Elliptic Double Zeta Values

We study an elliptic analogue of multiple zeta values, the elliptic multiple zeta values of Enriquez, which are the coefficients of the elliptic KZB associator. Originally defined by iterated integrals on a once-punctured complex elliptic curve, it turns out that they can also be expressed as certain linear combinations of indefinite iterated integrals of Eisenstein series and multiple zeta values. In this paper, we prove that the $\mathbb{Q}$-span of these elliptic multiple zeta values forms a $\mathbb{Q}$-algebra, which is naturally filtered by the length and is conjecturally graded by the weight. Our main result is a proof of a formula for the number of $\mathbb{Q}$-linearly independent elliptic multiple zeta values of lengths one and two for arbitrary weight.