Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

On equivariant global epsilon constants for certain dihedral extensions


Author: Manuel Breuning
Journal: Math. Comp. 73 (2004), 881-898
MSC (2000): Primary 11R33; Secondary 11R42, 11Y40
Published electronically: August 19, 2003
MathSciNet review: 2031413
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a conjecture of Bley and Burns which relates the epsilon constant of the equivariant Artin $L$-function of a Galois extension of number fields to certain natural algebraic invariants. For an odd prime number $p$, we describe an algorithm which either proves the conjecture for all degree $2p$ dihedral extensions of the rational numbers or finds a counterexample. We apply this to show the conjecture for all degree $6$dihedral extensions of $\mathbb Q$. The correctness of the algorithm follows from a finiteness property of the conjecture which we prove in full generality.


References [Enhancements On Off] (What's this?)

  • 1. E. Artin and J. Tate, Class field theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0223335
  • 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 and D. Burns, Étale cohomology and a generalisation of Hilbert’s Theorem 132, Math. Z. 239 (2002), no. 1, 1–25. MR 1879327, 10.1007/s002090100281
  • 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 1863302, 10.1023/A:1014502826745
  • 7. D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501–570 (electronic). MR 1884523
  • 8. Charles W. Curtis and Irving Reiner, 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 632548
  • 9. S. Y. Kim, On the Equivariant Tamagawa Number Conjecture for Quaternion fields, thesis, King's College London (2002).
  • 10. Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723
  • 11. J. Martinet, Character theory and Artin 𝐿-functions, Algebraic number fields: 𝐿-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 1–87. MR 0447187
  • 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. Dieter Pumplün, Über die Klassenzahl und die Grundeinheit des reellquadratischen Zahlkörpers, J. Reine Angew. Math. 230 (1968), 167–210 (German). MR 0224590
  • 15. Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380
  • 16. V. Snaith, Burns' equivariant Tamagawa invariant $T\Omega^{loc}(N/\mathbb{Q} ,1)$ for some quaternion fields, to appear in J. London Math. Soc.
  • 17. J. T. Tate, Local constants, Algebraic number fields: 𝐿-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 89–131. Prepared in collaboration with C. J. Bushnell and M. J. Taylor. MR 0457408

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11R33, 11R42, 11Y40

Retrieve articles in all journals with MSC (2000): 11R33, 11R42, 11Y40


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: http://dx.doi.org/10.1090/S0025-5718-03-01605-3
Keywords: Equivariant Tamagawa number conjecture, equivariant epsilon constants, dihedral extensions
Received by editor(s): November 25, 2002
Published electronically: August 19, 2003
Additional Notes: The author was supported by the DAAD and the EPSRC
Article copyright: © Copyright 2003 American Mathematical Society