Elliptic Double Zeta Values

TL;DR

This paper explores elliptic analogues of multiple zeta values, proving their algebraic structure and providing formulas for their linear independence at lengths one and two across all weights.

Contribution

It establishes that elliptic multiple zeta values form a filtered $Q$-algebra and derives explicit counts of linearly independent values at low lengths.

Findings

Elliptic multiple zeta values form a $Q$-algebra.

Explicit formulas for independence at lengths one and two.

The algebra is filtered by length and conjecturally graded by weight.

Abstract

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.