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On equivariant global epsilon constants for certain dihedral extensions
Author(s):
Manuel
Breuning.
Journal:
Math. Comp.
73
(2004),
881-898.
MSC (2000):
Primary 11R33;
Secondary 11R42, 11Y40
Posted:
August 19, 2003
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Abstract:
We consider a conjecture of Bley and Burns which relates the epsilon constant of the equivariant Artin  -function of a Galois extension of number fields to certain natural algebraic invariants. For an odd prime number  , we describe an algorithm which either proves the conjecture for all degree  dihedral extensions of the rational numbers or finds a counterexample. We apply this to show the conjecture for all degree  dihedral extensions of  . The correctness of the algorithm follows from a finiteness property of the conjecture which we prove in full generality.
References:
-
- 1.
- E. Artin, J. Tate, Class field theory, W.A. Benjamin, Inc., New York-Amsterdam, 1968. MR 36:6383
- 2.
- W. Bley, Computation of Stark-Tamagawa units, Math. Comp. 72 (2003), 1963-1974.
- 3.
- W. Bley, Numerical evidence for a conjectural generalization of Hilbert's Theorem 132, LMS J. Comput. Math. 6 (2003), 68-88 (electronic).
- 4.
- W. Bley, D. Burns, Étale cohomology and a generalisation of Hilbert's Theorem 132, Math. Z. 239 (2002), no. 1, 1-25. MR 2002j:11135
- 5.
- W. Bley, D. Burns, Equivariant epsilon constants, discriminants and étale cohomology, preprint 2001, to appear in Proc. London Math. Soc.
- 6.
- D. Burns, Equivariant Tamagawa numbers and Galois module theory I, Compositio Math. 129 (2001), no. 2, 203-237. MR 2002g:11152
- 7.
- D. Burns, M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501-570. MR 2002m:11055
- 8.
- C. W. Curtis, I. Reiner, Methods of representation theory. Vol. I, John Wiley & Sons, Inc, New York, 1981. MR 82i:20001
- 9.
- S. Y. Kim, On the Equivariant Tamagawa Number Conjecture for Quaternion fields, thesis, King's College London (2002).
- 10.
- S. Lang, Algebraic number theory, Second Edition, Graduate Texts in Mathematics 110, Springer-Verlag, New York, 1994. MR 95f:11085
- 11.
- J. Martinet, Character theory and Artin
 -functions, in: Algebraic number fields (ed. A. Fröhlich), pp. 1-87, Academic Press, London, 1977. MR 56:5502 - 12.
- J. Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992.
- 13.
- The Pari Group, PARI/GP, Version 2.1.4, 2000 Bordeaux, available from http:// www.parigp-home.de/.
- 14.
- D. Pumplün, Über die Klassenzahl und die Grundeinheit des reellquadratischen Zahlkörpers, J. Reine Angew. Math. 230 (1968), 167-210. MR 37:189
- 15.
- J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer-Verlag, New York-Heidelberg, 1977. MR 56:8675
- 16.
- V. Snaith, Burns' equivariant Tamagawa invariant
 for some quaternion fields, to appear in J. London Math. Soc. - 17.
- J. T. Tate, Local constants, in: Algebraic number fields (ed. A. Fröhlich), pp. 89-131, Academic Press, London, 1977. MR 56:15613
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Additional Information:
Manuel
Breuning
Affiliation:
Department of Mathematics, King's College London, Strand, London WC2R 2LS, United Kingdom
Email:
breuning@mth.kcl.ac.uk
DOI:
10.1090/S0025-5718-03-01605-3
PII:
S 0025-5718(03)01605-3
Keywords:
Equivariant Tamagawa number conjecture,
equivariant epsilon constants,
dihedral extensions
Received by editor(s):
November 25, 2002
Posted:
August 19, 2003
Additional Notes:
The author was supported by the DAAD and the EPSRC
Copyright of article:
Copyright
2003,
American Mathematical Society
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