The 1-eigenspace for matrices in GL2(Zℓ)

Fix some prime number ℓ and consider an open subgroup G either of GL2(Zℓ) or of the normalizer of a Cartan subgroup of GL2(Zℓ). The elements of G act on (Z/ℓnZ)2for every n ≥ 1 and 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 ℓ-adic representation and the Galois action on the torsion points of order a power of ℓ.