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On Poly(ana)logs I

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Compositio Mathematica

Abstract

We investigate a connection between the differential of polylogarithms (as considered by Cathelineau) and a finite variant of them. This allows to answer a question raised by Kontsevich concerning the construction of functional equations for the finite analogs, using in part the p-adic version of polylogarithms and recent work of Besser. Kontsevich's original unpublished note is supplied (with his kind permission) in an ‘Appendix’ at the end of the paper.

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References

  1. Aczél, J. and Dhombres, J.: Functional equations in several variables, in Encyclopedia of Math. Appl. 31, Cambridge Univ. Press, 1989.

  2. Besser, A.: Finite and p-adic polylogarithms, Compositio Math. 130 (2002), 215-223.

    Google Scholar 

  3. Bloch, S.: On the tangent space to Quillen K-theory, In: Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972, Lecture Notes in Math. 341, Springer, New York, 1973, pp. 205-210.

    Google Scholar 

  4. Bloch, S.: Higher regulators, algebraic K-theory, and zeta-functions of elliptic curves, Lecture Notes, U.C. Irvine, 1977.

  5. Bloch, S.: Applications of the dilogarithm function in algebraic K-theory and algebraic geometry, Proc. Int. Symp. Alg. Geom. (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo, 1978, pp. 103-114.

  6. Cathelineau, J.-L.: Sur l'homologie de SL2 àcoefficients dans l'action adjointe, Math. Scand. 63 (1988), 51-86.

    Google Scholar 

  7. Cathelineau, J.-L.: Homologie du groupe linéaire et polylogarithmes (d'après Goncharovet d'autres), Séminaire Bourbaki 772 (1992-93), Astérisque 216 (1993), 311-341.

    Google Scholar 

  8. Cathelineau, J.-L.: Remarques sur les différentielles des polylogarithmes uniformes, Ann. Inst. Fourier Grenoble, 46(5) (1996), 1327-1347.

    Google Scholar 

  9. Cathelineau, J.-L.: Infinitesimal polylogarithms, multiplicative presentations of Kähler differentials and Goncharovcompl exes, talk at theWorkshop on Polylogarithms, Essen, May 1997. (see http://www.exp-math.uni-essen.de/herbert/polyloquy.html)

  10. Coleman, R.: Dilogarithms, regulators and p-adic L-functions, Invent. Math. 69(2) (1982), 171-208.

    Google Scholar 

  11. Dupont, J. L. and Sah, C.-H.: Scissors congruences. II., J. Pure Appl. Algebra 25 (1982), 159-195.

    Google Scholar 

  12. Elbaz-Vincent, Ph.: The indecomposable K3 of rings and homology of SL2, J. Pure Appl. Algebra 132 (1998), 27-71.

    Google Scholar 

  13. Elbaz-Vincent, Ph.: Homology of Linear Groups with Coefficients in the Adjoint Action and K-theory, K-Theory 16 (1999), 35-50.

    Google Scholar 

  14. Elbaz-Vincent, Ph.: Arithmetic and infinitesimal variations on a polylogarithm theme (report on a work in progress with H. Gangl), talk at the Workshop on Polylogarithms, Essen, May 1997. (see http://www.exp-math.uni-essen.de/∼herbert/polyloquy.html)

  15. Elbaz-Vincent, Ph. and Gangl, H.: On Poly(ana)logs II, (in preparation).

  16. Elbaz-Vincent, Ph. and Gangl, H.: Arithmetical and Infinitesimal Variations on a Polylogarithmic Theme (in preparation).

  17. Gangl, H.: Funktionalgleichungen von Polylogarithmen, (German) Dissertation, Bonn, 1995, Bonner Mathematische Schriften 278, UniversitTt Bonn.

  18. Gangl, H.: Functional equations for higher logarithms (to appear).

  19. Goncharov, A. B.: Geometry of configurations, polylogarithms and motivic cohomology, Adv. Math. 114 (1995), 197-318.

    Google Scholar 

  20. Goncharov, A. B.: Polylogarithms and motivic Galois groups, Proc. of the Seattle conf. on motives, Seattle July 1991, Proc. Sympos. Pure Math. 55, (2), Amer. Math. Soc., Providence, 1994, pp. 43-96.

    Google Scholar 

  21. Goncharov, A. B.: Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5(4) (1998), 497-516.

    Google Scholar 

  22. Kontsevich, M.: The 1 12 -logarithm, Appendix to this paper (Compositio Math. 130 (2002), 211-214).

    Google Scholar 

  23. Lewin, L.: Polylogarithms andAs sociated Functions, North-Holland, Amsterdam, 1981.

  24. Lewin, L. (ed.): Structural Properties of Polylogarithms, Math. Surveys Monogr. 37, Amer. Math. Soc., Providence, 1991.

  25. Lewin, L.: The order-independence of the polylogarithmic ladder structureC implications for a new category of functional equations, AequationesMath. 30(1) (1986), 1-20.

    Google Scholar 

  26. Oesterlé, J. Polylogarithmes, Sém. Bourbaki 762 (1992-93), Astérisque 216 (1993), 49-67.

    Google Scholar 

  27. Ribenboim, P.: 13 Lectures on Fermat's last theorem, Springer-Verlag, New York (1979).

    Google Scholar 

  28. Rogers, L. J.: On function sum theorems connected with the series Σ ∞1 xn/n2, Proc. London Math. Soc. 4 (1907), 169-189.

    Google Scholar 

  29. Rosenberg, J.: Algebraic K-Theory andits Applications, Grad. Text. Math. 147, Springer, New York, 1994.

    Google Scholar 

  30. Suslin, A. A.: K3 of a field, and the Bloch group (Russian), Galois theory, rings, algebraic groups and their applications (Russian), TrudyMat. Inst. Steklov. 183 (1990), 180-199, 229.

    Google Scholar 

  31. Washington, L. C.: Introduction to Cyclotomic Fields, (2nd edn), Grad. Text. Math. 83, Springer, New York, 1997.

    Google Scholar 

  32. Weibel, C.: Private e-mail correspondence, October 1999.

  33. Wojtkowiak, Z.: A construction of analogs of the Bloch-Wigner function, Math. Scand. 65 (1989), 140-142.

    Google Scholar 

  34. Wojtkowiak, Z.: A note on functional equations of the p-adic polylogarithms, Bull. Soc. Math. France 119 (1991), 343-370.

    Google Scholar 

  35. Wojtkowiak, Z.: The basic structure of polylogarithmic functional equations, Chapter 10 in [24], 205-231.

  36. Zagier,D.: Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields, Proc. Texel Conf. on Arithm. Algebraic Geom. 1989, Birkhäuser, Boston, 1991, pp. 391-430.

    Google Scholar 

  37. Zagier, D.: Special values and functional equations of polylogarithms, Appendix A in [24], 377-400.

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Authors and Affiliations

  1. Laboratoire GTA, UMR CNRS 5030, CC 051, Université Montpellier II, 34095, Montpellier Cedex 5, France

    Philippe Elbaz-Vincent

  2. Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111, Bonn, Germany

    Herbert Gangl

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  1. Philippe Elbaz-Vincent
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  2. Herbert Gangl
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Elbaz-Vincent, P., Gangl, H. On Poly(ana)logs I. Compositio Mathematica 130, 161–214 (2002). https://doi.org/10.1023/A:1013757217319

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