Mathematics of Computation

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

 

 

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] Spencer Bloch, Applications of the dilogarithm function in algebraic 𝐾-theory and algebraic geometry, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) Kinokuniya Book Store, Tokyo, 1978, pp. 103–114. MR 578856
  • [Bo1] Armand Borel, Cohomologie de 𝑆𝐿_{𝑛} et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 4, 613–636 (French). MR 0506168
    Armand Borel, Errata: “Cohomologie de 𝑆𝐿_{𝑛} et valeurs de fonctions zêta aux points entiers” [Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 4, 613–636; MR 58 #22016], Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 2, 373 (French). MR 581146
  • [Bo2] Armand Borel, Values of zeta-functions at integers, cohomology and polylogarithms, Current trends in mathematics and physics, Narosa, New Delhi, 1995, pp. 1–44. MR 1354171
  • [B-82] Jerzy Browkin, The functor 𝐾₂ for the ring of integers of a number field, Universal algebra and applications (Warsaw, 1978) Banach Center Publ., vol. 9, PWN, Warsaw, 1982, pp. 187–195. MR 738813
  • [B-92] Jerzy Browkin, On the 𝑝-rank of the tame kernel of algebraic number fields, J. Reine Angew. Math. 432 (1992), 135–149. MR 1184763, 10.1515/crll.1992.432.135
  • [B-S] J. Browkin and A. Schinzel, On Sylow 2-subgroups of 𝐾₂𝑂_{𝐹} for quadratic number fields 𝐹, J. Reine Angew. Math. 331 (1982), 104–113. MR 647375, 10.1515/crll.1982.331.104
  • [C-H] P. E. Conner and J. Hurrelbrink, Class number parity, Series in Pure Mathematics, vol. 8, World Scientific Publishing Co., Singapore, 1988. MR 963648
  • [Ga] H. Gangl, Werte von Dedekindschen Zetafunktionen, Dilogarithmuswerte und Pflasterungen des hyperbolischen Raumes, Diplomarbeit Bonn, 1989.
  • [Gr] Daniel R. Grayson, Dilogarithm computations for 𝐾₃, Algebraic 𝐾-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980) Lecture Notes in Math., vol. 854, Springer, Berlin, 1981, pp. 168–178. MR 618304
  • [KNF] Manfred Kolster, Thong Nguyen Quang Do, and Vincent Fleckinger, Twisted 𝑆-units, 𝑝-adic class number formulas, and the Lichtenbaum conjectures, Duke Math. J. 84 (1996), no. 3, 679–717. MR 1408541, 10.1215/S0012-7094-96-08421-5
  • [Li] Stephen Lichtenbaum, Values of zeta-functions, étale cohomology, and algebraic 𝐾-theory, Algebraic 𝐾-theory, II: “Classical” algebraic 𝐾-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 489–501. Lecture Notes in Math., Vol. 342. MR 0406981
  • [M-W] B. Mazur and A. Wiles, Class fields of abelian extensions of 𝑄, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, 10.1007/BF01388599
  • [Q1] Hou Rong Qin, The 2-Sylow subgroups of the tame kernel of imaginary quadratic fields, Acta Arith. 69 (1995), no. 2, 153–169. MR 1316704
  • [Q2] Hou Rong Qin, Computation of 𝐾₂𝐙[√-6], J. Pure Appl. Algebra 96 (1994), no. 2, 133–146. MR 1303542, 10.1016/0022-4049(94)90124-4
  • [Q3] Hourong Qin, Computation of 𝐾₂𝑍[(1+√-35)/2], Chinese Ann. Math. Ser. B 17 (1996), no. 1, 63–72. A Chinese summary appears in Chinese Ann. Math. Ser. A 17 (1996), no. 1, 121. MR 1387182
  • [Sk] Mariusz Skałba, Generalization of Thue’s theorem and computation of the group 𝐾₂𝑂_{𝐹}, J. Number Theory 46 (1994), no. 3, 303–322. MR 1273447, 10.1006/jnth.1994.1016
  • [Su] A. A. Suslin, Algebraic 𝐾-theory of fields, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 222–244. MR 934225
  • [Ta] H. Bass and J. Tate, The Milnor ring of a global field, Algebraic 𝐾-theory, II: “Classical” algebraic 𝐾-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972) Springer, Berlin, 1973, pp. 349–446. Lecture Notes in Math., Vol. 342. MR 0442061

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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
Email: bro@mimuw.edu.pl

Herbert Gangl
Affiliation: Herbert Gangl, Institute for Experimental Mathematics, Ellernstr. 29, D-45326 Essen, Germany
Email: herbert@mpim-bonn.mpg.de

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01000-5
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