Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T12:32:59.638Z Has data issue: false hasContentIssue false
  • English
  • Français

The algebra of cell-zeta values

Part of: Computational aspects in algebraic geometry Computational number theory

Published online by Cambridge University Press:  10 March 2010

Francis Brown ,
Sarah Carr  and
Leila Schneps
Francis Brown
Affiliation:
CNRS and IMJ, 175 rue du Chevaleret, 75013 Paris, France (email: brown@math.jussieu.fr)
Sarah Carr
Affiliation:
Mathématiques Bât. 425, Université Paris-Sud, 91405 Orsay Cedex, France (email: sarah@math.jussieu.fr)
Leila Schneps
Affiliation:
CNRS and IMJ, 175 rue du Chevaleret, 75013 Paris, France (email: leila@math.jussieu.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we introduce cell-forms on 𝔐0,n, which are top-dimensional differential forms diverging along the boundary of exactly one cell (connected component) of the real moduli space 𝔐0,n(ℝ). We show that the cell-forms generate the top-dimensional cohomology group of 𝔐0,n, so that there is a natural duality between cells and cell-forms. In the heart of the paper, we determine an explicit basis for the subspace of differential forms which converge along a given cell X. The elements of this basis are called insertion forms; their integrals over X are real numbers, called cell-zeta values, which generate a ℚ-algebra called the cell-zeta algebra. By a result of F. Brown, the cell-zeta algebra is equal to the algebra of multizeta values. The cell-zeta values satisfy a family of simple quadratic relations coming from the geometry of moduli spaces, which leads to a natural definition of a formal version of the cell-zeta algebra, conjecturally isomorphic to the formal multizeta algebra defined by the much-studied double shuffle relations.

Keywords

multiple zeta valuesLie algebrascohomologymoduli spacespolygons

MSC classification

Secondary: 14Q15: Higher-dimensional varieties 11Y40: Algebraic number theory computations
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 3 , May 2010 , pp. 731 - 771
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Arnol’d, V. I., The cohomology ring of the coloured braid group, Mat. Zametki 5 (1969), 227–231; Math Notes 5 (1969), 138–140Google Scholar
[2]Bergström, J. and Brown, F., Inversion of series and the cohomology of the moduli spaces 𝔐δ0,n, in Motives, quantum field theory, and pseudodifferential operators, 2–13 June 2008, Boston University, Clay Mathematics Proceedings (American Mathematical Society, Providence, RI), to appear.Google Scholar
[3]Brown, F. C. S., Multiple zeta values and periods of moduli spaces , Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 373–491.Google Scholar
[4]Cartier, P., Fonctions polylogarithmes, nombres polyzêtas et groupes pro-unipotents, Séminaire Bourbaki, vol. 43, no. 885 (Séminaire Bourbaki, Paris, 2000-01), 137–173.Google Scholar
[5]Chen, K. T., Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), 831–879.CrossRefGoogle Scholar
[6]Deligne, P., Le groupe fondamental de la droite projective moins trois points, in Galois groups over , Berkeley, CA, 23–27 March 1987, eds Y. Ihara, K. Ribet and J.-P. Serre, Mathematical Sciences Research Institute Publications, vol. 16 (Springer, New York, NY, 1989), 79–297.CrossRefGoogle Scholar
[7]Deligne, P. and Goncharov, A. B., Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), 1–56.CrossRefGoogle Scholar
[8]Deligne, P. and Mumford, D., The irreducibility of the space of curves of a given genus, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 75–109.CrossRefGoogle Scholar
[9]Devadoss, S., Tesselations of moduli spaces and the mosaic operad, Contemp. Math. 239 (1999), 91–114.CrossRefGoogle Scholar
[10]Gangl, H., Goncharov, A. B. and Levin, A., Multiple polylogarithms, polygons, trees and algebraic cycles, Preprint (2005), arXiv:math.NT/0508066.Google Scholar
[11]Goncharov, A. B., Multiple polylogarithms and mixed Tate motives, Preprint (2001), arXiv:math.AG/0103059v4.Google Scholar
[12]Goncharov, A. B., Periods and mixed motives (2001), arXiv:math.AG/0202154.Google Scholar
[13]Goncharov, A. B. and Manin, Y. I., Multiple ζ-motives and moduli spaces , Compositio Math. 140 (2004), 1–14.CrossRefGoogle Scholar
[14]Hoffman, M. E., Quasi-shuffle products, J. Algebraic Combin. 11 (2000), 49–68.CrossRefGoogle Scholar
[15]Knudsen, F. F., The projectivity of the moduli space of stable curves II. The stacks , Math. Scand. 52 (1983), 163–199.CrossRefGoogle Scholar
[16]Kontsevich, M. and Zagier, D., Periods, in Mathematics unlimited; 2001 and beyond, eds Engquist, B. and Schmidt, W. (Springer, New York, 2001), 771–808.CrossRefGoogle Scholar
[17]Radford, D. E., A natural ring basis for shuffle algebra and an application to group schemes, J. Algebra 58 (1979), 432–454.CrossRefGoogle Scholar
[18]Reutenauer, C., Free Lie algebras, London Mathematical Society Monographs, New Series, vol. 7 (Clarendon Press, Oxford, 1993).CrossRefGoogle Scholar
[19]Salvatore, P. and Tauraso, R., The operad lie is free, February (2008), ArXiv:0802.3010v1.Google Scholar
[20]Soudères, I., Motivic double shuffle relations, Preprint (2008).Google Scholar
[21]Terasoma, T., Mixed Tate motives and multiple zeta values, Invent. Math. 149 (2002), 339–369.CrossRefGoogle Scholar
[22]Terasoma, T., Selberg integrals and multiple zeta values, Compositio Math. 133 (2002).CrossRefGoogle Scholar
[23]Waldschmidt, M., Valeurs zêtas multiples: une introduction, J. Théor. Nombres Bordeaux 12 (2002), 581–595.CrossRefGoogle Scholar