On -adic Iterated Integrals V: Linear Independence, Properties of -adic Polylogarithms, -adic Sheaves

Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 2)

Abstract

In a series of papers we have introduced and studied -adic polylogarithms and -adic iterated integrals which are analogues of the classical complex polylogarithms and iterated integrals in -adic Galois realizations. In this note we shall show that in the generic case -adic iterated integrals are linearly independent over \({\mathbb{Q}}_{\mathcal{l}}\). In particular they are non trivial. This result can be viewed as analogous of the statement that the classical iterated integrals from 0 to z of sequences of one forms \(\frac{dz} {z}\) and \(\frac{dz} {z-1}\) are linearly independent over \(\mathbb{Q}\). We also study ramification properties of -adic polylogarithms and the minimal quotient subgroup of the absolute Galois group G K of a number field K on which -adic polylogarithms are defined. In the final sections of the paper we study -adic sheaves and their relations with -adic polylogarithms. We show that if an -adic sheaf has the same monodromy representation as the classical complex polylogarithms then the action of G K in stalks is given by -adic polylogarithms.

Keywords

Fundamental Group Algebraic Variety Galois Group Galois Representation Iterate Integral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BD94.
    A. A. Beilinson and P. Deligne. Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs. In U. Jannsen, S. L. Kleiman, J.-P. Serre, Motives, Proc. of Sym. in Pure Math., pages 97–121, 55, Part II AMS 1994.Google Scholar
  2. BL94.
    A. A. Beilinson and A. Levin. The elliptic polylogarithm. In U. Jannsen, S. L. Kleiman, J.-P. Serre, Motives, Proc. of Sym. in Pure Math., pages 123–190, 55, Part II AMS 1994.Google Scholar
  3. Che75.
    K. T. Chen Iterated integrals, fundamental groups and covering spaces. Trans. of the Amer. Math. Soc., 206:83–98, 1975.Google Scholar
  4. DW04.
    J.-C. Douai and Z. Wojtkowiak. On the Galois actions on the fundamental group of \({\mathbb{P}}_{\mathbb{Q}({\mu }_{n})}^{1} \setminus \{ 0, 1,\infty \}\). Tokyo Journal of Math., 27(1):21–34, 2004.CrossRefMATHMathSciNetGoogle Scholar
  5. GM88.
    M. Goresky and R. MacPherson. Stratified Morse theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 14, Springer 1988.Google Scholar
  6. NW02.
    H. Nakamura and Z. Wojtkowiak. On the explicit formulae for -adic polylogarithms. In Arithmetic Fundamental Groups and Noncommutative Algebra, Proc. Symp. Pure Math., (AMS) 70:285–294, 2002.Google Scholar
  7. SGA1.
    A. Grothendieck. Revêtements étale et groupe fondamental (SGA 1). Séminaire de géométrie algébrique du Bois Marie 1960-61, directed by A. Grothendieck, augmented by two papers by Mme M. Raynaud, Lecture Notes in Math. 224, Springer-Verlag, Berlin-New York, 1971. Updated and annotated new edition: Documents Mathématiques 3, Société Mathématique de France, Paris, 2003.Google Scholar
  8. Sul77.
    D. Sullivan. Infinitesimal computations in topology. Publications Mathématiques, Institut des Hautes Études Scientifiques, 47:269–332, 1977.Google Scholar
  9. Woj91.
    Z. Wojtkowiak. The basic structure of polylogarithmic functional equations. In L. Lewin, Structural Properties of Polylogarithms, Mathematical Surveys and Monographs, pages 205–231, Vol 37, 1991.Google Scholar
  10. Woj93.
    Z. Wojtkowiak. Cosimplicial objects in algebraic geometry. In Algebraic K-theory and Algebraic Topology, Kluwer Academic Publishers, 407:287–327, 1993.Google Scholar
  11. Woj02.
    Z. Wojtkowiak. Mixed Hodge structures and iterated integrals, I. In F. Bogomolov and L. Katzarkov, Motives, Polylogarithms and Hodge Theory. Part I: Motives and Polylogarithms, International Press Lectures Series , Vol.3:121–208, 2002.Google Scholar
  12. Woj04.
    Z. Wojtkowiak. On -adic iterated integrals, I. Analog of Zagier conjecture. Nagoya Math. Journals, 176:113–158, 2004.Google Scholar
  13. Woj05a.
    Z. Wojtkowiak. On -adic iterated integrals, II. Functional equations and l-adic polylogarithms. Nagoya Math. Journals, 177:117–153, 2005.Google Scholar
  14. Woj05b.
    Z. Wojtkowiak. On -adic iterated integrals, III. Galois actions on fundamental groups. Nagoya Math. Journals, 178:1–36, 2005.Google Scholar
  15. Woj07.
    Z. Wojtkowiak. On the Galois actions on torsors of paths I, Descent of Galois representations. J. Math. Sci. Univ. Tokyo, 14:177–259, 2007.MATHMathSciNetGoogle Scholar
  16. Woj09.
    Z. Wojtkowiak. On -adic Galois Periods, relations between coefficients of Galois representations on fundamental groups of a projective line minus a finite number of points. Publ. Math. de Besançon, Algèbra et Théorie des Nombres, 155–174, 2007–2009, Février 2009.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Nice-Sophia Antipolis, Laboratoire Jean Alexandre DieudonnéNice Cedex 2France

Personalised recommendations