Nagoya Mathematical Journal

p-adic Eisenstein–Kronecker series for CM elliptic curves and the Kronecker limit formulas

Kenichi Bannai, Hidekazu Furusho, and Shinichi Kobayashi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Consider an elliptic curve defined over an imaginary quadratic field K with good reduction at the primes above p5 and with complex multiplication by the full ring of integers OK of K. In this paper, we construct p-adic analogues of the Eisenstein–Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove p-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

Article information

Source
Nagoya Math. J., Volume 219 (2015), 269-302.

Dates
Received: 28 June 2012
Revised: 18 June 2014
Accepted: 14 August 2014
First available in Project Euclid: 20 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1445345522

Digital Object Identifier
doi:10.1215/00277630-2891995

Mathematical Reviews number (MathSciNet)
MR3413578

Zentralblatt MATH identifier
1336.11050

Subjects
Primary: 11G55: Polylogarithms and relations with $K$-theory
Secondary: 11G07: Elliptic curves over local fields [See also 14G20, 14H52] 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 14F30: $p$-adic cohomology, crystalline cohomology 14G10: Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture)

Keywords
Eisenstein–Kronecker series Kronecker limit formula distribution relation Coleman’s $p$-adic integration

Citation

Bannai, Kenichi; Furusho, Hidekazu; Kobayashi, Shinichi. $p$ -adic Eisenstein–Kronecker series for CM elliptic curves and the Kronecker limit formulas. Nagoya Math. J. 219 (2015), 269--302. doi:10.1215/00277630-2891995. https://projecteuclid.org/euclid.nmj/1445345522


Export citation

References

  • [1] K. Bannai, Rigid syntomic cohomology and $p$-adic polylogarithms, J. Reine Angew. Math. 529 (2000), 205–237.
  • [2] K. Bannai, On the $p$-adic realization of elliptic polylogarithms for CM-elliptic curves, Duke Math. J. 113 (2002), 193–236.
  • [3] K. Bannai, Specialization of the $p$-adic polylogarithm to $p$-th power roots of unity, Doc. Math. 2003, Extra Vol., 73–97.
  • [4] K. Bannai and S. Kobayashi, “Algebraic theta functions and Eisenstein-Kronecker numbers” in Proceedings of the Symposium on Algebraic Number Theory and Related Topics, RIMS Kôkyûroku Bessatsu B4, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007, 63–77.
  • [5] K. Bannai and S. Kobayashi, Algebraic theta functions and $p$-adic interpolation of Eisenstein-Kronecker numbers, Duke Math. J. 153 (2010), 229–295.
  • [6] K. Bannai, S. Kobayashi, and T. Tsuji, “Realizations of the elliptic polylogarithm for CM elliptic curves” in Algebraic Number Theory and Related Topics (2007), RIMS Kôkyûroku Bessatsu B12, Res. Inst. Math. Sci. (RIMS), Kyoto, 2009, 33–50.
  • [7] K. Bannai, S. Kobayashi, and T. Tsuji, On the de Rham and $p$-adic realizations of the elliptic polylogarithm for CM elliptic curves, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 185–234.
  • [8] A. Beilinson and A. Levin, “The elliptic polylogarithm” in Motives (Seattle, 1991), Proc. Sympos. Pure Math. 55, Amer. Math. Soc., Providence, 1994, 123–190.
  • [9] P. Berthelot, Finitude et pureté cohomologique en cohomologie rigide, with an appendix by A. J. de Jong, Invent. Math. 128 (1997), 329–377.
  • [10] A. Besser, “Syntomic regulators and $p$-adic integration, I: Rigid syntomic regulators” in Conference on $p$-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998), Israel J. Math. 120, 2000, 291–334.
  • [11] A. Besser, “Syntomic regulators and $p$-adic integration, II: $K_{2}$ of curves” in Conference on $p$-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998), Israel J. Math. 120, 2000, 335–359.
  • [12] A. Besser, Coleman integration using the Tannakian formalism, Math. Ann. 322 (2002), 19–48.
  • [13] A. Besser and R. de Jeu, The syntomic regulator for the $K$-theory of fields, Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), 867–924.
  • [14] R. F. Coleman, Dilogarithms, regulators and $p$-adic $L$-functions, Invent. Math. 69 (1982), 171–208.
  • [15] R. Damerell, $L$-functions of elliptic curves with complex multiplication, I. Acta Arith. 17 (1970) 287–301.
  • [16] R. Damerell, $L$-functions of elliptic curves with complex multiplication, II. Acta Arith. 19 (1971), 311–317.
  • [17] P. Deligne, “Le groupe fondamental de la droite projective moins trois points” in Galois Groups Over $\mathbb{Q}$ (Berkeley, 1987), Math. Sci. Res. Inst. Publ. 16, Springer, New York, 1989, 79–297.
  • [18] E. de Shalit, Iwasawa Theory of Elliptic Curves with Complex Multiplication: $p$-adic $L$ Functions, Perspect. Math. 3, Academic Press, Boston, 1987.
  • [19] H. Furusho, $p$-adic multiple zeta values, I: $p$-adic multiple polylogarithms and the $p$-adic KZ equation, Invent. Math. 55 (2004), 253–286.
  • [20] N. M. Katz, $p$-adic interpolation of real analytic Eisenstein series, Ann. of Math. (2) 104 (1976), 459–571.
  • [21] A. Levin, Elliptic polylogarithms: An analytic theory, Compos. Math. 106 (1997), 267–282.
  • [22] G. Robert, Unités elliptiques et formules pour le nombre de classes des extensions abéliennes d’un corps quadratique imaginaire, Mém. Soc. Math. France 36, Soc. Math. France, Paris, 1973.
  • [23] M. Somekawa, Log-syntomic regulators and $p$-adic polylogarithms, $K$-Theory 17 (1999), 265–294.
  • [24] A. Weil, Elliptic Functions According to Eisenstein and Kronecker, Ergeb. Math. Grenzgeb (3) 88, Springer, Berlin, 1976.
  • [25] J. Wildeshaus, On an elliptic analogue of Zagier’s conjecture, Duke Math. J. 87 (1997), 355–407.
  • [26] D. Zagier, Periods of modular forms and Jacobi theta functions, Invent. Math. 104 (1991), 449–465.