Arithmetic monodromy actions on pro-metabelian fundamental groups of once-punctured elliptic curves

TL;DR

This paper explores the structure of G-structures on elliptic curves with finite metabelian groups, revealing connections to congruence subgroups and automorphism groups of free profinite metabelian groups.

Contribution

It provides a group-theoretic framework for understanding G-structures on elliptic curves and decomposes the automorphism group of a free profinite metabelian group.

Findings

G-structures correspond to congruence structures of level H

Decomposition of Out(\widehat{M}) into semi-direct product

All IA-automorphisms stabilize every open normal subgroup

Abstract

We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has exponent $e$, then there is a subgroup $H\le GL_2(\mathbb{Z}/e)$ such that $G$-structures on elliptic curves $E$ are equivalent to "congruence structures of level $H$". Our methods are almost entirely group theoretic. Let $\widehat{M}$ denote the free profinite metabelian group of rank 2, then along the way we prove a decomposition of $Out(\widehat{M})$ as an internal semi-direct product of the subgroup of "braid-like outer automorphisms" with the subgroup of "IA" outer automorphisms which induce the identity on the abelianization. We also show a surprising result that all IA-automorphisms leave every open normal subgroup stable.