Osaka Journal of Mathematics

Polylogarithmic analogue of the Coleman-Ihara formula, I

Hiroaki Nakamura, Kenji Sakugawa, and Zdzisław Wojtkowiak

Full-text: Open access

Abstract

The Coleman-Ihara formula expresses Soule's $p$-adic characters restricted to $p$-local Galois group as the Coates-Wiles homomorphism multiplied by $p$-adic $L$-values at positive integers. In this paper, we show an analogous formula that $\ell$-adic polylogarithmic characters for $\ell=p$ restrict to the Coates-Wiles homomorphism multiplied by Coleman's $p$-adic polylogarithms at any roots of unity of order prime to $p$.

Article information

Source
Osaka J. Math., Volume 54, Number 1 (2017), 55-74.

Dates
First available in Project Euclid: 3 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1488531784

Mathematical Reviews number (MathSciNet)
MR3619748

Zentralblatt MATH identifier
06705680

Subjects
Primary: 11G55: Polylogarithms and relations with $K$-theory
Secondary: 14H30: Coverings, fundamental group [See also 14E20, 14F35] 11S31: Class field theory; $p$-adic formal groups [See also 14L05] 11R18: Cyclotomic extensions

Citation

Nakamura, Hiroaki; Sakugawa, Kenji; Wojtkowiak, Zdzisław. Polylogarithmic analogue of the Coleman-Ihara formula, I. Osaka J. Math. 54 (2017), no. 1, 55--74. https://projecteuclid.org/euclid.ojm/1488531784


Export citation

References

  • G. Anderson: The hyperadelic gamma function, Invent. Math. 95 (1989), 63–131.
  • S. Bloch and K. Kato: $L$-functions and Tamagawa numbers of motives, The Grothendieck Festschrift Vol. I, (P.Cartier et. al. eds.), Progr. Math. 86 (1990), 333–400.
  • R. Coleman: Division values in local fields, Invent. Math. 53 (1979), 91–116.
  • R. Coleman: The dilogarithm and the norm residue symbol, Bull. Soc. Math. France 109 (1981), 373–402.
  • R. Coleman: Dilogarithms, regulators, and $p$-adic L-functions, Invent. Math. 69 (1982), 171–208.
  • R. Coleman: Local units modulo circular units, Proc. Amer. Math. Soc. 89 (1983), 1–7.
  • R. Coleman: Anderson-Ihara theory: Gauss sums and circular units, in “Algebraic number theory – in honor of K.Iwasawa” (J.Coates, R.Greenberg, B.Mazur, I.Satake eds.), Adv. Studies in Pure Math. 17 (1989), 55–72.
  • J.W.S. Cassels and A. Fröhlich: Algebraic Number Theory, 2nd edition, London Mathematical Society, 2010.
  • J. Coates and R. Sujatha: Cyclotomic fields and zeta values, Springer-Verlag, Berlin, 2006.
  • P. Deligne: Le groupe fondamental de la droite projective moins trois points, in “Galois groups over $\mathbb{Q}$” (Y.Ihara, K.Ribet, J.-P.Serre eds.), Math. Sci. Res. Inst. Publ. 16 (1989), 79–297.
  • F. Ichimura and K. Sakaguchi: The nonvanishing of a certain Kummer character $\chi_m$ (after C. Soulé), and some related topics, in “Galois representations and arithmetic algebraic geometry” (Y. Ihara ed.), Adv. Studies in Pure Math. 12 (1987), 53–64.
  • Y. Ihara: Profinite braid groups, Galois representations and complex multiplications, Ann. of Math. 123 (1986), 43–106.
  • Y. Ihara: Braids, Galois groups, and some arithmetic functions, Proc. Intern. Congress of Math. Kyoto, 99–120, 1990.
  • Y. Ihara, M. Kaneko and A. Yukinari: On some properties of the universal power series for Jacobi sums, in “Galois representations and arithmetic algebraic geometry” (Y. Ihara ed.), Adv. Studies in Pure Math. 12 (1987), 65–86.
  • K. Iwasawa: On explicit formulas for the norm residue symbol, J. Math. Soc. Japan 20 (1968), 151–165.
  • H. Furusho: $p$-adic multiple zeta values I: $p$-adic multiple polylogarithms and the $p$-adic KZ equation, Invent. Math. 155 (2004), 223–286; II: Tannakian interpretations, Amer. Journal of Math. 129 (2007), 1105–1144.
  • M. Gros: Régulateurs syntomiques et valeurs de fonctions $L$ $p$-adiques, I, (with an appendix by Masato Kurihara), Invent. Math. 99 (1990), 293–320; II, Invent. Math. 115 (1994), 61–79.
  • M. Kim: The unipotent Albanese map and Selmer varieties for curves, Publ. Res. Inst. Math. Sci. 45 (2009), 89–133.
  • N. Koblitz: A new proof of certain formulas for $p$-adic L-functions, Duke Math. J. 46 (1979), 455–468.
  • M. Kolster and T. Nguyen Quang Do: Syntomic regulators and special values of $p$-adic $L$-functions, Invent. math. 133 (1998), 417–447.
  • M. Kurihara: Computation of the syntomic regulator in the cyclotomic case, Appendix to [Gr, I].
  • H. Nakamura, K. Sakugawa and Z. Wojtkowiak: Polylogarithmic analogue of the Coleman-Ihara formula, II, RIMS Kôkyûroku Bessatsu (to appear).
  • H. Nakamura and Z. Wojtkowiak: On explicit formulae for l-adic polylogarithms, in “Arithmetic fundamental groups and noncommutative algebra”, (M.Fried, Y.Ihara eds.) Proc. Sympos. Pure Math. 70 (2002), 285–294.
  • H. Nakamura and Z. Wojtkowiak: Tensor and homotopy criteria for functional equations of $l$-adic and classical iterated integrals, in “Non-abelian fundamental groups and Iwasawa theory” (J.Coates, M.Kim, F.Pop, M.Saïdi, P.Schneider eds.), London Math. Soc. Lecture Note Ser. 393 (2012), 258–310.
  • M.C. Olsson: Towards Non-Abelian p-adic Hodge Theory in the Good Reduction Case, Memoirs of A.M.S. 210, 2011.
  • K. Sakugawa: On a non-abelian generalization of the Bloch-Kato exponential map, Math. J. Okayama Univ. 59 (2017), 41–70.
  • S. Sen: On explicit reciprocity laws, J. reine anew. math. 313 (1980), 1–26; Part II, J. reine anew. math. 323 (1981), 68–87.
  • C. Soulé: On higher p-adic regulators, Lecture Notes in Mathematics 854 (1981), 372–401.
  • L.C. Washington: Introduction to Cyclotomic Fields, 2nd Edition, Springer 1997.
  • Z. Wojtkowiak: On $l$-adic iterated integrals I: Analog of Zagier conjecture, Nagoya Math. J. 176 (2004), 113–158.
  • Z. Wojtkowiak: On $l$-adic iterated integrals II: Functional equations and l-adic polylogarithms, Nagoya Math. J. 177 (2005), 117–153.
  • Z. Wojtkowiak: On $l$-adic Galois L-functions, (preprint 2014) arXiv:1403.2209.