Algebra & Number Theory

The algebraic de Rham realization of the elliptic polylogarithm via the Poincaré bundle

Johannes Sprang

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

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


We describe the algebraic de Rham realization of the elliptic polylogarithm for arbitrary families of elliptic curves in terms of the Poincaré bundle. Our work builds on previous work of Scheider and generalizes results of Bannai, Kobayashi and Tsuji, and Scheider. As an application, we compute the de Rham–Eisenstein classes explicitly in terms of certain algebraic Eisenstein series.

Article information

Algebra Number Theory, Volume 14, Number 3 (2020), 545-585.

Received: 26 February 2018
Revised: 25 June 2019
Accepted: 8 November 2019
First available in Project Euclid: 2 July 2020

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 11G55: Polylogarithms and relations with $K$-theory
Secondary: 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]

de Rham cohomology polylogarithm Eisenstein classes


Sprang, Johannes. The algebraic de Rham realization of the elliptic polylogarithm via the Poincaré bundle. Algebra Number Theory 14 (2020), no. 3, 545--585. doi:10.2140/ant.2020.14.545.

Export citation


  • K. Bannai and G. Kings, “$p$-adic elliptic polylogarithm, $p$-adic Eisenstein series and Katz measure”, Amer. J. Math. 132:6 (2010), 1609–1654.
  • 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. $(4)$ 43:2 (2010), 185–234.
  • A. A. Beilinson, “Higher regulators and values of $L$-functions”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh. 24 (1984), 181–238. In Russian; translated in 30:2 (1985), 2036–2070}.
  • A. Beilinson and A. Levin, “The elliptic polylogarithm”, pp. 123–190 in Motives (Seattle, WA, 1991), edited by U. Jannsen et al., Proc. Sympos. Pure Math. 55, Amer. Math. Soc., Providence, RI, 1994.
  • P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton University Press, 1978.
  • N. Bourbaki, Algebra, vol. II: Chapters 4–7, Springer, 1990.
  • C. Deninger, “Higher regulators and Hecke $L$-series of imaginary quadratic fields, I”, Invent. Math. 96:1 (1989), 1–69.
  • G. Faltings and C.-L. Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete $(3)$ 22, Springer, 1990.
  • K. Kato, “Lectures on the approach to Iwasawa theory for Hasse–Weil $L$-functions via $B_{\rm dR}$, I”, pp. 50–163 in Arithmetic algebraic geometry (Trento, 1991), edited by E. Ballico, Lecture Notes in Math. 1553, Springer, 1993.
  • K. Kato, “$p$-adic Hodge theory and values of zeta functions of modular forms”, pp. 117–290 in Cohomologies $p$-adiques et applications arithmétiques, vol. III, edited by P. Berthelot et al., Astérisque 295, Société Mathématique de France, Paris, 2004.
  • N. M. Katz, “Nilpotent connections and the monodromy theorem: applications of a result of Turrittin”, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175–232.
  • N. M. Katz, “$p$-adic properties of modular schemes and modular forms”, pp. 69–190 in Modular functions of one variable (Antwerp, 1972), vol. III, edited by W. Kuyk and J.-P. Serre, Lecture Notes in Mathematics 350, 1973.
  • N. M. Katz, “The Eisenstein measure and $p$-adic interpolation”, Amer. J. Math. 99:2 (1977), 238–311.
  • G. Kings, “The Tamagawa number conjecture for CM elliptic curves”, Invent. Math. 143:3 (2001), 571–627.
  • G. Laumon, “Transformation de Fourier généralisée”, preprint, 1996.
  • B. Mazur and W. Messing, Universal extensions and one dimensional crystalline cohomology, Lecture Notes in Mathematics 370, Springer, 1974.
  • R. A. Scheider, The de Rham realization of the elliptic polylogarithm in families, Ph.D. thesis, Universität Regensburg, 2014,
  • J. Sprang, Eisenstein series via the Poincaré bundle and applications, Ph.D. thesis, Universität Regensburg, 2017,
  • J. Sprang, “Real-analytic Eisenstein series via the Poincaré bundle”, preprint, 2018.
  • J. Sprang, “The syntomic realization of the elliptic polylogarithm via the Poincaré bundle”, Doc. Math. 24 (2019), 1099–1134.
  • T. Tsuji, “Explicit reciprocity law and formal moduli for Lubin–Tate formal groups”, J. Reine Angew. Math. 569 (2004), 103–173.