Hostname: page-component-6bf8c574d5-mggfc Total loading time: 0 Render date: 2025-02-23T05:28:48.427Z Has data issue: false hasContentIssue false
  • English
  • Français

Elliptic hyperlogarithms

Part of: Spaces and algebras of analytic functions Other special functions

Published online by Cambridge University Press:  14 January 2025

Benjamin Enriquez  and
Federico Zerbini
Benjamin Enriquez*
Affiliation:
IRMA (UMR 7501) et Département de Mathématiques, Université de Strasbourg, 7 rue René-Descartes, 67084 Strasbourg, France
Federico Zerbini
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter (550), Woodstock Road, Oxford, OX2 6GG (UK) e-mail: [email protected]
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let ${\mathcal {E}}$ be a complex elliptic curve and S be a non-empty finite subset of ${\mathcal {E}}$. We show that the functions $\tilde {\Gamma }$ introduced in [BDDT] out of string theory motivations give rise to a basis (as a vector space) of the minimal algebra $A_{{\mathcal {E}}{\smallsetminus } S}$ of holomorphic multivalued functions on ${\mathcal {E}}{\smallsetminus } S$ which is stable under integration, introduced in [EZ]; this basis is alternative to the basis of $A_{{\mathcal {E}}{\smallsetminus } S}$ constructed in loc. cit. using elliptic analogs of the hyperlogarithm functions.

Keywords

Hyperlogarithmselliptic polylogarithmsiterated integrals

MSC classification

Primary: 33E05: Elliptic functions and integrals
Secondary: 30H50: Algebras of analytic functions
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 36
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bloch, S., Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, CRM Monograph Series, 11, American Mathematical Society, Providence, RI, 2000.Google Scholar
Bowman, D. and Bradley, D., Multiple polylogarithms: A brief survey . In: B. C. Berndt and K. Ono (eds.), q-Series with Applications to Combinatorics, Number Theory, and Physics, Contemporary Mathematics, 291, American Mathematical Society, Providence, RI, 2001, pp. 71–92.Google Scholar
Broedel, J., Duhr, C., Dulat, F., and Tancredi, L., Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: General formalism . J. High Energy Phys. 12(2018), no. 5, 093.CrossRefGoogle Scholar
Broedel, J. and Kaderli, A., Amplitude recursions with an extra marked point . Commun. Number Theory Phys. 16(2022), no. 1, 75–158.CrossRefGoogle Scholar
Broedel, J., Mafra, C. R., Matthes, N., and Schlotterer, O., Elliptic multiple zeta values and one-loop superstring amplitudes . J. High Energy Phys. 2017(2015), no. 7, 112.CrossRefGoogle Scholar
Brown, F., Multiple zeta values and periods of moduli spaces ${\overline{\mathfrak{M}}}_{0,n}$ . Ann. Sci. Éc. Norm. Supér. (4). 42(2009), no. 3, 371–489.CrossRefGoogle Scholar
Brown, F. and Levin, A., Multiple elliptic polylogarithms. Preprint, 2011. arXiv:1110.6917.Google Scholar
Burgos Gil, J. and Fresan, J., Multiple zeta values: from numbers to motives . Clay Math. Proc., to appear.Google Scholar
Calaque, D., Enriquez, B., and Etingof, P., Universal KZB equations: The elliptic case . In: Y. Tschinkel and Y. Zarhin (eds.), Algebra, Arithmetic, and Geometry, In Honor of Yu. I. Manin, Progress in Mathematics, 269 Birkhäuser Boston, Ltd., Boston, MA, 2009, pp. 165–266.Google Scholar
D’Hoker, E., Hidding, M., and Schlotterer, O., Constructing polylogarithms on higher-genus Riemann surfaces. Preprint, 2023. arXiv:2306.08644.Google Scholar
Deligne, P., Le groupe fondamental de la droite projective moins trois points . In: Y. Ihara, K. Ribet, and J.-P. Serre (eds.), Galois Groups over $\mathbb{Q}$ , Mathematical Sciences Research Institute Publications, 16, Springer-Verlag, New York, 1989, pp. 79–297.Google Scholar
Deneufchâtel, M., Duchamp, G. H. E., Minh, V. H. N., and Solomon, A. I., Independence of hyperlogarithms over function fields via algebraic combinatorics . In: Winkler, F. (ed.), Algebraic informatics, Lecture Notes in Computer Science, 6742, Springer, Linz, 2011, pp. 127–139.CrossRefGoogle Scholar
Enriquez, B. and Zerbini, F., Analogues of hyperlogarithm functions on affine complex curves. Preprint, 2022. arXiv:2212.03119.Google Scholar
Knapp, A., Elliptic curves, Mathematical Notes, 40, Princeton University Press, Princeton, NJ, 1992.Google Scholar
Lappo-Danilevsky, J. A., Mémoires sur la théorie des systèmes des équations différentielles linéaires. Chelsea Publishing Co., New York, 1953.Google Scholar
Levin, A.. Elliptic polylogarithms: An analytic theory . Compos. Math., 106(1997), no. 3, 267–282.CrossRefGoogle Scholar
Levin, A. and Racinet, G., Towards multiple elliptic polylogarithms. Preprint, 2007. arXiv:math/0703237.Google Scholar
Matthes, N., Elliptic Multiple Zeta Values. Ph.D. thesis, Fakultät für Mathematik Informatik und Naturwissenschaften, Universität Hamburg, 2016.Google Scholar
Panzer, E., Feynman integrals and hyperlogarithms. Ph.D. thesis, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universität zu Berlin, 2015.Google Scholar
Poincaré, H., Sur les groupes des équations linéaires . Acta Math. 4(1884), no. 1, 201–312.CrossRefGoogle Scholar
Weil, A., Elliptic functions according to Eisenstein and Kronecker, Classics of Mathematics, Springer-Verlag, Berlin, 1999.Google Scholar
Zagier, D., The Bloch–Wigner–Ramakrishnan polylogarithm function . Math. Ann. 286(1990), nos. 1–3, 613–624.CrossRefGoogle Scholar
Zagier, D., Periods of modular forms and Jacobi theta functions , Invent. Math. 104(1991), no. 3, 449–465.CrossRefGoogle Scholar