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SINGLE-VALUED MOTIVIC PERIODS AND MULTIPLE ZETA VALUES

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  10 October 2014

FRANCIS BROWN
FRANCIS BROWN*
Affiliation:
CNRS-IHES, France; [email protected]

Abstract

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The values at 1 of single-valued multiple polylogarithms span a certain subalgebra of multiple zeta values. The properties of this algebra are studied from the point of view of motivic periods.

MSC classification

Secondary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e25
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2014

References

André, Y., Galois theory, motives, and transcendental number theory, arXiv:0805.2569.Google Scholar
André, Y., ‘Une introduction aux motifs’, inPanoramas et Synthèses, Vol. 17 (SMF, 2004).Google Scholar
Beilinson, A. and Deligne, P., ‘Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs’, Proc Sympos Pure Math., 55, Part 2 (Amer. Math. Soc., Providence, RI, 1994).Google Scholar
Besser, A., ‘Coleman integration using the Tannakian formalism’, Math. Ann. 322 (1) (2002), 19–48.CrossRefGoogle Scholar
Beilinson, A., Goncharov, A. , Schechtman, V.  and Varchenko, A. , ‘Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of a pair of triangles in the plane’, inThe Grothendieck Feschtrift (Birkhauser, 1990), 131–172.Google Scholar
Beilinson, A., Macpherson, R. and Schechtman, V., ‘Notes on motivic cohomology’, Duke Math. J. 55 (1987), 679–710.Google Scholar
Del Duca, V., Dixon, L. , Duhr, C. and Pennington, J., The BFKL equation, Mueller-Navelet jets and single-valued harmonic polylogarithms, arXiv:1309.6647 (2013).CrossRefGoogle Scholar
Broadhurst, D. and Kreimer, D. , ‘Assocation of multiple zeta values with positive knots via Feynman diagrams up to 9 loops’, Phys. Lett. B 393 (1997), 403–412.Google Scholar
Brown, F., ‘Mixed Tate motives over ℤ’, Ann. of Math. (2) 175 (1) (2012).CrossRefGoogle Scholar
Brown, F., ‘On the decomposition of Motivic Multiple Zeta Values’, inGalois–Teichmüller Theory and Arithmetic Geometry, Advanced Studies in Pure Mathematics, 63 (2012), 31–58.Google Scholar
Brown, F., ‘Single-valued multiple polylogarithms in one variable’, C. R. Acad. Sci. Paris, Ser. I 338 (2004), 527–532.Google Scholar
Brown, F., Depth-graded motivic multiple zeta values, http://arxiv.org/abs/1301.3053.Google Scholar
Brown, F., ‘Motivic Periods and the projective line minus three points’, Proceedings of the ICM (2014).Google Scholar
Deligne, P., ‘Catégories Tannakiennes’, inGrothendieck Festschrift, Vol. II, Birkhäuser Progr. Math., 87 (1990), 111–195.Google Scholar
Deligne, P., ‘Le groupe fondamental de la droite projective moins trois points’, Galois groups over Q, Berkeley, CA, 1987, Math. Sci. Res. Inst. Publ., 16 (Springer, New York, 1989), 79–297.Google Scholar
Deligne, P., ‘Multizêtas’, Séminaire Bourbaki (2012).Google Scholar
Deligne, P., Letter to Brown and Zagier, 28 April 2012.Google Scholar
Deligne, P. and Goncharov, A. B., ‘Groupes fondamentaux motiviques de Tate mixte’, Ann. Sci. Éc. Norm. Supér. 38 (2005), 1–56.Google Scholar
Dixon, L., Duhr, C. and Pennington, J., Single-valued harmonic polylogarithms and the multi-Regge limit, arXiv:1207.0186.Google Scholar
Chavez, F. and Duhr, C., Three-mass triangle integrals and single-valued polylogarithms, arXiv:1209:2722.Google Scholar
Furusho, H., ‘ p-adic multiple zeta values II: Tannakian interpretations’, Amer. J. Math. 129 (4) (2007), 1105–1144.Google Scholar
Goncharov, A. B., Multiple polylogarithms and mixed Tate motives, preprint arXiv:math.AG/0103059.Google Scholar
Goncharov, A. B., ‘Volumes of hyperbolic manifolds and mixed Tate motives’, J. Amer. Math. Soc. 12 (2) (1999), 569–618.Google Scholar
Leurent, S. and Volin, D., Multiple zeta functions and double wrapping in planar $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}N=4$ SYM, arXiv:1302.1135.Google Scholar
Levine, M., ‘Tate motives and the vanishing conjectures for algebraic K-theory’, Algebraic K-theory and Algebraic Topology, Lake Louise, AB, 1991, 167–188.CrossRefGoogle Scholar
Pennington, J., The six-point remainder function to all loop orders in the multi-Regge limit, arXiv:1209.5357.Google Scholar
Schnetz, O., Graphical functions and single-valued multiple polylogarithms, arXiv:1302.6445.Google Scholar
Schnetz, O., Zeta procedures, http://www.mathematik.hu-berlin.de/∼kreimer/index.php?section=program.Google Scholar
Schlotterer, O. and Stieberger, S., Motivic multiple zeta values and superstring amplitudes, arXiv:1205.1516.Google Scholar
Stieberger, S. and Taylor, T., Closed string amplitudes as single-valued open string amplitudes, arXiv:1401.1218 (2014).Google Scholar
Wojtkowiak, Z., ‘A construction of analogs of the Bloch–Wigner function’, Math. Scand. 65 (1) (1989), 140–142.CrossRefGoogle Scholar
Zagier, D., ‘The Bloch–Wigner–Ramakrishnan polylogarithm function’, Math. Ann. 286 (13) (1990), 613–624.CrossRefGoogle Scholar