Sign functions of imaginary quadratic fields and applications
[Fonction signe des corps quadratiques imaginaires et applications]
Annales de l'Institut Fourier, Tome 55 (2005) no. 3, pp. 753-772.

Nous proposons une définition du signe pour les corps quadratiques imaginaires. Nous donnons un exemple de telles fonctions et l'utilisons pour définir de nouveaux invariants qui sont racines des invariants de Ramachandra classiques. D'autre part nous introduisons les distributions ordinaires signées et calculons leur cohomologie à l'aide de la theéorie d'Anderson dite du double complexe.

We propose a definition of sign of imaginary quadratic fields. We give an example of such functions, and use it to define new invariants that are roots of the classical Ramachandra invariants. Also we introduce signed ordinary distributions and compute their signed cohomology by using Anderson's theory of double complex.

DOI : 10.5802/aif.2113
Classification : 11G16, 14K22, 11R21
Keywords: sign function, narrow ray class field, Shimura reciprocity law, ordianary $s$-ditributions, Anderson’s resolution, spectrales sequences
Mot clés : fonction signe, corps de rayon restreints, loi de réciprocité de Shimura, $s$-distributions ordinaires, résolution d’Anderson, suites spectrales
Oukhaba, Hassan 1

1 Université de Franche-Comte, laboratoire de mathématique, 16 Route de Gray, 25030 Besançon cedex (France)
@article{AIF_2005__55_3_753_0,
     author = {Oukhaba, Hassan},
     title = {Sign functions of imaginary quadratic fields and applications},
     journal = {Annales de l'Institut Fourier},
     pages = {753--772},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {3},
     year = {2005},
     doi = {10.5802/aif.2113},
     mrnumber = {2149402},
     zbl = {02171524},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2113/}
}
TY  - JOUR
AU  - Oukhaba, Hassan
TI  - Sign functions of imaginary quadratic fields and applications
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 753
EP  - 772
VL  - 55
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2113/
DO  - 10.5802/aif.2113
LA  - en
ID  - AIF_2005__55_3_753_0
ER  - 
%0 Journal Article
%A Oukhaba, Hassan
%T Sign functions of imaginary quadratic fields and applications
%J Annales de l'Institut Fourier
%D 2005
%P 753-772
%V 55
%N 3
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2113/
%R 10.5802/aif.2113
%G en
%F AIF_2005__55_3_753_0
Oukhaba, Hassan. Sign functions of imaginary quadratic fields and applications. Annales de l'Institut Fourier, Tome 55 (2005) no. 3, pp. 753-772. doi : 10.5802/aif.2113. http://archive.numdam.org/articles/10.5802/aif.2113/

[1] G. W. Anderson A double complex for computing the sign-cohomology of the universal ordinary distribution, Recent progress in algebra (Taejon/Seoul, 1997), Volume 199 (1997), pp. 1-27 | Zbl

[2] G. W. Anderson Kronecker-Weber plus epsilon, Duke Math. J., Volume 114 (2002) no. 3, pp. 439-475 | DOI | MR | Zbl

[3] S. Bae; L. Yin Epsilon extensions over global function fields, Manuscripta Math., Volume 110 (2003) no. 3, pp. 313-324 | DOI | MR | Zbl

[4] J.-R. Belliard; H. Oukhaba Sur la torsion de la distribution ordinaire universelle attachée à un corps de nombres, Manuscripta Math., Volume 106 (2001) no. 1, pp. 117-130 | DOI | MR | Zbl

[5] K. S. Brown Cohomology of groups, Graduate Texts in Mathematics, 87, Springer-Verlag, New-York, 1982 | MR | Zbl

[6] F. Hajir; F. R. Villegas Explicit elliptic units, I, Duke Math. J., Volume 90 (1997) no. 3, pp. 495-521 | DOI | MR | Zbl

[7] D. R. Hayes Stickelberger elements in function fields, Compositio Math., Volume 55 (1985) no. 2, pp. 209-239 | Numdam | MR | Zbl

[8] D. R. Hayes A brief introduction to Drinfel'd modules, The arithmetic of function fields (Columbus, OH, 1991) (Ohio State Univ. Math. Res. Inst. Publ.), Volume 2 (1991), pp. 1-32 | Zbl

[9] A. Hayward Congruences satisfied by Stark units (2004) (PhD thesis, King's College, London)

[10] P. J. Hilton; U. Stammbach A course in homological algebra, Graduate Texts in Mathematics, 4, Springer-Verlag, New York, 1971 | MR | Zbl

[11] D. Kubert The universal ordinary distribution, Bull. Soc. Math. France, Volume 107 (1979) no. 2, pp. 179-202 | Numdam | MR | Zbl

[12] D. S. Kubert Product formulae on elliptic curves, Invent. Math., Volume 117 (1994) no. 2, pp. 227-273 | MR | Zbl

[13] D. S. Kubert; S. Lang Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematicsl Science], 244, Springer-Verlag, New York, 1981 | MR | Zbl

[14] H. Oukhaba Index formulas for ramified elliptic units, Compositio Math., Volume 137 (2003) no. 1, pp. 1-22 | DOI | MR | Zbl

[15] Y. Ouyang Group cohomology of the universal ordinary distribution, J. Reine Angew. Math., Volume 537 (2001), pp. 1-32 | MR | Zbl

[16] Y. Ouyang The universal norm distribution and Sinnott's index formula, Proc. Amer. Math. Soc., Volume 130 (2002) no. 8, pp. 2203-2213 | DOI | MR | Zbl

[17] G. Robert Unités elliptiques, Mémoires, 36, Bull. Soc. Math. France, 1973 | Numdam | MR | Zbl

[18] G. Robert La racine 12-ième canonique Δ(L) [L ̲:L] /Δ(L ̲) (Séminaire de Théorie des Nombres, Paris, 1989-90) (1992), pp. 209-232 | Zbl

[19] R. Schertz Niedere Potenzen elliptischer Einheiten, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986) (1986), pp. 67-88 | Zbl

[20] W. Sinnott On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math. 2, Volume 108 (1978) no. 1, pp. 107-134 | MR | Zbl

[21] H. M. Stark L-functions at s=1, IV, First derivatives at s=0, Adv. in Math., Volume 35 (1980) no. 3, pp. 197-235 | DOI | MR | Zbl

[22] L. Yin Index-class number formulas over global function fields, Compositio Math., Volume 109 (1997) no. 1, pp. 49-66 | DOI | MR | Zbl

[23] L. Yin On the index of cyclotomic units in characteristic p and its applications, J. Number Theory, Volume 63 (1997) no. 2, pp. 302-324 | DOI | MR | Zbl

[24] L. Yin Distributions on a global field, J. Number Theory, Volume 80 (2000) no. 1, pp. 154-167 | DOI | MR | Zbl

Cité par Sources :