Cet article se propose de démontrer une version analogue de la conjecture de Gras pour un corps abélien et un nombre premier qui divise le degré . On fait l’hypothèse que la -partie du groupe est cyclique.
The aim of this paper is to prove an analog of Gras’ conjecture for an abelian field and an odd prime dividing the degree assuming that the -part of group is cyclic.
Classification : 11R20, 11R29
Mots clés : conjecture de Gras, unités cyclotomiques, groupe des classes, systèmes d’Euler, annulateurs du groupe des classes
@article{AIF_2014__64_5_2165_0, author = {Greither, Cornelius and Ku\v cera, Radan}, title = {Eigenspaces of the ideal class group}, journal = {Annales de l'Institut Fourier}, pages = {2165--2203}, publisher = {Association des Annales de l'institut Fourier}, volume = {64}, number = {5}, year = {2014}, doi = {10.5802/aif.2908}, mrnumber = {3330935}, zbl = {06387335}, language = {en}, url = {archive.numdam.org/item/AIF_2014__64_5_2165_0/} }
Greither, Cornelius; Kučera, Radan. Eigenspaces of the ideal class group. Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 2165-2203. doi : 10.5802/aif.2908. http://archive.numdam.org/item/AIF_2014__64_5_2165_0/
[1] Formules de classes pour les corps abéliens réels, Ann. Inst. Fourier (Grenoble), Volume 51 (2001) no. 4, pp. 903-937 http://aif.cedram.org/item?id=AIF_2001__51_4_903_0 | Article | Numdam | MR 1849210 | Zbl 1007.11063
[2] Kolyvagin systems of Stark units, J. Reine Angew. Math., Volume 631 (2009), pp. 85-107 | Article | MR 2542218 | Zbl 1216.11102
[3] On -adic -functions and cyclotomic fields. II, Nagoya Math. J., Volume 67 (1977), pp. 139-158 | MR 444614 | Zbl 0373.12007
[4] Annihilators for the class group of a cyclic field of prime power degree. II, Canad. J. Math., Volume 58 (2006) no. 3, pp. 580-599 | Article | MR 2223457 | Zbl 1155.11054
[5] Linear forms on Sinnott’s module, J. Number Theory, Volume 141 (2014), pp. 324-342 | Article | MR 3195403
[6] Commutative Rings, Polygonal Publishing House, 1994 (Washington, NJ)
[7] Circular units and class groups of abelian fields, Ann. Sci. Math. Québec, Volume 28 (2004) no. 1-2, p. 121-136 (2005) | MR 2183100 | Zbl 1103.11031
[8] On formulas for the class number of real abelian fields, Izv. Ross. Akad. Nauk Ser. Mat., Volume 60 (1996) no. 4, pp. 43-110 | Article | MR 1416925 | Zbl 1007.11065
[9] Class fields of abelian extensions of , Invent. Math., Volume 76 (1984) no. 2, pp. 179-330 | Article | MR 742853 | Zbl 0545.12005
[10] The Main Conjecture, Appendix in S. Lang, Cyclotomic Fields I and II, second ed (Graduate Texts in Mathematics) Volume 121, Springer, New York, 1990
[11] On the Stickelberger ideal and the circular units of an abelian field, Invent. Math., Volume 62 (1980/81) no. 2, pp. 181-234 | Article | MR 595586 | Zbl 0465.12001
[12] On the ideal class groups of real abelian number fields, Ann. of Math. (2), Volume 128 (1988) no. 1, pp. 1-18 | Article | MR 951505 | Zbl 0665.12003
[13] Introduction to cyclotomic fields, Graduate Texts in Mathematics, Volume 83, Springer-Verlag, New York, 1997, pp. xiv+487 | Article | MR 1421575 | Zbl 0484.12001