Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Bounds for computing the tame kernel

Author: Richard P. Groenewegen
Journal: Math. Comp. 73 (2004), 1443-1458
MSC (2000): Primary 11R70; Secondary 11Y40, 19C20
Published electronically: July 29, 2003
MathSciNet review: 2047095
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The tame kernel of the $K_2$ of a number field $F$ is the kernel of some explicit map $K_2F\to \bigoplus k_v^*$, where the product runs over all finite primes $v$ of $F$ and $k_v$ is the residue class field at $v$. When $S$ is a set of primes of $F$, containing the infinite ones, we can consider the $S$-unit group $U_S$ of $F$. Then $U_S\otimes U_S$ has a natural image in $K_2F$. The tame kernel is contained in this image if $S$ contains all finite primes of $F$ up to some bound. This is a theorem due to Bass and Tate. An explicit bound for imaginary quadratic fields was given by Browkin. In this article we give a bound, valid for any number field, that is smaller than Browkin's bound in the imaginary quadratic case and has better asymptotics. A simplified version of this bound says that we only have to include in $S$ all primes with norm up to  $4\vert\Delta\vert^{3/2}$, where $\Delta$ is the discriminant of $F$. Using this bound, one can find explicit generators for the tame kernel, and a ``long enough'' search would also yield all relations. Unfortunately, we have no explicit formula to describe what ``long enough'' means. However, using theorems from Keune, we can show that the tame kernel is computable.

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

  • 1. H. Bass and J. Tate, The Milnor ring of a global field, Lecture Notes in Mathematics, Vol. 342, pp. 349-446, Springer, Berlin (1973). MR 56:449
  • 2. J. Browkin, Computing the tame kernel of quadratic imaginary fields. With an appendix by Karim Belabas and Herbert Gangl, Mathematics of Computation 69 (2000), no. 232, pp. 1667-1683. MR 2001a:11189
  • 3. Y. Bugeaud and G. Kálmán, Bounds for the solutions of unit equations, Acta Arithmetica 74 (1996), no. 1, 67-80. MR 97b:11045
  • 4. H. Cohen, A Course in Computational Algebraic Number Theory, second corrected printing, Graduate Texts in Mathematics, 138, Springer (1995). MR 97e:14001
  • 5. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Grundlehren der mathematischen Wissenschaften, Vol. 290, Springer (1991). MR 93h:11069
  • 6. H. Garland, A finiteness theorem for $K\sb 2$ of a number field, Annals of Mathematics (2), 94, (1971), pp. 534-548. MR 45:6785
  • 7. E. Hlawka, Ausfüllung und Überdeckung durch Zylinder, Anzeiger der Österreichischen Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse, 85 (1948), nr. 11, pp. 116-119. MR 11:12c
  • 8. F. Keune, On the Structure of the $K_2$ of the Ring of Integers in a Number Field, $K$-Theory 2, (1989), pp. 625-645. MR 90g:11162
  • 9. K. Knopp, Theorie und Anwendung der unendlichen Reihen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 2, fünfte Auflage, Springer (1964). MR 10:446a
  • 10. S. Lang, Algebraic Number Theory, second edition, Graduate Texts in Mathematics, 110, Springer (1994). MR 95f:11085
  • 11. H. W. Lenstra, Jr., Algorithms in algebraic number theory, Bulletin of the American Mathematical Society (New Series) 26, (1992), no. 2, pp. 211-244. MR 93g:11131
  • 12. J. Milnor, Introduction to algebraic $K$-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, (1971). MR 50:2304
  • 13. D. Quillen, Higher algebraic $K$-theory I. In: Algebraic $K$-Theory I. Lecture Notes in Mathematics, 341, 85-147, Springer (1973). MR 49:2895
  • 14. C. A. Rogers, Packing and covering. Cambridge Tracts in Mathematics and Mathematical Physics, no. 54, Cambridge University Press, New York (1964). MR 30:2405
  • 15. J. Tate, Fourier Analysis in Number Fields and Hecke's Zeta-functions, Thesis Princeton (1950). In: J. W. S. Cassels, A. Fröhlich, Algebraic Number Theory, Thompson, Washington D.C. (1967). MR 36:121

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11R70, 11Y40, 19C20

Retrieve articles in all journals with MSC (2000): 11R70, 11Y40, 19C20

Additional Information

Richard P. Groenewegen
Affiliation: Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands
Address at time of publication: ABN AMRO, Gustav Mahlerlaan 10, HQ 1056, 1082 PP Amsterdam, The Netherlands

Keywords: $K$-theory, tame kernel, calculations, $S$-units
Received by editor(s): April 18, 2002
Received by editor(s) in revised form: December 6, 2002
Published electronically: July 29, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society