The 1-eigenspace for matrices in $\operatorname{GL}_2(\mathbb{Z}_\ell)$

TL;DR

This paper analyzes the structure of 1-eigenspaces for matrices in certain subgroups of 2(Z_l) and provides a finite procedure to compute the Haar measure of related sets, with applications to elliptic curves over number fields.

Contribution

It introduces a method to evaluate the Haar measure of sets defined by 1-eigenspace structures in subgroups of 2(Z_l), applicable to elliptic curve Galois representations.

Findings

Finite procedure for Haar measure computation

Partitioning of groups based on 1-eigenspace structures

Application to elliptic curves over number fields

Abstract

Fix some prime number $\ell$ and consider an open subgroup $G$ either of $\operatorname{GL}_2(\mathbb{Z}_\ell)$ or of the normalizer of a Cartan subgroup of $\operatorname{GL}_2(\mathbb{Z}_\ell)$. The elements of $G$ act on $(\mathbb{Z}/\ell^n \mathbb{Z})^2$ for every $n\geqslant 1$ and hence also on the direct limit, and we call 1-eigenspace the group of fixed points. We partition $G$ by considering the possible group structures for the 1-eigenspace and show how to evaluate with a finite procedure the Haar measure of all sets in the partition. The results apply to all elliptic curves defined over a number field, where we consider the image of the $\ell$-adic representation and the Galois action on the torsion points of order a power of $\ell$.