Tame and wild kernels of quadratic imaginary number fields
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- by Jerzy Browkin and Herbert Gangl PDF
- Math. Comp. 68 (1999), 291-305 Request permission
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
- C. Bernardi. D. Batut, H. Cohen and M. Olivier, GP–PARI, a computer package.
- 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
- Armand Borel, Cohomologie de $\textrm {SL}_{n}$ et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 4, 613–636 (French). MR 506168
- 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
- Jerzy Browkin, The functor $K_{2}$ 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
- Jerzy Browkin, On the $p$-rank of the tame kernel of algebraic number fields, J. Reine Angew. Math. 432 (1992), 135–149. MR 1184763, DOI 10.1515/crll.1992.432.135
- J. Browkin and A. Schinzel, On Sylow $2$-subgroups of $K_{2}O_{F}$ for quadratic number fields $F$, J. Reine Angew. Math. 331 (1982), 104–113. MR 647375, DOI 10.1515/crll.1982.331.104
- P. E. Conner and J. Hurrelbrink, Class number parity, Series in Pure Mathematics, vol. 8, World Scientific Publishing Co., Singapore, 1988. MR 963648, DOI 10.1142/0663
- H. Gangl, Werte von Dedekindschen Zetafunktionen, Dilogarithmuswerte und Pflasterungen des hyperbolischen Raumes, Diplomarbeit Bonn, 1989.
- Daniel R. Grayson, Dilogarithm computations for $K_{3}$, 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, DOI 10.1007/BFb0089521
- 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, DOI 10.1215/S0012-7094-96-08421-5
- 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) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 489–501. MR 0406981
- B. Mazur and A. Wiles, Class fields of abelian extensions of $\textbf {Q}$, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, DOI 10.1007/BF01388599
- 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, DOI 10.4064/aa-69-2-153-169
- Hou Rong Qin, Computation of $K_2\mathbf Z[\sqrt {-6}]$, J. Pure Appl. Algebra 96 (1994), no. 2, 133–146. MR 1303542, DOI 10.1016/0022-4049(94)90124-4
- Hourong Qin, Computation of $K_2\textbf {Z}[(1+\sqrt {-35})/2]$, 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
- Mariusz Skałba, Generalization of Thue’s theorem and computation of the group $K_2O_F$, J. Number Theory 46 (1994), no. 3, 303–322. MR 1273447, DOI 10.1006/jnth.1994.1016
- 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
- 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., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 349–446. MR 0442061, DOI 10.1007/BFb0073733
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
- Received by editor(s): January 3, 1997
- Additional Notes: The second author was supported by the Deutsche Forschungsgemeinschaft.
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 68 (1999), 291-305
- MSC (1991): Primary 11R11; Secondary 11R70, 11Y40, 19C99, 19F27
- DOI: https://doi.org/10.1090/S0025-5718-99-01000-5
- MathSciNet review: 1604336