Catalogue of modular relations for double zeta values

TL;DR

This paper compiles and reviews known modular relations for multiple zeta values, especially double zeta values, highlighting their origins from modular forms and outlining future research directions.

Contribution

It provides a comprehensive overview of existing modular relations for double zeta values and suggests potential avenues for further exploration in the field.

Findings

Multiple types of modular relations for double zeta values identified

Relations originate from modular forms on subgroups of the modular group

The paper outlines future research directions in the area

Abstract

This note is a compilation of related research on modular relations for multiple zeta values. Roughly speaking, modular relations are (homogeneous) linear relations of multiple zeta values of fixed weight whose coefficients are `originated' from modular forms on (a subgroup of) the full modular group. Several kinds of such relations, in particular, for double zeta values have been found up to now. It is our purpose to review these results and also succinctly outline potential avenues for future research projects.