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

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 $Y_1(N)$.