# Elliptic multizetas and the elliptic double shuffle relations

## TL;DR

This paper introduces elliptic multizetas, explores their algebraic relations, and demonstrates that elliptic double shuffle relations encapsulate all algebraic relations among these values, extending the theory of multiple zeta values to the elliptic setting.

## Contribution

It defines elliptic multizetas, studies their algebraic structure, and establishes elliptic double shuffle relations as a comprehensive set of relations, extending multiple zeta value theory.

## Key findings

- Elliptic multizetas generate an elliptic analogue of multiple zeta value algebra.
- Elliptic multizetas satisfy double shuffle type algebraic relations.
- Elliptic double shuffle relations potentially encompass all algebraic relations among elliptic multizetas.

## Abstract

We define an elliptic generating series whose coefficients, the elliptic multizetas, are related to the elliptic analogues of multiple zeta values introduced by Enriquez as the coefficients of his elliptic associator; both sets of coefficients lie in $\mathcal{O}(\mathfrak{H})$, the ring of functions on the Poincar\'e upper half-plane $\mathfrak H$. The elliptic multizetas generate a $\mathbb Q$-algebra $\mathcal{E}$ which is an elliptic analogue of the algebra of multiple zeta values. Working modulo $2\pi i$, we show that the algebra $\mathcal{E}$ decomposes into a geometric and an arithmetic part and study the precise relationship between the elliptic generating series and the elliptic associator defined by Enriquez. We show that the elliptic multizetas satisfy a double shuffle type family of algebraic relations similar to the double shuffle relations satisfied by multiple zeta values. We prove that these elliptic double shuffle relations give all algebraic relations among elliptic multizetas if (a) the classical double shuffle relations give all algebraic relations among multiple zeta values and (b) the elliptic double shuffle Lie algebra has a certain natural semi-direct product structure analogous to that established by Enriquez for the elliptic Grothendieck-Teichm\"uller Lie algebra.