On graded Lie algebras associated to once-punctured elliptic curves with complex multiplication

TL;DR

This paper investigates a graded Lie algebra linked to the Galois action on the fundamental group of a once-punctured elliptic curve with complex multiplication, providing a minimal generating set under certain assumptions.

Contribution

It introduces a minimal generating set for the rationalized Lie algebra associated with such elliptic curves, extending weighted completion techniques.

Findings

Identified a minimal generating set for the Lie algebra

Applied weighted completion theory to this geometric context

Enhanced understanding of Galois actions on fundamental groups

Abstract

We study a graded Lie algebra arising from the Galois action on the pro-$p$ fundamental group of a once-punctured elliptic curve with complex multiplication. Among other things, we provide a minimal generating set of the rationalized Lie algebra under suitable assumptions. The proof is based on a slight variant of the theory of weighted completion of profinite groups developed by Hain and Matsumoto.