Advertisement

Overview on Elliptic Multiple Zeta Values

  • Nils MatthesEmail author
Conference paper
  • 35 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 314)

Abstract

We give an overview of some work on elliptic multiple zeta values. First defined by Enriquez as the coefficients of the elliptic KZB associator, elliptic multiple zeta values are also special values of multiple elliptic polylogarithms in the sense of Brown and Levin. Common to both approaches to elliptic multiple zeta values is their representation as iterated integrals on a once-punctured elliptic curve. Having compared the two approaches, we survey various recent results about the algebraic structure of elliptic multiple zeta values, as well as indicating their relation to iterated integrals of Eisenstein series, and to a special algebra of derivations.

Keywords

Elliptic multiple zeta values Elliptic KZB equation Multiple elliptic polylogarithms 

Notes

Acknowledgements

Many thanks to the organizers of the Research Trimester on Multiple Zeta Values, held September-December 2014 at ICMAT, Madrid, where part of this research was carried out. This paper contains results obtained in joint work with Johannes Broedel, Carlos Mafra and Oliver Schlotterer, and I would like to thank them very much. Incidentally, that collaboration started after the author gave a talk at the ICMAT in September 2014, as part of this research trimester. Also, many thanks to Henrik Bachmann, Johannes Broedel, Ulf Kühn and Oliver Schlotterer for helpful comments, as well as the Albert-Einstein-Institute in Potsdam, the Department of Applied Mathematics and Theoretical Physics in Cambridge and the Mainz Institute for Theoretical Physics for hospitality. This work is part of the author’s PhD thesis at Universität Hamburg, and I would like to thank my advisor Ulf Kühn for his constant support of my work and for his encouragement.

