Motivic double zeta values of odd weight

Abstract

For odd \(N\ge 5\), we establish a short exact sequence about motivic double zeta values \(\zeta ^{\mathfrak {m}}(r,N-r)\) with \(r\ge 3\) odd, \(N-r\ge 2\). From this we classify all the relations among depth-graded motivic double zeta values \(\zeta ^{\mathfrak {m}}(r,N-r)\) with \(r\ge 3\) odd, \(N-r\ge 2\). As a corollary, we confirm a conjecture of Zagier on the rank of a matrix which concerns relations among multiple zeta values of odd weight.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Baumard, S., Schneps, L.: Period polynomial relations between double zeta values. Ramanujian J. 32(1), 83–100 (2013)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Brown, F.: Depth-graded motivic multiple zeta value, arXiv:1301.3053

  3. 3.

    Brown, F.: Mixed Tate motives over \(\mathbb{Z}\). Ann. Math. 175(2), 949–976 (2012)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Burgos Gil, J., Fresán, J.: Multiple zeta values: from numbers to motives, Clay Math. in Proceedings, to appear

  5. 5.

    Deligne, P., Goncharov, A.B.: Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. École Norm. Sup. 38, 1–56 (2005)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Gangl, H., Kaneko, M., Zagier, D.: Double zeta values and modular forms, Automorphic forms and zeta functions. in Proceedings of the Conference in Memory of Tsuneo Arakawa, World Scientific, pp. 71–106, (2006)

  7. 7.

    Li, J.: The depth structure of motivic multiple zeta values. Math. Ann. 374, 179–209 (2019)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Ma, D.: Period polynomial relations between formal double zeta values of odd weight. Mathematische Annalen 365(1–2), 345–362 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Soudères, I.: Motivic double shuffle. Int. J. Number Theory 6, 339–370 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Zagier, D.: Evaluation of the multiple zeta values \(\zeta (2,\ldots,2,3,2,\ldots,2)\). Ann. Math. 175, 977–1000 (2012)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to express their sincere gratitude to the anonymous referee for his/her detailed comments to improve this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jiangtao Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, J., Liu, F. Motivic double zeta values of odd weight. manuscripta math. (2020). https://doi.org/10.1007/s00229-020-01222-1

Download citation

Mathematics Subject Classification

  • Primary 11F32
  • Secondary 11F67