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Modular phenomena for regularized double zeta values

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Abstract

In this paper, we investigate linear relations among regularized motivic iterated integrals on ℙ1 ∖ {0, 1, ∞} of depth two, which we call regularized motivic double zeta values. Some mysterious connections between motivic multiple zeta values and modular forms are known, e.g., Gangl–Kaneko–Zagier relation for the totally odd double zeta values and Ihara–Takao relation for the depth graded motivic Lie algebra. In this paper, we investigate so-called non-admissible cases and give many new Gangl–Kaneko–Zagier type and Ihara–Takao type relations for regularized motivic double zeta values. Specifically, we construct linear relations among a certain family of regularized motivic double zeta values from odd period polynomials of modular forms for the unique index two congruence subgroup of the full modular group. This gives the first non-trivial example of a construction of the relations among multiple zeta values (or their analogues) from modular forms for a congruence subgroup other than the SL2(ℤ).

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Numbers, JP18J00982 and JP18K13392. The author would like to thank Koji Tasaka and Nobuo Sato for useful comments on the draft of this paper.

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  1. Institute for Advanced Research, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan

    Minoru Hirose

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  1. Minoru Hirose
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Correspondence to Minoru Hirose.

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Hirose, M. Modular phenomena for regularized double zeta values. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2587-4

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  • DOI: https://doi.org/10.1007/s11856-023-2587-4

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