References

  1. 1.
    Bannai, K., Kobayashi, S., Tsuji, T.: On the de Rham and p-adic realizations of the elliptic polylogarithm for CM elliptic curves. Ann. Sci. École. Norm. Sup. (4) 43(2), 185–234 (2010)Google Scholar
  2. 2.
    Baumard, S., Schneps, L.: On the derivation representation of the fundamental Lie algebra of mixed elliptic motives. Ann. Math. Qué. 41(1), 43–62 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bloch, S.J.: Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. CRM Monograph Series, vol. 11. American Mathematical Society, Providence, RI (2000)Google Scholar
  4. 4.
    Broadhurst, D.J., Kreimer, D.: Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Phys. Lett. B 393(3–4), 403–412 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Broedel, J., Mafra, C.R., Matthes, N., Schlotterer, O.: Elliptic multiple zeta values and one-loop superstring amplitudes. J. High Energy Phys. 7, 112, front matter+41 pp 2015Google Scholar
  6. 6.
    Broedel, J., Matthes, N., Schlotterer, O.: Relations between elliptic multiple zeta values and a special derivation algebra. J. Phys. A 49(15), 155203, 49 pp (2016)Google Scholar
  7. 7.
    Broedel, J., Schlotterer, O., Stieberger, S.: Polylogarithms, multiple zeta values and superstring amplitudes. Fortschr. Phys. 61(9), 812–870 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Broedel, J., Schlotterer, O., Stieberger, S., Terasoma, T.: All order \(\alpha ^{\prime }\)-expansion of superstring trees from the Drinfeld associator. Phys. Rev. D 89(6), 066014 (2014)CrossRefGoogle Scholar
  9. 9.
    Brown, F.: Mixed Tate motives over \({\mathbb{Z}}\). Ann. of Math. (2) 175(2), 949–976 (2012)Google Scholar
  10. 10.
    Brown, F.: Iterated Integrals in Quantum Field Theory. Geometric and Topological Methods for Quantum Field Theory, pp. 188–240. Cambridge University Press, Cambridge (2013)Google Scholar
  11. 11.
    Brown, F.: Depth-graded motivic multiple zeta values. arXiv:1301.3053
  12. 12.
    Brown, F.: Multiple modular values and the relative completion of the fundamental group of \({\mathscr {M}{}_{1,1}}\). arXiv:1407.5167v3
  13. 13.
    Brown, F.: Zeta elements in depth \(3\) and the fundamental Lie algebra of the infinitesimal Tate curve. Forum Math. Sigma, 5:e1(56) (2017)Google Scholar
  14. 14.
    Brown, F.: Anatomy of an associator. arXiv:1709.02765
  15. 15.
    Brown, F., Levin, A.: Multiple elliptic polylogarithms. arXiv:1110.6917
  16. 16.
    Calaque, D., Enriquez, B., Etingof, P.: Universal KZB equations: the elliptic case. In: Yu. I. (ed.) Manin Algebra, arithmetic, and geometry: in honor of Vol. I, volume 269 of Progr. Math., pages 165–266. Birkhäuser Boston, Inc., Boston, MA (2009)Google Scholar
  17. 17.
    Chen, K.T.: Iterated path integrals. Bull. Amer. Math. Soc. 83(5), 831–879 (1977)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Deligne, P.: Le groupe fondamental de la droite projective moins trois points. In Galois groups over \({\mathbb{Q}}\) (Berkeley, CA, 1987), volume 16 of Math. Sci. Res. Inst. Publ., pages 79–297. Springer, New York (1989)Google Scholar
  19. 19.
    Deligne, P., Goncharov, A. B.: Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. École Norm. Sup. (4) 38(1), 1–56 (2005)Google Scholar
  20. 20.
    Drinfel’d, V.G.: On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \({\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\). Leningrad Math. J. 2(4), 829–860 (1991)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Enriquez, B.: Elliptic associators. Selecta Math. (N.S.) 20 (2014), no. 2, 491–584Google Scholar
  22. 22.
    Enriquez, B.: Analogues elliptiques des nombres multizétas. Bull. Soc. Math. France 144(3), 395–427 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Furusho, H.: Double shuffle relation for associators. Ann. Math. (2) 174(1), 341–360 (2011)Google Scholar
  24. 24.
    Gangl, H., Kaneko, M., Zagier, D.: Double zeta values and modular forms. In: Automorphic Forms and Zeta Functions, pp. 71–106. World Scientific Publishing, Hackensack, NJ (2006)Google Scholar
  25. 25.
    Goncharov, A.B.: Multiple polylogarithms, cyclotomy and modular complexes. Math. Res. Lett. 5(4), 497–516 (1998)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Goncharov, A.B., Manin, Y.I.: Multiple \(\zeta \)-motives and moduli spaces \(\mathscr {M}_{0, n}\). Compos. Math. 140(1), 1–14 (2004)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hain, R.M.: The geometry of the mixed Hodge structure on the fundamental group. In: Algebraic geometry, Bowdoin, 1985 Brunswick, Maine, 1985, volume 46 of Proc. Sympos. Pure Math., pp. 247–282. Amer. Math. Soc., Providence, RI (1987)Google Scholar
  28. 28.
    Hain, R., Matsumoto M.: Universal mixed elliptic motives. J. Inst. Math. Jussieu 1–104 (2018). https://doi.org/10.1017/S1474748018000130
  29. 29.
    Knizhnik, V.G., Zamolodchikov, A.B.: Current algebra and Wess-Zumino model in two dimensions. Nuclear Phys. B 247(1), 83–103 (1984)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Le, T.T.Q., Murakami, J.: Kontsevich’s integral for the Kauffman polynomial. Nagoya Math. J. 142, 39–65 (1996)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Levin, A.: Elliptic polylogarithms: an analytic theory. Compositio Math. 106(3), 267–282 (1997)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Levin, A., Racinet, G.: Towards multiple elliptic polylogarithms. arXiv:math/0703237
  33. 33.
    Lochak, P., Matthes, N., Schneps, L.: Elliptic multizetas and the elliptic double shuffle relations, arXiv:1703.09410
  34. 34.
    Manin, Y. I.: Iterated integrals of modular forms and noncommutative modular symbols. In: Algebraic geometry and number theory, vol. 253 of Progr. Math., pages 565–597. Birkhäuser Boston, Boston, MA (2006)Google Scholar
  35. 35.
    Matthes, N.: Elliptic multiple zeta values. Ph.D. thesis, Universität Hamburg (2016)Google Scholar
  36. 36.
    Matthes, N.: Elliptic double zeta values. J. Number Theory 171, 227–251 (2017)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Pollack, A.: Relations between derivations arising from modular forms. Master’s thesis, Duke University (2009)Google Scholar
  38. 38.
    Racinet, G.: Doubles mélanges des polylogarithmes multiples aux racines de l’unité. Publ. Math. Inst. Hautes Études Sci. 95, 185–231 (2002)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Ree, R.: Lie elements and an algebra associated with shuffles. Ann. Math. 2(68), 210–2220 (1958)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Schlotterer, O., Stieberger, S.: Motivic multiple zeta values and superstring amplitudes. J. Phys. A 46(47), 475401, 37 (2013)Google Scholar
  41. 41.
    Terasoma, T.: Geometry of multiple zeta values. In: International Congress of Mathematicians. Vol. II, pages 627–635. Eur. Math. Soc., Zürich (2006)Google Scholar
  42. 42.
    Weil, A.: Elliptic functions according to Eisenstein and Kronecker. Springer, Berlin-New York. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 88 (1976)Google Scholar
  43. 43.
    Zagier, D.: The Bloch-Wigner-Ramakrishnan polylogarithm function. Math. Ann. 286(1–3), 613–624 (1990)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Zagier, D.: Periods of modular forms and Jacobi theta functions. Invent. Math. 104(3), 449–465 (1991)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Zagier, D.: Values of zeta functions and their applications. In: First European Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages 497–512. Birkhäuser, Basel (1994)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Fachbereich Mathematik (AZ)Universität HamburgHamburgGermany
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations