Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T06:36:54.298Z Has data issue: false hasContentIssue false
  • English
  • Français

Indépendance algébrique de logarithmes en caractéristique P

Published online by Cambridge University Press:  17 April 2009

Laurent Denis
Laurent Denis
Affiliation:
Laboratoire Paul Painlevé UMR CNRS 8524, U.F.R. de Mathématiques Pures et Appliquées, Bât. M2, Université des Sciences et Technologies de Lille 1, 59665 Villeneuve d'Ascq Cedex, France, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let k be the rational function field over the field with q elements with characteristic p. Since the work of Carlitz we know in this situation the function ζ analog of the Riemann zeta function and the function Logφ analog of the usual logarithm. We will show two main results. Firstly, if ξ denotes the fundamental period of Carlitz module, we prove that ξ, ζ(1),…, ζ(p – 2) are algebraically independent over k. Secondly if α1,…, αn are rational elements (of degree less than q/(q − 1) to ensure convergence of the logarithm) such that Logφ α1,…, Logφ αn are linearly independent over k then they are algebraically independent over k. The point is to find suitable functions taking these values and for which Mahler's method can be used.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 74 , Issue 3 , December 2006 , pp. 461 - 470
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Becker, P.G., ‘Algebraic independence of the values of certain series by Mahler's method’, Monatsh. Math. 114 (1992), 183–198.CrossRefGoogle Scholar
[2]Berthé, V., ‘Automates et valeurs de transcendance du logarithme de Carlitz’, Acta Arith. 66 (1994), 369–390.CrossRefGoogle Scholar
[3]Carlitz, L., ‘On certain functions connected with polynomials in a Galois field’, Duke Math. J. 1 (1935), 137–168.CrossRefGoogle Scholar
[4]Denis, L., ‘Théorème de Baker et modules de Drinfeld’, J. Number Theory 43 (1993), 203–215.CrossRefGoogle Scholar
[5]Denis, L., ‘Dérivées d'un module de Drinfeld et transcendance’, Duke Math. J. 80 (1995), 1–13.CrossRefGoogle Scholar
[6]Denis, L., ‘Méthodes fonctionnelles pour la transcendance en caractéristique finie’, Bull. Austral. Math. Soc. 50 (1994), 273–286.CrossRefGoogle Scholar
[7]Denis, L., ‘Indépendance algébrique des dérivées d'une période du module de Carlitz’, J. Austral. Math. Soc. 69 (2000), 8–18.CrossRefGoogle Scholar
[8]Denis, L., ‘Indépendance algébrique de différents π’, C.R. Acad. Sci. Paris, Sér. 1 Math. 327 (1998), 711–714.CrossRefGoogle Scholar
[9]Damamme, G. et Hellegouarch, Y., ‘Transcendence of the values of the Carlitz Zeta function by Wade's method’, J. Number Theory 39 (1991), 257–278.CrossRefGoogle Scholar
[10]de Mathan, B., ‘Un critère de transcendance en caractéristique positive’, C.R. Acad. Sci. Paris, Sér. 1 Math. 319 (1994), 427–432.Google Scholar
[11]Papanikolas, M.A., ‘Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms’, (preprint 2005).Google Scholar
[12]Wade, L.I., ‘Certain quantities transcendental over GF(pn, x)’, Duke Math. J. 8 (1941), 701–720.CrossRefGoogle Scholar
[13]Yu, J., ‘Analytic homomorphism into Drinfeld modules’, Ann. of Math. (2) 145 (1997), 215–233.CrossRefGoogle Scholar
[14]Yu, J., ‘Transcendance and Drinfeld modules: Several variables’, Duke Math. J. 58 (1989), 559–575.CrossRefGoogle Scholar