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Tame and wild kernels
of quadratic imaginary number fields

Authors: Jerzy Browkin and Herbert Gangl
Journal: Math. Comp. 68 (1999), 291-305
MSC (1991): Primary 11R11; Secondary 11R70, 11Y40, 19C99, 19F27
MathSciNet review: 1604336
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Abstract | References | Similar Articles | Additional Information

Abstract: For all quadratic imaginary number fields $F$ of discriminant
$d>-5000,$ we give the conjectural value of the order of Milnor's group (the tame kernel) $K_{2}O_{F},$ where $O_{F}$ is the ring of integers of $F.$ Assuming that the order is correct, we determine the structure of the group $K_{2}O_{F}$ and of its subgroup $W_{F}$ (the wild kernel). It turns out that the odd part of the tame kernel is cyclic (with one exception, $d=-3387$).

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

  • [BBCO] C. Bernardi. D. Batut, H. Cohen and M. Olivier, GP-PARI, a computer package.
  • [Bl] S.Bloch, Applications of the dilogarithm function in algebraic $K$-theory and algebraic geometry, Proc. Int. Symp. Alg. Geom., Kyoto, Kinokuniya, 1977, pp. 103-114. MR 82f:14009
  • [Bo1] A. Borel, Cohomologie de $\text{SL}\sb {n}$ et valeurs de fonctions zêta aux points entiers, Ann. Sc. Norm. Sup. Pisa (4) 4, no. 4 (1977), 613-636; errata 7, no. 2 (1980), 373. MR 58:22016; MR 81k:12012
  • [Bo2] A. Borel, Values of zeta-functions at integers, cohomology and polylogarithms, Current Trends in Mathematics and Physics, 1-44, Narosa, New Delhi, 1995. MR 97a:19005
  • [B-82] J. Browkin, The functor $K_{2}$ for the ring of integers of a number field, Universal Algebra and Applications, (Warsaw, 1978), Banach Center Publications, vol. 9, PWN, Warsaw, 1982, pp. 187-195. MR 85f:11084
  • [B-92] J. Browkin, On the $p-$rank of the tame kernel of algebraic number fields, Journ. Reine Angew. Math., 432 (1992), 135-149. MR 93j:11077
  • [B-S] J. Browkin and A. Schinzel, On Sylow 2-subgroups of $K_{2} O_{F}$ for quadratic number fields $F\,$, Journ. Reine Angew. Math., 331 (1982), 104-113. MR 83g:12011
  • [C-H] P. E. Conner and J. Hurrelbrink, Class number parity, Series in Pure Math. 8, World Scientific Publ, Singapore, 1988. MR 90f:11092
  • [Ga] H. Gangl, Werte von Dedekindschen Zetafunktionen, Dilogarithmuswerte und Pflasterungen des hyperbolischen Raumes, Diplomarbeit Bonn, 1989.
  • [Gr] D. Grayson, Dilogarithm computations for $K_{3}$ in: Algebraic $K-$theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math. 854 (1981), 168-178. MR 82i:12012
  • [KNF] M. Kolster, T. Nguyen Quang Do, V. Fleckinger, Twisted $S-$units, $p-$adic class number formulas, and the Lichtenbaum conjectures, Duke Math. J., 84 (1996), 679-717; errata 90 (1997), 641-643. MR 97g:11136; CMP 98:04
  • [Li] S. Lichtenbaum, Values of zeta-functions, étale cohomology, and algebraic $K$-theory, Lecture Notes in Math. 342 (1973), 489-501 Springer, Berlin. MR 53:10765
  • [M-W] B. Mazur, A. Wiles, Class fields of abelian extensions of ${{\Bbb Q}}$, Invent. Math. 76, no. 2 (1984), 179-330. MR 85m:11069
  • [Q1] Qin Hourong, The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields, Acta Arith., 69 (1995), 153-169. MR 96a:11132
  • [Q2] Qin Hourong, Computation of $K_{2} Z[\sqrt {-6}]$, Journ. Pure Appl. Algebra 96 (1994), 133-146. MR 95i:11135
  • [Q3] Qin Hourong, Computation of $K_{2} Z[{\frac{1+\sqrt {-35}}{2}}]$, Chin. Ann. of Math., 17B, 1 (1996), 63-72. MR 97a:19004
  • [Sk] M. Ska{\l}ba, Generalization of Thue's theorem and computation of the group $K_{2} O_{F}$, J. Number Theory 46 (1994), 303-322. MR 95d:19001
  • [Su] A.A. Suslin, Algebraic $K$-theory of fields, in: Proceedings of the International Congress of Mathematicians, Berkeley, CA, 1986, Vol.I, AMS, Providence, RI, 1987, pp. 222-244. MR 89k:12010
  • [Ta] J. Tate, Appendix to ``The Milnor ring of a global field" by H. Bass and J. Tate in: Algebraic $K-$theory, II: ``Classical" algebraic $K-$theory and connections with arithmetic (Proc. Conf., Seattle Res. Center, Battelle Memorial Inst., 1971), Lecture Notes in Math. 342 (1973), 429-446. MR 56:449

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Additional Information

Jerzy Browkin
Affiliation: Jerzy Browkin, Institute of Mathematics, University of Warsaw, ul. Banacha 2, PL-02-097 Warszawa, Poland

Herbert Gangl
Affiliation: Herbert Gangl, Institute for Experimental Mathematics, Ellernstr. 29, D-45326 Essen, Germany

Keywords: Tame kernel, wild kernel, quadratic imaginary fields, Lichtenbaum's conjecture
Received by editor(s): January 3, 1997
Additional Notes: The second author was supported by the Deutsche Forschungsgemeinschaft.
Article copyright: © Copyright 1999 American Mathematical Society

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