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

Abstract

Fix some prime number $\ell$ and consider an open subgroup $G$ either of ${\mathrm{GL}}_2({\mathbb{Z}}_{\ell })$ or of the normalizer of a Cartan subgroup of ${\mathrm{GL}}_2({\mathbb{Z}}_{\ell })$. The elements of $G$ act on $(\mathbb{Z}/{\ell }^n\mathbb{Z}{)}^2$ for every $n⩾1$ 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 $\ell$.

Minor changes and other updates to incorporate referee comments. Final version

References in corpus (1)

• Galois theory of iterated endomorphisms