# Functional relations for elliptic polylogarithms

## TL;DR

This paper investigates functional relations among elliptic polylogarithms, linking them to elliptic Bloch groups and symbol formalism, revealing a complex landscape of identities beyond the classical five-term relation.

## Contribution

It introduces a new framework connecting elliptic polylogarithm relations with elliptic Bloch groups and symbol formalism, expanding understanding of elliptic identities.

## Key findings

- Linked elliptic polylogarithm relations to elliptic Bloch group considerations.
- Established an alternative proof of the elliptic Bloch relation using symbol formalism.
- Identified a class of elliptic identities beyond the classical five-term relation.

## Abstract

Numerous examples of functional relations for multiple polylogarithms are known. For elliptic polylogarithms, however, tools for the exploration of functional relations are available, but only very few relations are identified. Starting from an approach of Zagier and Gangl, which in turn is based on considerations about an elliptic version of the Bloch group, we explore functional relations between elliptic polylogarithms and link them to the relations which can be derived using the elliptic symbol formalism. The elliptic symbol formalism in turn allows for an alternative proof of the validity of the elliptic Bloch relation. While the five-term identity is the prime example of a functional identity for multiple polylogarithms and implies many dilogarithm identities, the situation in the elliptic setup is more involved: there is no simple elliptic analogue, but rather a whole class of elliptic identities.