On Distribution Formulas for Complex and l-adic Polylogarithms

  • Hiroaki NakamuraEmail author
  • Zdzisław Wojtkowiak
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 314)


We study an l-adic Galois analogue of the distribution formulas for polylogarithms with special emphasis on path dependency and arithmetic behaviors. As a goal, we obtain a notion of certain universal Kummer–Heisenberg measures that enable interpolating the l-adic polylogarithmic distribution relations for all degrees.


Arithmetic fundamental group Galois actions on étale paths Functional equations of polylogarithms 



This work was partially supported by JSPS KAKENHI Grant Number JP26287006.


  1. 1.
    Beilinson, A., Deligne, P.: Interprétation motivique de la conjecture de Zagier reliant polylogarithms et régulateurs. Proc. Symp. in Pure Math. (AMS) 55–2, 97–121 (1994)Google Scholar
  2. 2.
    Deligne, P.: Le groupe fondamental de la droite projective moins trois points. In Ihara, Y., Ribet, K., Serre, J.-P. (eds.) Galois group over \({\mathbb{Q}}\), vol. 16, pp. 79–297. MSRI Publications (1989)Google Scholar
  3. 3.
    Douai, J.-C., Wojtkowiak, Z.: Descent for \(\ell \)-adic polylogarithms. Nagoya Math. J. 192, 59–88 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gangl, H.: Families of functional equations for polylogarithms. Comtemp. Math. (AMS) 199, 83–105 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Huber, A., Kings, G.: Polylogarithms for families of commutative group schemes. arXiv preprint arXiv:1505.04574
  6. 6.
    Hain, R.: On a generalization of Hilbert’s 21st problem. Annales scientifiques de l’École Normale Supérieure, vol. 19, pp. 609–627 (1986)Google Scholar
  7. 7.
    Huber, A., Wildeshaus, J.: Classical motivic polylogarithm according to Beilinson and Deligne. Doc. Math. 3, 27–133 (1998). Correction Doc. Math. 3, 297–299 (1998)Google Scholar
  8. 8.
    Lang, S.: Cyclotomic fields I and II. In: Graduate Texts in Math. vol. 121. Springer (1990)Google Scholar
  9. 9.
    Lewin, L.: Polylogarithms and associated functions. North Holland (1981)Google Scholar
  10. 10.
    Milnor, J.: On polylogarithms Hurwitz zeta functions, and the Kubert identities. L’Enseignement Math. 29, 281–322 (1983)Google Scholar
  11. 11.
    Nakamura, H., Sakugawa, K., Wojtkowiak, Z.: Polylogarithmic analogue of the Coleman-Ihara formula II. RIMS Kôkyûroku Bessatsu B 64, 33–54 (2017)Google Scholar
  12. 12.
    Nakamura, H., Wojtkowiak, Z.: On explicit formulae for \(\ell \)-adic polylogarithms. Proc. Symp. Pure Math. (AMS) 70, 285–294 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Nakamura, H., Wojtkowiak, Z.: Tensor and homotopy criterions for functional equations of \(\ell \)-adic and classical iterated integrals. In: Coates, J., et al. (eds.) Non-abelian Fundamental Groups and Iwasawa Theory. London Mathematical Society Lecture Note Series, vol. 393, pp. 258–310 (2012)Google Scholar
  14. 14.
    Soulé, C.: On higher p-adic regulators. Lecture Notes in Math, vol. 854, pp. 372–401. Springer (1981)Google Scholar
  15. 15.
    Wojtkowiak, Z.: A note on functional equations of the p-adic polylogarithms. Bull. Soc. Math. Fr. 119, 343–370 (1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wojtkowiak, Z.: Functional equations of iterated integrals with regular singularities. Nagoya Math. J. 142, 145–159 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wojtkowiak, Z.: Monodromy of iterated integrals and non-abelian unipotent periods, Geometric Galois Actions II. In: London Mathematical Society Lecture Note Series, vol. 243, pp. 219–289 (1997)Google Scholar
  18. 18.
    Wojtkowiak, Z.: On \({\ell }\)-adic iterated integrals, I Analog of Zagier Conjecture. Nagoya Math. J. 176, 113–158 (2004)Google Scholar
  19. 19.
    Wojtkowiak, Z.: A note on functional equations of \({\ell }\)-adic polylogarithms. J. Inst. Math. Jussieu 3, 461–471 (2004)Google Scholar
  20. 20.
    Wojtkowiak, Z.: On \(\ell \)-adic iterated integrals, II—Functional equations and \(\ell \)-adic polylogarithms. Nagoya Math. J. 177, 117–153 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wojtkowiak, Z.: On \(\ell \)-adic iterated integrals, III—Galois actions on fundamental groups. Nagoya Math. J. 178, 1–36 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wojtkowiak, Z.: On \({\ell }\)-adic iterated integrals V, linear independence, properties of \({\ell }\)-adic polylogarithms, \({\ell }\)-adic sheave. In: Jacob, S. (ed.) The Arithmetic of Fundamental Groups, PIA 2010, pp. 339–374. Springer (2012)Google Scholar
  23. 23.
    Wojtkowiak, Z.: On \(\ell \)-adic Galois L-functions. In: Algebraic Geometry and Number Theory, Summer School, Galatasaray University, Istanbul, 2014. Progress in Math. Birkhauser, vol. 321, pp. 161–209, Springer International Publishing (2017). arXiv:1403.2209v1 10 March (2014)

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsOsaka UniversityOsakaJapan
  2. 2.Départment of Mathematics, Laboratoire Jean Alexandre Dieudonné, U.R.A. au C.N.R.S.Université de Nice-Sophia AntipolisNice Cedex 2France

Personalised recommendations