Abstract

This paper draws connections between the double shuffle equations and structure of associators; Hain and Matsumoto’s universal mixed elliptic motives; and the Rankin–Selberg method for modular forms for $\text{SL}_{2}(\mathbb{Z})$. We write down explicit formulae for zeta elements $\unicode[STIX]{x1D70E}_{2n-1}$ (generators of the Tannaka Lie algebra of the category of mixed Tate motives over $\mathbb{Z}$) in depths up to four, give applications to the Broadhurst–Kreimer conjecture, and solve the double shuffle equations for multiple zeta values in depths two and three.

Highlights

  • Using three different techniques (via double shuffle equations, the unipotent fundamental group of the punctured Tate curve, and the relative completion of SL2(Z)) we show, surprisingly, that there is an explicit way to write down canonical elements σ2cn+1 to the order, namely depth 3

  • Every solution to the regularized double shuffle equations in depths 4 and depths 3 can be expressed using the explicit elements σ2cn+1 and the element τ c. This theorem can be applied to the method of [5] for decomposing motivic multiple zeta values into a basis, which involved a numerical computation at each step

  • Tate curve, which is the fibre of the universal elliptic curve over M1,1 with respect to the tangential base point ∂/∂q

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Summary

Introduction

Using three different techniques (via double shuffle equations, the unipotent fundamental group of the punctured Tate curve, and the relative completion of SL2(Z)) we show, surprisingly, that there is an explicit way to write down canonical elements σ2cn+1 (respectively τ c) to the order, namely depth 3 (respectively depth 2) Their coefficients involve products of Bernoulli numbers, which can be thought of as a higher-depth version of Euler’s formula expressing even zeta values as multiples of powers of π. Every solution to the regularized double shuffle equations in depths 4 (odd weight) and depths 3 (even weight) can be expressed using the explicit elements σ2cn+1 and the element τ c This theorem can be applied to the method of [5] for decomposing motivic multiple zeta values into a basis, which involved a numerical computation at each step. The fundamental Lie algebra of the first-order Tate curve Background material for this section can be found in [23], [24], and [25]

Background
Commutative power series
Zeta elements in depth 3 via anatomical construction
Cuspidal elements and the Broadhurst–Kreimer conjecture
Description of pls in depths
We give a
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