logo ANNALES DE L'INSTITUT FOURIER

With cedram.org
Table of contents for this issue | Previous article | Next article
Hassan Oukhaba
Sign functions of imaginary quadratic fields and applications
(Fonction signe des corps quadratiques imaginaires et applications)
Annales de l'institut Fourier, 55 no. 3 (2005), p. 753-772, doi: 10.5802/aif.2113
Article PDF | Reviews MR 2149402 | Zbl 02171524 | 1 citation in Cedram
Class. Math.: 11G16, 14K22, 11R21
Keywords: sign function, narrow ray class field, Shimura reciprocity law, ordianary $s$-ditributions, Anderson's resolution, spectrales sequences

Résumé - Abstract

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.

Bibliography

[1] G. W. Anderson, A double complex for computing the sign-cohomology of the universal ordinary distribution, Amer. Math. Soc., 1997, p. 1-27  Zbl 0939.11035
[2] G. W. Anderson, “Kronecker-Weber plus epsilon”, Duke Math. J. 114 (2002) no. 3, p. 439-475 Article |  MR 1924570 |  Zbl 1056.11060
[3] S. Bae & L. Yin, “Epsilon extensions over global function fields”, Manuscripta Math. 110 (2003) no. 3, p. 313-324 Article |  MR 1969003 |  Zbl 01945277
[4] J.-R. Belliard & H. Oukhaba, “Sur la torsion de la distribution ordinaire universelle attachée à un corps de nombres”, Manuscripta Math. 106 (2001) no. 1, p. 117-130 Article |  MR 1860983 |  Zbl 1011.11076
[5] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer-Verlag, New-York, 1982  MR 672956 |  Zbl 0584.20036
[6] F. Hajir & F. R. Villegas, “Explicit elliptic units, I”, Duke Math. J. 90 (1997) no. 3, p. 495-521 Article |  MR 1480544 |  Zbl 0898.11025
[7] D. R. Hayes, “Stickelberger elements in function fields”, Compositio Math. 55 (1985) no. 2, p. 209-239 Numdam |  MR 795715 |  Zbl 0569.12008
[8] D. R. Hayes, A brief introduction to Drinfel'd modules, Ohio State Univ. Math. Res. Inst. Publ., Gruyter, 1991, p. 1-32  Zbl 0793.11015
[9] A. Hayward, “Congruences satisfied by Stark units”, PhD thesis, King's College, London, 2004
[10] P. J. Hilton & U. Stammbach, A course in homological algebra, Graduate Texts in Mathematics 4, Springer-Verlag, New York, 1971  MR 346025 |  Zbl 0238.18006
[11] D. Kubert, “The universal ordinary distribution”, Bull. Soc. Math. France 107 (1979) no. 2, p. 179-202 Numdam |  MR 545171 |  Zbl 0409.12021
[12] D. S. Kubert, “Product formulae on elliptic curves”, Invent. Math. 117 (1994) no. 2, p. 227-273  MR 1273265 |  Zbl 0834.14016
[13] D. S. Kubert & S. Lang, Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematicsl Science] 244, Springer-Verlag, New York, 1981  MR 648603 |  Zbl 0492.12002
[14] H. Oukhaba, “Index formulas for ramified elliptic units”, Compositio Math. 137 (2003) no. 1, p. 1-22 Article |  MR 1981934 |  Zbl 1045.11043
[15] Y. Ouyang, “Group cohomology of the universal ordinary distribution”, J. Reine Angew. Math. 537 (2001), p. 1-32  MR 1856256 |  Zbl 1008.11042
[16] Y. Ouyang, “The universal norm distribution and Sinnott's index formula”, Proc. Amer. Math. Soc. 130 (2002) no. 8, p. 2203-2213 Article |  MR 1896399 |  Zbl 0997.11089
[17] G. Robert, Unités elliptiques, Mémoires 36, Bull. Soc. Math. France, 1973 Numdam |  MR 469889 |  Zbl 0314.12006
[18] G. Robert, La racine 12-ième canonique $\Delta (L)^{[{\underline L}:L]}/\Delta ({\underline L})$, Séminaire de Théorie des Nombres, Paris, 1989-90, Birkhäuser, 1992, p. 209-232  Zbl 0751.11025
[19] R. Schertz, Niedere Potenzen elliptischer Einheiten, Nagoya university, 1986, p. 67-88  Zbl 0615.12013
[20] W. Sinnott, “On the Stickelberger ideal and the circular units of a cyclotomic field”, Ann. of Math. 2 108 (1978) no. 1, p. 107-134  MR 485778 |  Zbl 0395.12014
[21] H. M. Stark, “$L$-functions at $s=1$, IV, First derivatives at $s = 0$”, Adv. in Math. 35 (1980) no. 3, p. 197-235 Article |  MR 563924 |  Zbl 0475.12018
[22] L. Yin, “Index-class number formulas over global function fields”, Compositio Math. 109 (1997) no. 1, p. 49-66 Article |  MR 1473605 |  Zbl 0902.11023
[23] L. Yin, “On the index of cyclotomic units in characteristic $p$ and its applications”, J. Number Theory 63 (1997) no. 2, p. 302-324 Article |  MR 1443764 |  Zbl 0896.11023
[24] L. Yin, “Distributions on a global field”, J. Number Theory 80 (2000) no. 1, p. 154-167 Article |  MR 1735653 |  Zbl 1005.11062
top