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Overview on Elliptic Multiple Zeta Values

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Periods in Quantum Field Theory and Arithmetic (ICMAT-MZV 2014)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 314))

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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.

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Notes

  1. 1.

    See also Sect. 1.4.

  2. 2.

    Iterated integrals starting or ending with \(\omega ^{(1)}\) diverge, and need to be regularized, such that the shuffle product formula remains valid. See [22, 36] for details on this regularization procedure.

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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.

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  1. Fachbereich Mathematik (AZ), Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Nils Matthes

  2. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter Woodstock Road, Oxford, OX2 6GG, UK

    Nils Matthes

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  1. Institute of Mathematical Sciences (ICMAT), Spanish National Research Council (CSIC), Madrid, Spain

    José Ignacio Burgos Gil

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  3. Department of Mathematical Sciences, Durham University, Durham, UK

    Herbert Gangl

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Matthes, N. (2020). Overview on Elliptic Multiple Zeta Values. In: Burgos Gil, J., Ebrahimi-Fard, K., Gangl, H. (eds) Periods in Quantum Field Theory and Arithmetic. ICMAT-MZV 2014. Springer Proceedings in Mathematics & Statistics, vol 314. Springer, Cham. https://doi.org/10.1007/978-3-030-37031-2_5

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