On non-abelian Stark-type conjectures
Annales de l'Institut Fourier, Volume 61 (2011) no. 6, pp. 2577-2608.

We introduce non-abelian generalizations of Brumer’s conjecture, the Brumer-Stark conjecture and the strong Brumer-Stark property attached to a Galois CM-extension of number fields. Moreover, we discuss how they are related to the equivariant Tamagawa number conjecture, the strong Stark conjecture and a non-abelian generalization of Rubin’s conjecture due to D. Burns.

Nous présentons des généralisations non abéliennes de la conjecture de Brumer, de la conjecture de Brumer-Stark et de la propriété forte de Brumer-Stark, qui sont associées à une CM-extension galoisienne de corps de nombres. De plus, nous étudions les liens avec la conjecture équivariante sur les nombres de Tamagawa, la conjecture forte de Stark et la généralisation non abélienne d’une conjecture de Rubin due à D. Burns.

DOI: 10.5802/aif.2683
Classification: 11R42,  11R29
Keywords: Stark conjectures, L-values, class groups
Nickel, Andreas 1

1 Universität Regensburg Fakultät für Mathematik Universitätsstr. 31 93053 Regensburg, Germany
@article{AIF_2011__61_6_2577_0,
     author = {Nickel, Andreas},
     title = {On non-abelian {Stark-type} conjectures},
     journal = {Annales de l'Institut Fourier},
     pages = {2577--2608},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {6},
     year = {2011},
     doi = {10.5802/aif.2683},
     mrnumber = {2976321},
     zbl = {1246.11176},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2683/}
}
TY  - JOUR
AU  - Nickel, Andreas
TI  - On non-abelian Stark-type conjectures
JO  - Annales de l'Institut Fourier
PY  - 2011
DA  - 2011///
SP  - 2577
EP  - 2608
VL  - 61
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2683/
UR  - https://www.ams.org/mathscinet-getitem?mr=2976321
UR  - https://zbmath.org/?q=an%3A1246.11176
UR  - https://doi.org/10.5802/aif.2683
DO  - 10.5802/aif.2683
LA  - en
ID  - AIF_2011__61_6_2577_0
ER  - 
%0 Journal Article
%A Nickel, Andreas
%T On non-abelian Stark-type conjectures
%J Annales de l'Institut Fourier
%D 2011
%P 2577-2608
%V 61
%N 6
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2683
%R 10.5802/aif.2683
%G en
%F AIF_2011__61_6_2577_0
Nickel, Andreas. On non-abelian Stark-type conjectures. Annales de l'Institut Fourier, Volume 61 (2011) no. 6, pp. 2577-2608. doi : 10.5802/aif.2683. http://archive.numdam.org/articles/10.5802/aif.2683/

[1] Barsky, Daniel Fonctions zeta p-adiques d’une classe de rayon des corps de nombres totalement réels, Groupe d’Etude d’Analyse Ultramétrique (5e année: 1977/78), Secrétariat Math., Paris, 1978, pp. Exp. No. 16, 23 | Numdam | MR | Zbl

[2] Burns, D. Equivariant Tamagawa numbers and Galois module theory. I, Compositio Math., Volume 129 (2001) no. 2, pp. 203-237 | DOI | MR | Zbl

[3] Burns, D. On refined Stark conjectures in the non-abelian case, Math. Res. Lett., Volume 15 (2008) no. 5, pp. 841-856 | MR | Zbl

[4] Burns, D. On main conjectures in non-commutative Iwasawa theory and related conjectures, 2010 (http://www.mth.kcl.ac.uk/staff/dj_burns/newdbpublist.html)

[5] Burns, D. On derivatives of Artin L-series, Invent. Math., to appear (http://www.mth.kcl.ac.uk/staff/dj_burns/newdbpublist.html) | MR

[6] Burns, D.; Flach, M. Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math., Volume 6 (2001), p. 501-570 (electronic) | MR | Zbl

[7] Burns, D.; Johnston, H. A non-abelian Stickelberger Theorem, Compositio Math., Volume 147 (2011), pp. 35-55 | DOI | MR

[8] Cassou-Noguès, Pierrette Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques, Invent. Math., Volume 51 (1979) no. 1, pp. 29-59 | DOI | MR | Zbl

[9] Chinburg, T. On the Galois structure of algebraic integers and S-units, Invent. Math., Volume 74 (1983) no. 3, pp. 321-349 | DOI | MR | Zbl

