Algorithmic proof of the epsilon constant conjecture
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- by Werner Bley and Ruben Debeerst PDF
- Math. Comp. 82 (2013), 2363-2387 Request permission
Abstract:
In this paper we will algorithmically prove the global epsilon constant conjecture for all Galois extensions $L/\mathbb {Q}$ of degree at most $15$. In fact, we will obtain a slightly more general result whose proof is based on an algorithmic proof of the local epsilon constant conjecture for Galois extensions $E/\mathbb {Q}_p$ of small degree. To this end we will present an efficient algorithm for the computation of local fundamental classes and address several other problems arising in the algorithmic proof of the local conjecture.References
- Vincenzo Acciaro and Jürgen Klüners, Computing local Artin maps, and solvability of norm equations, J. Symbolic Comput. 30 (2000), no. 3, 239–252. MR 1775936, DOI 10.1006/jsco.2000.0361
- W. Bley, Numerical evidence for a conjectural generalization of Hilbert’s Theorem 132, LMS J. Comput. Math. 6 (2003), 68–88. With an appendix by D. Kusnezow. MR 1971493, DOI 10.1112/S1461157000000383
- Werner Bley and Manuel Breuning, Exact algorithms for $p$-adic fields and epsilon constant conjectures, Illinois J. Math. 52 (2008), no. 3, 773–797. MR 2546007
- W. Bley and D. Burns, Equivariant epsilon constants, discriminants and étale cohomology, Proc. London Math. Soc. (3) 87 (2003), no. 3, 545–590. MR 2005875, DOI 10.1112/S0024611503014217
- Werner Bley and Stephen M. J. Wilson, Computations in relative algebraic $K$-groups, LMS J. Comput. Math. 12 (2009), 166–194. MR 2564571, DOI 10.1112/S1461157000001480
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- M. Breuning. Equivariant epsilon constants for Galois extensions of number fields and $p$-adic fields. Ph.D. thesis, King’s College, London, May 2004.
- Manuel Breuning, Equivariant local epsilon constants and étale cohomology, J. London Math. Soc. (2) 70 (2004), no. 2, 289–306. MR 2078894, DOI 10.1112/S002461070400554X
- Manuel Breuning, On equivariant global epsilon constants for certain dihedral extensions, Math. Comp. 73 (2004), no. 246, 881–898. MR 2031413, DOI 10.1090/S0025-5718-03-01605-3
- Manuel Breuning and David Burns, Leading terms of Artin $L$-functions at $s=0$ and $s=1$, Compos. Math. 143 (2007), no. 6, 1427–1464. MR 2371375, DOI 10.1112/S0010437X07002874
- Manuel Breuning and David Burns, On equivariant Dedekind zeta-functions at $s=1$, Doc. Math. Extra vol.: Andrei A. Suslin sixtieth birthday (2010), 119–146. MR 2804251
- David Burns, Equivariant Whitehead torsion and refined Euler characteristics, Number theory, CRM Proc. Lecture Notes, vol. 36, Amer. Math. Soc., Providence, RI, 2004, pp. 35–59. MR 2076565, DOI 10.1090/crmp/036/04
- D. Burns and M. Flach, Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math. 6 (2001), 501–570. MR 1884523
- David Burns and Matthias Flach, On the equivariant Tamagawa number conjecture for Tate motives. II, Doc. Math. Extra Vol. (2006), 133–163. MR 2290586
- Ted Chinburg, Exact sequences and Galois module structure, Ann. of Math. (2) 121 (1985), no. 2, 351–376. MR 786352, DOI 10.2307/1971177
- Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313, DOI 10.1007/978-1-4419-8489-0
- R. Debeerst. Algorithms for Tamagawa Number Conjectures. Ph.D. thesis, University of Kassel, 2011. URL http://kobra.bibliothek.uni-kassel.de/handle/urn:nbn:de:hebis:34-2011060937825.
- Claus Fieker, Applications of the class field theory of global fields, Discovering mathematics with Magma, Algorithms Comput. Math., vol. 19, Springer, Berlin, 2006, pp. 31–62. MR 2278922, DOI 10.1007/978-3-540-37634-7_{2}
- Albrecht Fröhlich, Galois module structure of algebraic integers, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 1, Springer-Verlag, Berlin, 1983. MR 717033, DOI 10.1007/978-3-642-68816-4
- Christian Greve. Galoisgruppen von Eisensteinpolynomen über p-adischen Körpern. Ph.D. thesis, Universität Paderborn, Oct. 2010.
- Guy Henniart, Relèvement global d’extensions locales: quelques problèmes de plongement, Math. Ann. 319 (2001), no. 1, 75–87 (French, with English and French summaries). MR 1812820, DOI 10.1007/PL00004431
- Derek F. Holt, Cohomology and group extensions in Magma, Discovering mathematics with Magma, Algorithms Comput. Math., vol. 19, Springer, Berlin, 2006, pp. 221–241. MR 2278930, DOI 10.1007/978-3-540-37634-7_{1}0
- Christian U. Jensen, Arne Ledet, and Noriko Yui, Generic polynomials, Mathematical Sciences Research Institute Publications, vol. 45, Cambridge University Press, Cambridge, 2002. Constructive aspects of the inverse Galois problem. MR 1969648
- J. W. Jones and D. P. Roberts. Database of local fields. URL http://math.la.asu.edu/~jj/localfields/.
- Jürgen Klüners and Gunter Malle, A database for field extensions of the rationals, LMS J. Comput. Math. 4 (2001), 182–196. MR 1901356, DOI 10.1112/S1461157000000851
- F. Lorenz. Einführung in die Algebra II. Spektrum Akademischer Verlag, Heidelberg, 1997.
- 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
- Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
- Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR 1737196
- Sebastian Pauli and Xavier-François Roblot, On the computation of all extensions of a $p$-adic field of a given degree, Math. Comp. 70 (2001), no. 236, 1641–1659. MR 1836924, DOI 10.1090/S0025-5718-01-01306-0
- I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003. Corrected reprint of the 1975 original; With a foreword by M. J. Taylor. MR 1972204
- Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
- Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
- R. G. Swan, Algebraic $K$-theory, Lecture Notes in Mathematics, No. 76, Springer-Verlag, Berlin-New York, 1968. MR 0245634
- M. J. Taylor, On Fröhlich’s conjecture for rings of integers of tame extensions, Invent. Math. 63 (1981), no. 1, 41–79. MR 608528, DOI 10.1007/BF01389193
Additional Information
- Werner Bley
- Affiliation: Universität München, Theresienstr. 39, 80333 München, Germany
- Email: bley@math.lmu.de
- Ruben Debeerst
- Affiliation: Universität Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany
- Address at time of publication: Heidelberger Landstraße 101B, 64 297 Darmstadt, Germany
- Email: ruben.debeerst@gmx.de
- Received by editor(s): October 7, 2011
- Received by editor(s) in revised form: February 23, 2012
- Published electronically: April 1, 2013
- Additional Notes: The second author was supported by DFG grant BL 395/3-1
- © Copyright 2013 American Mathematical Society
- Journal: Math. Comp. 82 (2013), 2363-2387
- MSC (2010): Primary 11Y40; Secondary 11R33, 11S25
- DOI: https://doi.org/10.1090/S0025-5718-2013-02691-9
- MathSciNet review: 3073206