Multiple zeta values and iterated Eisenstein integrals

Abstract

Brown showed that the affine ring of the motivic path torsor ${\pi }_1^{\text{mot}}({\mathbb{P}}^1\backslash \left\{0,1,\infty \right\},{\vec{1}}_0,-{\vec{1}}_1)$, whose periods are multiple zeta values, generates the Tannakian category $\mathsf{MT}(\mathbb{Z})$ of mixed Tate motives over $\mathbb{Z}$. Brown also introduced multiple modular values, which are periods of the relative completion of the fundamental group of the moduli stack ${\mathcal{M}}_{1,1}$ of elliptic curves. We prove that all motivic multiple zeta values may be expressed as $\mathbb{Q}[2\pi i]$-linear combinations of motivic iterated Eisenstein integrals along elements of ${\pi }_1({\mathcal{M}}_{1,1})\cong S{L}_2(\mathbb{Z})$, which are examples of motivic multiple modular values. This provides a new modular generator for $\mathsf{MT}(\mathbb{Z})$. We also explain how the coefficients in this linear combination may be partially determined using the motivic coaction.

References in corpus (1)

• From the Deligne-Ihara conjecture to Multiple Modular Values

Cited by in corpus (1)

• Poincaré series for modular graph forms at depth two. II. Iterated integrals of cusp forms