[10] Chinburg, T. Exact sequences and Galois module structure, Ann. of Math. (2), Volume 121 (1985) no. 2, pp. 351-376 | DOI | MR | Zbl

[11] Curtis, Charles W.; Reiner, Irving Methods of representation theory. Vol. I, John Wiley & Sons Inc., New York, 1981 (With applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication) | MR | Zbl

[12] Curtis, Charles W.; Reiner, Irving Methods of representation theory. Vol. II, Pure and Applied Mathematics (New York), John Wiley & Sons Inc., New York, 1987 (With applications to finite groups and orders, A Wiley-Interscience Publication) | MR | Zbl

[13] Deligne, Pierre; Ribet, Kenneth A. Values of abelian L-functions at negative integers over totally real fields, Invent. Math., Volume 59 (1980) no. 3, pp. 227-286 | DOI | EuDML | MR | Zbl

[14] Greither, C.; Burns, D.; Popescu, C.; Sands, J.; Solomon, D. Arithmetic annihilators and Stark-type conjectures, Stark’s Conjectures: Recent work and new directions, Papers from the international conference on Stark’s Conjectures and related topics (Contemporary Math.), Volume 358 (2004), pp. 55-78 | MR | Zbl

[15] Greither, C. Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture, Compositio Math., Volume 143 (2007) no. 6, pp. 1399-1426 | DOI | MR | Zbl

[16] Greither, C.; Kurihara, Masato Stickelberger elements, Fitting ideals of class groups of CM-fields, and dualisation, Math. Z., Volume 260 (2008) no. 4, pp. 905-930 | DOI | MR | Zbl

[17] Greither, C.; Roblot, Xavier-François; Tangedal, Brett A. The Brumer-Stark conjecture in some families of extensions of specified degree, Math. Comp., Volume 73 (2004) no. 245, p. 297-315 (electronic) | DOI | MR | Zbl

[18] Gruenberg, K. W.; Ritter, J.; Weiss, A. A local approach to Chinburg’s root number conjecture, Proc. London Math. Soc. (3), Volume 79 (1999) no. 1, pp. 47-80 | DOI | MR | Zbl

[19] Nickel, A. Non-commutative Fitting invariants and annihilation of class groups, J. Algebra, Volume 323 (2010) no. 10, pp. 2756-2778 | DOI | MR | Zbl

[20] Nickel, A. On the Equivariant Tamagawa Number Conjecture in tame CM-extensions, Math. Z., 2010 (DOI 10.1007/s00209-009-0658-9) | MR | Zbl

[21] Nickel, A. On the Equivariant Tamagawa Number Conjecture in tame CM-extensions, II, Compositio Math., to appear (http://www.mathematik.uni-regensburg.de/Nickel/english.html) | MR | Zbl

[22] Parker, A. Equivariant Tamagawa Numbers and non-commutative Fitting invariants, King’s College London (2007) (Ph. D. Thesis)

[23] Ritter, J.; Weiss, A. A Tate sequence for global units, Compositio Math., Volume 102 (1996) no. 2, pp. 147-178 | EuDML | Numdam | MR | Zbl

[24] Ritter, J.; Weiss, A. On the ’main conjecture’ of equivariant Iwasawa theory, preprint, 2010 (arXiv:1004.2578v2) | MR | Zbl

[25] Rubin, Karl A Stark conjecture “over Z” for abelian L-functions with multiple zeros, Ann. Inst. Fourier (Grenoble), Volume 46 (1996) no. 1, pp. 33-62 | DOI | EuDML | Numdam | MR | Zbl

[26] Swan, R. G. Algebraic K-theory, Lecture Notes in Mathematics, No. 76, Springer-Verlag, Berlin, 1968 | MR | Zbl

[27] Tate, J. The cohomology groups of tori in finite Galois extensions of number fields, Nagoya Math. J., Volume 27 (1966), pp. 709-719 | MR | Zbl

[28] Tate, J. Les conjectures de Stark sur les fonctions L d’Artin en s=0, Progress in Mathematics, 47, Birkhäuser Boston Inc., Boston, MA, 1984 (Lecture notes edited by Dominique Bernardi and Norbert Schappacher) | MR | Zbl

[29] Weiss, A. Multiplicative Galois module structure, Fields Institute Monographs, 5, American Mathematical Society, Providence, RI, 1996 | MR | Zbl

Cited by Sources: