Elliptic Multizetas and the Elliptic Double Shuffle Relations

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 Poincaré 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 (1) the classical double shuffle relations give all algebraic relations among multiple zeta values and (2) the elliptic double shuffle Lie algebra has a certain natural semi-direct product structure analogous to that established by Enriquez for the elliptic Grothendieck–Teichmüller Lie algebra.