New York Journal of Mathematics
Volume 23 (2017) 897-925


Davide Lombardo and Antonella Perucca

The 1-eigenspace for matrices in GL2(Z)

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Published: July 30, 2017
Keywords: Haar measure, general linear group, Cartan subgroup, ℓ-adic representation, elliptic curve
Subject: 28C10, 16S50, 11G05, 11F80

Fix some prime number ℓ and consider an open subgroup G either of GL2(Z) or of the normalizer of a Cartan subgroup of GL2(Z). The elements of G act on (Z/ℓnZ)2 for 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.


The second author gratefully acknowledges financial support from the SFB-Higher Invariants at the University of Regensburg.

Author information

Davide Lombardo:
Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy

Antonella Perucca:
Universität Regensburg, Universitätsstrasse 31, 93053 Regensburg, Germany