Trivialité du 2 -rang du noyau hilbertien

Hervé Thomas

Journal de théorie des nombres de Bordeaux (1994)

  • Volume: 6, Issue: 2, page 459-483
  • ISSN: 1246-7405

Abstract

top
We give exhaustive list of biquadratic fields K = ℚ ( i , m ) and K = ℚ ( 2 , m ) without 2 -exotic symbol, i.e. for which the 2 -rank of the Hilbert kernel (or wild kernel) is zero. Such K = ℚ ( i , m ) are logarithmic principals [J3]. We detail an exemple of this technical numerical exploration and quote the family of theories and results we utilize. The 2 -rank of tame, regular and wild kernel of K -theory are connected with local and global problem of embedding in a Z 2 -extension. Global class field theory can describe the 2 -rank of the Hilbert kernel and reveals existence of symbols on K not given by local class field theory.

How to cite

top

Thomas, Hervé. "Trivialité du $2$-rang du noyau hilbertien." Journal de théorie des nombres de Bordeaux 6.2 (1994): 459-483. <http://eudml.org/doc/93613>.

@article{Thomas1994,
author = {Thomas, Hervé},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Milnor -group; Hilbert symbols; class field theory; biquadratic fields; Hilbert kernel; regular kernel},
language = {fre},
number = {2},
pages = {459-483},
publisher = {Université Bordeaux I},
title = {Trivialité du $2$-rang du noyau hilbertien},
url = {http://eudml.org/doc/93613},
volume = {6},
year = {1994},
}

TY - JOUR
AU - Thomas, Hervé
TI - Trivialité du $2$-rang du noyau hilbertien
JO - Journal de théorie des nombres de Bordeaux
PY - 1994
PB - Université Bordeaux I
VL - 6
IS - 2
SP - 459
EP - 483
LA - fre
KW - Milnor -group; Hilbert symbols; class field theory; biquadratic fields; Hilbert kernel; regular kernel
UR - http://eudml.org/doc/93613
ER -

References

top
  1. [BT] H. Bass and J. Tate, The Milnor ring of a global field, (with an appendix by J. Tate), in Algebraic K-theory II. Lecture Notes in Mathematics, 342, Springer-Verlag, 1973. Berlin-Heidelberg- New York. Zbl0299.12013MR442061
  2. [BP] F. Bertrandias et J.-J. Payan, Γ-extensions et invariants cyclotomiques, Ann. scient. Éc. Norm. Sup.5 (1972), 517-548. Zbl0246.12005
  3. [BS] J. Browkin and A. Schinzel, On Sylow 2-subgroups of K2OF for quadratic number fields F, J. reine angew. Math.331 (1982), 104-113. Zbl0493.12013MR647375
  4. [Br] A. Brumer, On the units of algebraic number field, Mathematika14 (1967), 121-124. Zbl0171.01105MR220694
  5. [Co] J. Coates, p-adic L-functions and Iwasawa theory, in Durham symposium in algebraic number field, (A. Frôlich editor), Academic Press, 1977. New York, London. Zbl0393.12027MR460282
  6. [CH] P.E. Conner and J. Hurrelbrink, A comparison theorem for the 2-rank of K2D, Contemporary Mathematics55, Part II (1986), 411-420. Zbl0598.12012MR862645
  7. [Ga] H. Garland, A finiteness theorem for K2 of a number field, Annals of Math.94 (1971), 534-548. Zbl0247.12103MR297733
  8. [Gi] R. Gillard, Formulations de la conjecture de Leopoldt et étude d'une condition suffisante, Abh. Math. Sem. Hambourg48 (1979), 125-138. Zbl0396.12008MR537453
  9. [G1] G. Gras, Groupe de Galois de la p-extension abélienne p-ramifiée maximale d'un corps de nombres, J. reine angew. Math.333 (1982), 86-132. Zbl0477.12009MR660786
  10. [G2] G. Gras, Plongements kummeriens dans les Zp-extensions, Compositio Math.55 (1985), 383-396. Zbl0584.12004MR799822
  11. [GJ] G. Gras et J.-F. Jaulent, Sur les corps de nombres réguliers, Math. Z.202 (1989), 343-365. Zbl0704.11040MR1017575
  12. [J1] J.-F. Jaulent, L'arithmétique des l-extensions, (thèse) Pub. Math. Fac. Sci. Besançon, Théor. Nombres 1984-1985 & 1985-1986 (1986), 1-348. Zbl0601.12002MR898668
  13. [J2] J.-F. Jaulent, Sur les conjectures de Leopoldt et Gross, in Journées arithmétiques de Besançon, Astérisque147-148 (1987), 107-120. Zbl0623.12003MR891423
  14. [J3] J.-F. Jaulent, La théorie de Kummer et le K2 des corps de nombres, J. Théor. Nombres Bordeaux6 (1994). 
  15. [J4] J.-F. Jaulent, Sur le noyau sauvage des corps de nombres, Acta Arith.67 (1994), 335-348. Zbl0835.11042MR1301823
  16. [KC] K. Kramer et A. Candiotti, On K2 and Zl- extensions of numbers fields, Amer. J. Math.100 (1978), 177-196. Zbl0388.12004MR485369
  17. [Ma] H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. scient. Éc. Norm. Sup.42 (1969), 1-62. Zbl0261.20025MR240214
  18. [MN] A. Movahhedi & T. Nguyen Quang Do, Sur l'arithmétique des corps de nombres p-rationnels, Sém. Th. des Nbres Paris (1987/1988), Prog. Math.102 (1990), 155-197. Zbl0703.11059MR1042770
  19. [Ti] J. Tate, Symbols in arithmetics, Actes Congrès intern. math., Tome 1 (1970), 201-211. Zbl0229.12013MR422212
  20. [T2] J. Tate, Relation between K2 and Galois cohomology, Invent. Math.36 (1976), 257-274. Zbl0359.12011MR429837
  21. [Th] H. Thomas, Premier étage d'une Zl-extension, Manuscripta Math.81 (1993), 413-435. Zbl0799.11047MR1248764
  22. [Wh] K.S. Williams, Integers of biquadratic fields, Canad. Math. Bull.13 (1970), 519-528. Zbl0205.35401MR279069
  23. [Wi] A. Wiles, The Iwasawa conjecture for totally real fields, Annals of Math.131 (1990), 493-540. Zbl0719.11071MR1053488

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.