On the derivation representation of the fundamental Lie algebra of mixed elliptic motives

TL;DR

This paper explores the derivation representation of the fundamental Lie algebra of mixed elliptic motives, proving a conjecture about relations in all depths and weights using Ecalle's mould theory.

Contribution

It establishes the existence of relations in the derivation algebra for all depths and weights, and proves this in depth 3, connecting to period polynomials of cusp forms.

Findings

Relations in the derivation algebra exist in all depths and weights.

Proved the conjecture in depth 3 for all weights.

Connected relations to period polynomials of cusp forms.

Abstract

Richard Hain and Makoto Matsumoto constructed a category of universal mixed elliptic motives, and described the fundamental Lie algebra of this category: it is a semi-direct product of the fundamental Lie algebra Lie$\,\pi_1(MTM)$ of the category of mixed Tate motives over ${\bf Z}$ with a filtered and graded Lie algebra $u$. This Lie algebra, and in particular the subspace $u$, admits a representation as derivations of the free Lie algebra on two generators. In this paper we study the image $E$ of this representation of $u$, starting from some results by Aaron Pollack, who determined all the relations in a certain filtered quotient of $E$, and gave several examples of relations in low weights in $E$ that are connected to period polynomials of cusp forms on $SL_2({\bf Z})$. Pollack's examples lead to a conjecture on the existence of such relations in all depths and all weights, that we state in this article and prove in depth 3 in all weights. The proof follows quite naturally from Ecalle's theory moulds, to which we give a brief introduction. We prove two useful general theorems on moulds in the appendices.