Polylogarithm Variations and Motivic Extensions of $\mathbb{Q}$ by $\mathbb{Q}(m)$

TL;DR

This paper advances the understanding of motivic Galois groups and mixed Tate motives related to polylogarithms and elliptic curves, generalizing previous results to all levels N and connecting to Eisenstein series and Hecke algebras.

Contribution

It generalizes the quadratic relations of the motivic Galois group to all N ≥ 1 by linking derivations on fundamental groups with Eisenstein series and K-groups, extending prior work.

Findings

Realized generators of the motivic Galois group via derivations on Lie algebras.

Connected K-groups of rings to spaces of Eisenstein series.

Computed periods of elliptic polylogarithm variations.

Abstract

Deligne and Goncharov constructed a neutral tannakian category of mixed Tate motives unramified over $\mathbb{Z}[\mu_N,1/N]$. Brown and Hain--Matsumoto computed the depth 2 quadratic relations of the motivic Galois group of this category for $N = 1$. We take the first steps in generalizing their results to all $N \ge 1$ by realizing the generators of the motivic Galois group by derivations on the Lie algebra of the unipotent fundamental group of a restriction of the Tate elliptic curve. This representation is compatible with a natural identification of the odd rational $K$-groups of the rings $\mathbb{Z}[\mu_N,1/N]$ with spaces of $\Gamma_1(N)$ Eisenstein series, thus inducing a natural action of the prime to $N$ part of the Hecke algebra on the $K$-groups. We establish these results by first showing the inclusion of $\mathbb{P}^1 - \{0,\mu_N,\infty\}$ into the nodal elliptic curve with a cyclic subgroup of order $N$ removed induces a morphism of mixed Tate motives on unipotent fundamental groups and then by computing the periods of the limit mixed Hodge structure of an elliptic polylogarithm variation of MHS over the universal elliptic curve of $Y_1(N)$.