Nous montrons des raffinements -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 -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.
Classification : 11R23, 11R29, 11R18
Mots clés : groupes de classes, fonctions -adiques, théorie d’Iwasawa
@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}, zbl = {1007.11063}, mrnumber = {1849210}, language = {fr}, url = {http://www.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 DA - 2001/// SP - 903 EP - 937 VL - 51 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1840/ UR - https://zbmath.org/?q=an%3A1007.11063 UR - https://www.ams.org/mathscinet-getitem?mr=1849210 UR - https://doi.org/10.5802/aif.1840 DO - 10.5802/aif.1840 LA - fr ID - AIF_2001__51_4_903_0 ER -
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://www.numdam.org/articles/10.5802/aif.1840/
[BB] Equivariant Tamagawa numbers, Fitting ideals and Iwasawa theory (2000) (A paraître dans Compositio Math.) | MR 1760494 | Zbl 0987.11069
[Be] Sur la structure galoisienne des unités circulaires dans les -extensions, J. of Number Theory, Volume 69 (1998), pp. 16-49 | Article | MR 1611081 | Zbl 0911.11051
[Coa] -adic -functions and Iwasawa's theory, Proc. Sympos., Univ. Durham, Durham (1975) (Algebraic number fields: ) (1977), pp. 269-353 | Zbl 0393.12027
[Cor] Fitting ideals of class groups in a -extension, Acta Arithm., Volume 87 (1998) no. 1, pp. 79-88 | MR 1659155 | Zbl 0926.11084
[FG] Regulators and Iwasawa modules. With an appendix by W. Sinnott, Invent. Math., Volume 62 (1981) no. 3, pp. 443-457 | MR 604838 | Zbl 0468.12005
[Gi1] Unités cyclotomiques, unités semi-locales et -extensions, Ann. Inst. Fourier, Grenoble, Volume 29 (1979) no. 1, pp. 49-79 | Article | Numdam | MR 526777 | Zbl 0387.12002
[Gi2] Unités cyclotomiques, unités semi-locales et -extensions II, Ann. Inst. Fourier, Grenoble, Volume 29 (1979) no. 4, pp. 1-15 | Article | Numdam | MR 558585 | Zbl 0403.12006
[Gi3] Remarques sur les unités cyclotomiques et elliptiques, J. of Number Theory, Volume 11 (1979), pp. 21-48 | Article | MR 527759 | Zbl 0405.12008
[GJ] Sur la capitulation dans une -extension, J. Reine Angew. Math., Volume 362 (1985), pp. 213-217 | Article | MR 809976 | Zbl 0564.12011
[Gra1] Classes d'idéaux des corps abéliens et nombres de Bernoulli généralisés, Ann. Inst. Fourier, Volume 27 (1977) no. 1, pp. 1-66 | Article | Numdam | MR 450238 | Zbl 0336.12004
[Gra2] Canonical divisibilities of values of -adic -functions, Journées Arithmétiques d'Exeter (1980) | Zbl 0494.12006
[Gree] On the Iwasawa invariants of totally real number fields, Amer. J. Math., Volume 98 (1976), pp. 263-284 | Article | MR 401702 | Zbl 0334.12013
[Grei1] Class groups of abelian fields and the Main Conjecture, Ann. Inst. Fourier, Volume 42 (1992) no. 3, pp. 449-499 | Article | Numdam | MR 1182638 | Zbl 0729.11053
[Grei2] The structure of some minus class groups, and Chinburg's third conjecture for abelian fields, Math. Zeit., Volume 229 (1998), pp. 107-136 | Article | MR 1649330 | Zbl 0919.11072
[I] On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan, Volume 16 (1964) no. 1, pp. 42-82 | Article | MR 215811 | Zbl 0125.29207
[J] Classes logarithmiques des corps de nombres, J. Théor. Nombres, Bordeaux, Volume 6 (1994) no. 2, pp. 301-325 | Article | Numdam | MR 1360648 | Zbl 0827.11064
[K1] The Tate module of algebraic number fields, Math. USSR-Izv, Volume 6 (1972), pp. 263-321 | Article | MR 304353 | Zbl 0257.12003
[K2] On formulas for the class number of real abelian fields, Math. USSR-Izv, Volume 60 (1996) no. 4, pp. 695-761 | MR 1416925 | Zbl 1007.11065
[KNF] Twisted -units, -adic class number formulas, and the Lichtenbaum conjectures, Duke Math. J., Volume 84 (1996), pp. 679-717 | Article | MR 1408541 | Zbl 0863.19003
[KS] 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 0840.11043
[La] Introduction to Cyclotomic Fields I and II, GTM, 121, Springer-Verlag, New-York, 1990 | MR 1029028 | Zbl 0704.11038
[Le1] The ring of integers of an abelian number field, J. reine angew. Math., Volume 404 (1990), pp. 162-170 | Article | MR 1037435 | Zbl 0703.11060
[Le2] Relative Galois module structure of integers of local abelian fields, Acta Arithm., Volume 85 (1998) no. 3, pp. 235-248 | MR 1627831 | Zbl 0910.11050
[MW] Class fields of abelian extensions of , Invent. Math., Volume 76 (1984) no. 2, pp. 179-330 | Article | MR 742853 | Zbl 0545.12005
[O1] 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 | Article | MR 1453211 | Zbl 0879.11058
[O2] 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 | Article | MR 1453211 | Zbl 0879.11059
[Si1] On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math., Volume 108 (1978) no. 1, pp. 107-134 | Article | MR 485778 | Zbl 0395.12014
[Si2] On the Stickelberger ideal and the circular units of an abelian field, Invent. Math., Volume 62 (1981), pp. 181-234 | Article | MR 595586 | Zbl 0465.12001
[So1] On a construction of p-units in abelian fields, Invent. Math., Volume 109 (1992), pp. 329-350 | Article | MR 1172694 | Zbl 0772.11043
[So2] Galois relations for cyclotomic numbers and p-units, J. Number Theory, Volume 78 (1994), pp. 1-26 | MR 1269250 | Zbl 0807.11054
[T] Semi-local units modulo cyclotomic units, J. Number Theory, Volume 46 (1999), pp. 158-178 | MR 1706941 | Zbl 0948.11042
[V] Etude du quotient des unités semi-locales par les unités cyclotomiques dans les -extensions des corps de nombres abéliens réels (1981) (thèse, Orsay) | MR 627614 | Zbl 0473.12003
[Wa] Introduction to Cyclotomic Fields, GTM, 83, Springer-Verlag, 1982 | MR 718674 | Zbl 0484.12001
[Wil] The Iwasawa conjecture for totally real fields, Ann. of Math., Volume 131 (1990), pp. 493-540 | Article | MR 1053488 | Zbl 0719.11071
[Win] Duality theorems for -extensions of algebraic number fields, Compositio Math., Volume 55 (1985), pp. 333-381 | Numdam | MR 799821 | Zbl 0608.12012
Cité par Sources :