Modular phenomena for regularized double zeta values

2020/03/11 by Minoru Hirose, Hirose, Minoru
Mathematics · #11C20 (Secondary) #11M32 (Primary) 11F67 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2003.05236

openalex publication_date 2020/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \rm SL₂(ℤ).

Citations

• The universal Vassiliev-Kontsevich invariant for framed oriented links
• Polylogarithms, Multiple Zeta Values and Superstring Amplitudes
• Relationships between multiple zeta values of depths 2 and 3 and period polynomials

Related

• Depth-graded motivic multiple zeta values
• Colored double zeta values and modular forms of general level
• Motivic multiple zeta values reletive to μ₂
• The depth structure of motivic multiple zeta values
• Multiple Modular Values and the relative completion of the fundamental group of M_1,1