Colored double zeta values and modular forms of general level

TL;DR

This paper extends known linear relations among double zeta values from the classical modular forms to colored double zeta values associated with modular forms of higher level, revealing new algebraic structures.

Contribution

It generalizes the relations among double zeta values to colored double zeta values of level N, linked to modular forms for congruence subgroups.

Findings

Established linear relations among colored double zeta values of level N.

Connected these relations to modular forms of higher level.

Extended previous results from level 1 to arbitrary levels.

Abstract

Gangl, Kaneko, and Zagier gave explicit linear relations among double zeta values of odd indices coming from the period polynomials of modular forms for ${\rm SL}(2,\mathbb{Z})$. In this paper, we generalize their result to the linear relations among colored double zeta values of level $N$ coming from the modular forms for level $N$ congruence subgroups.