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A non-abelian Stickelberger theorem

Part of:Algebraic number theory: global fields

Published online by Cambridge University Press:  01 July 2010

David Burns and
Henri Johnston
David Burns
Affiliation:
Department of Mathematics, King’s College London, London WC2R 2LS, UK (email: david.burns@kcl.ac.uk)
Henri Johnston
Affiliation:
St John’s College, University of Cambridge, St John’s Street, Cambridge CB2 1TP, UK (email: H.Johnston@dpmms.cam.ac.uk)
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Abstract

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Let L/k be a finite Galois extension of number fields with Galois group G. For every odd prime p satisfying certain mild technical hypotheses, we use values of Artin L-functions to construct an element in the centre of the group ring ℤ(p)[G] that annihilates the p-part of the class group of L.

Keywords

equivariant L-valuesclass groups

MSC classification

Secondary:11R29: Class numbers, class groups, discriminants11R42: Zeta functions and $L$-functions of number fields11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 1 , January 2011, pp. 35 - 55
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Bley, W., Equivariant Tamagawa number conjecture for abelian extensions of a quadratic imaginary field, Doc. Math. 11 (2006), 73–118 (electronic); MR 2226270(2007a:11153).CrossRefGoogle Scholar
[2]Breuning, M., Equivariant epsilon constants for Galois extensions of number fields and p-adic fields, PhD thesis, Department of Mathematics, King’s College London (2004).Google Scholar
[3]Breuning, M. and Burns, D., Leading terms of Artin L-functions at s=0 and s=1, Compositio Math. 143 (2007), 1427–1464; MR 2371375(2009a:11232).Google Scholar
[4]Burns, D., Equivariant Tamagawa numbers and Galois module theory I, Compositio Math. 129 (2001), 203–237; MR 1863302(2002g:11152).CrossRefGoogle Scholar
[5]Burns, D., Equivariant Whitehead torsion and refined Euler characteristics, in Number theory, CRM Proceedings & Lecture Notes, vol. 36 (American Mathematical Society, Providence, RI, 2004), 35–59; MR 2076565(2005d:19002).CrossRefGoogle Scholar
[6]Burns, D., On refined Stark conjectures in the non-abelian case, Math. Res. Lett. 15 (2008), 841–856; MR 2443986.CrossRefGoogle Scholar
[7]Burns, D., On derivatives of Artin L-series, Preprint (2009)http://www.mth.kcl.ac.uk/staff/dj_burns/newdbpublist.html.Google Scholar
[8]Burns, D. and Flach, M., Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501–570; MR 1884523(2002m:11055).CrossRefGoogle Scholar
[9]Burns, D. and Flach, M., Tamagawa numbers for motives with (noncommutative) coefficients II, Amer. J. Math. 125 (2003), 475–512; MR 1981031(2004c:11111).CrossRefGoogle Scholar
[10]Cassou-Noguès, P., Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta p-adiques, Invent. Math. 51 (1979), 29–59; MR 524276(80h:12009b).CrossRefGoogle Scholar
[11]Chinburg, T., On the Galois structure of algebraic integers and S-units, Invent. Math. 74 (1983), 321–349; MR 724009(86c:11096).Google Scholar
[12]Curtis, C. W. and Reiner, I., Methods of representation theory: with applications to finite groups and orders, vol. I, Pure and Applied Mathematics (Wiley-Interscience/John Wiley & Sons Inc., New York, 1981); MR 632548(82i:20001).Google Scholar
[13]Curtis, C. W. and Reiner, I., Methods of representation theory: with applications to finite groups and orders, vol. II, Pure and Applied Mathematics (Wiley-Interscience/John Wiley & Sons Inc., New York, 1987); MR 892316(88f:20002).Google Scholar
[14]Deligne, P. and Ribet, K. A., Values of abelian L-functions at negative integers over totally real fields, Invent. Math. 59 (1980), 227–286; MR 579702(81m:12019).CrossRefGoogle Scholar
[15]Greither, C., Arithmetic annihilators and Stark-type conjectures, in Stark’s conjectures: recent work and new directions, Contemporary Mathematics, vol. 358 (American Mathematical Society, Providence, RI, 2004), 55–78; MR 2088712(2005h:11259).Google Scholar
[16]Greither, C., Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture, Compositio Math. 143 (2007), 1399–1426; MR 2371374(2009a:11226).Google Scholar
[17]Nickel, A., On the equivariant Tamagawa number conjecture in tame CM-extensions, DOI 10.1007/s00209-009-0658-9, Math. Z., to appear.Google Scholar
[18]Nickel, A., The lifted root number conjecture for small sets of places, J. Lond. Math. Soc. 80 (2009), 446–470.CrossRefGoogle Scholar
[19]Reiner, I., Maximal orders, London Mathematical Society Monographs New Series, vol. 28 (Clarendon/Oxford University Press, Oxford, 2003), Corrected reprint of the 1975 original, with a foreword by M. J. Taylor; MR 1972204(2004c:16026).CrossRefGoogle Scholar
[20]Ritter, J. and Weiss, A., A Tate sequence for global units, Compositio Math. 102 (1996), 147–178; MR 1394524(97d:11170).Google Scholar
[21]Ritter, J. and Weiss, A., Cohomology of units and L-values at zero, J. Amer. Math. Soc. 10 (1997), 513–552; MR 1423032(98a:11150).CrossRefGoogle Scholar
[22]Tate, J., The cohomology groups of tori in finite Galois extensions of number fields, Nagoya Math. J. 27 (1966), 709–719; MR 0207680(34#7495).Google Scholar
[23]Tate, J., Les conjectures de Stark sur les fonctions L d’Artin en s=0, Progress in Mathematics, vol. 47 (Birkhäuser Boston Inc., Boston, MA, 1984), Lecture notes edited by Dominique Bernardi and Norbert Schappacher; MR 782485(86e:11112).Google Scholar