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Computations in Relative Algebraic K-Groups

Published online by Cambridge University Press:  01 February 2010

Werner Bley  and
Stephen M. J. Wilson
Werner Bley
Affiliation:
Fachbereich für Mathematik der Universität Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany, [email protected]
Stephen M. J. Wilson
Affiliation:
Heilbronn Institute for Mathematical Research, University Of Bristol, Royal Fort Annexe, Clifton, Bristol BS8 1TW, United Kingdom, [email protected]

Abstract

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Let G be finite group and K a number field or a p-adic field with ring of integers OK. In the first part of the manuscript we present an algorithm that computes the relative algebraic K-group K0(OK[G], K) as an abstract abelian group. We also give algorithms to solve the discrete logarithm problems in K0(OK[G], K) and in the locally free class group cl(OK[G]). All algorithms have been implemented in Magma for the case K = Q.

In the second part of the manuscript we prove formulae for the torsion subgroup of K0(Z[G], Q) for large classes of dihedral and quaternion groups.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 12 , 2009 , pp. 166 - 194
Copyright
Copyright © London Mathematical Society 2009

References

1.Bass, H., Algebraic K-theory, Benjamin, New York 1968.Google Scholar
2.Bley, W., Boltje, R., Computation of locally free class groups, in Hess, F., Pauli, S., Pohst, M. (Eds.), Algorithmic Number Theory, Lecture Notes in Computer Science 4076, Springer (2006), 72–86.Google Scholar
3.Bley, W., Breuning, M., Exact algorithms for p-adic fields and epsilon constant conjectures, preprint 2006, to appear in Illinois Journal of Mathematics.Google Scholar
4.Bley, W., Burns, D., Equivariant epsilon constants, discriminants and étale cohomology, Proc. London Math. Soc. 87 (2003), 545–590.CrossRefGoogle Scholar
5.Breuning, M., Equivariant epsilon constants for Galois extensions of number fields and p-adic fields, Phd thesis, King's College London, 2004.Google Scholar
6.Breuning, M., Burns, D., Leading terms of Artin L-functions at s = 0 and s = 1, Compositio Math. 143 (2007), 1427–1464.CrossRefGoogle Scholar
7.Burns, D., Equivariant Tamagawa numbers and Galois module theory I, Compositio Math. 129 (2001), 203–237.CrossRefGoogle Scholar
8.Curtis, C., Reiner, I., Methods of representation theory, volume I and II. Wiley, 1981 and 1987.Google Scholar
9.Chinburg, T., Exact sequences and Galois module structure, Ann. of Math. 121 (1985), 351–376.CrossRefGoogle Scholar
10.Cohen, H., A course in computational algebraic number theory, Springer Verlag (1993).CrossRefGoogle Scholar
11.Cohen, H., Advanced topics in computational number theory, Springer Verlag (2000).CrossRefGoogle Scholar
12.Eberly, W., Computations for Algebras and Group Representations, Phd thesis, University of Toronto, 1989.Google Scholar
13.Friedl, K., Rónyai, L., Polynomial time solutions for some problems in computational algebra, in Proceedings, 17th ACM Symposium on Theory of Computing, Providence, 1985, 153–162.CrossRefGoogle Scholar
14.Magma, Version V2.14–9, Sydney.Google Scholar
15.Nakayama, T. and Matsushima, Y., Über die multiplikative Gruppe einer p-adischen Divisionsalgebra, Proc. Imp. Acad. Tokyo 19 (1943) 622–628.Google Scholar
16.Reiner, I., Maximal orders, Academic Press, London 1975.Google Scholar
17.Swan, R. G., Algebraic K -theory, Lecture Notes in Mathematics 76, Springer Verlag (1968).CrossRefGoogle Scholar
18.Wang, S., On the commutator group of a simple algebra, Amer. J. Math. 72 (1950) 323–334.CrossRefGoogle Scholar
19.Taylor, M. J., On Fröhlich's conjecture for rings of integers of tame extensions, Inventiones Mathematicae, 63 (1981), 41–79.CrossRefGoogle Scholar
20.Weil, André, Basic Number Theory, Springer-Verlag (1974).CrossRefGoogle Scholar
21.Wilson, S. M. J., Twisted group rings and ramification, Proc. London Math. Soc. 31 (1975), 311–330.CrossRefGoogle Scholar