTM, Tome 83 (1982) | MR 718674 | Zbl 0484.12001

[Wil] A. Wiles The Iwasawa conjecture for totally real fields, Ann. of Math., Tome 131 (1990), pp. 493-540 | Article | MR 1053488 | Zbl 0719.11071

[Win] K. Wingberg Duality theorems for Γ -extensions of algebraic number fields, Compositio Math., Tome 55 (1985), pp. 333-381 | Numdam | MR 799821 | Zbl 0608.12012