Eigenspaces of the ideal class group

Greither, Cornelius; Kučera, Radan

doi:10.5802/aif.2908

Eigenspaces of the ideal class group
[Parties

χ

-invariantes dans les groupes de classes]

Greither, Cornelius ¹ ; Kučera, Radan ²

¹ Universität der Bundeswehr München Fakultät für Informatik Institut für theoretische Informatik, Mathematik und OR 85577 Neubiberg (Germany)
² Masaryk University Faculty of Science 611 37 Brno (Czech Republic)

Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 2165-2203.

Résumé
Abstract

Cet article se propose de démontrer une version analogue de la conjecture de Gras pour un corps abélien $F$ et un nombre premier $p > 2$ qui divise le degré $[F : ℚ]$ . On fait l’hypothèse que la $p$ -partie du groupe $Gal (F / ℚ)$ est cyclique.

The aim of this paper is to prove an analog of Gras’ conjecture for an abelian field $F$ and an odd prime $p$ dividing the degree $[F : ℚ]$ assuming that the $p$ -part of $Gal (F / ℚ)$ group is cyclic.

MR Zbl

DOI : 10.5802/aif.2908

Classification : 11R20, 11R29
Keywords: Gras’ conjecture, circular (cyclotomic) units, ideal class group, Euler system, annihilators of the class group
Mot clés : conjecture de Gras, unitÃ©s cyclotomiques, groupe des classes, systèmes d’Euler, annulateurs du groupe des classes

Affiliations des auteurs :

Greither, Cornelius ¹ ; Kučera, Radan ²

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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://www.numdam.org/articles/10.5802/aif.2908/

Bibliographie
Cité par

[1] Belliard, J.-R.; Nguyen Quang Do, T. Formules de classes pour les corps abéliens réels, Ann. Inst. Fourier (Grenoble), Volume 51 (2001) no. 4, pp. 903-937 | DOI | Numdam | MR | Zbl

[2] Büyükboduk, Kâzim Kolyvagin systems of Stark units, J. Reine Angew. Math., Volume 631 (2009), pp. 85-107 | DOI | MR | Zbl

[3] Greenberg, Ralph On $p$ -adic $L$ -functions and cyclotomic fields. II, Nagoya Math. J., Volume 67 (1977), pp. 139-158 | MR | Zbl

[4] Greither, Cornelius; Kučera, Radan 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 | DOI | MR | Zbl

[5] Greither, Cornelius; Kučera, Radan Linear forms on Sinnott’s module, J. Number Theory, Volume 141 (2014), pp. 324-342 | DOI | MR

[6] Kaplansky, I. Commutative Rings, Polygonal Publishing House, 1994 (Washington, NJ)

[7] Kučera, Radan Circular units and class groups of abelian fields, Ann. Sci. Math. Québec, Volume 28 (2004) no. 1-2, p. 121-136 (2005) | MR | Zbl

[8] Kuzmin, L. V. On formulas for the class number of real abelian fields, Izv. Ross. Akad. Nauk Ser. Mat., Volume 60 (1996) no. 4, pp. 43-110 | DOI | MR | Zbl

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

[10] Rubin, K. 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] Sinnott, W. On the Stickelberger ideal and the circular units of an abelian field, Invent. Math., Volume 62 (1980/81) no. 2, pp. 181-234 | DOI | MR | Zbl

[12] Thaine, Francisco On the ideal class groups of real abelian number fields, Ann. of Math. (2), Volume 128 (1988) no. 1, pp. 1-18 | DOI | MR | Zbl

[13] Washington, Lawrence C. Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1997, pp. xiv+487 | DOI | MR | Zbl

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