 

Davide Lombardo and Antonella Perucca
The 1eigenspace for matrices in GL_{2}(Z_{ℓ}) view print


Published: 
July 30, 2017

Keywords: 
Haar measure, general linear group, Cartan subgroup, ℓadic representation, elliptic curve 
Subject: 
28C10, 16S50, 11G05, 11F80 


Abstract
Fix some prime number ℓ and consider an open subgroup G either
of GL_{2}(Z_{ℓ}) or of the normalizer of a Cartan subgroup
of GL_{2}(Z_{ℓ}). The elements of G act
on
(Z/ℓ^{n}Z)^{2} for every n ≧ 1 and also on the direct limit, and we call 1eigenspace the group of fixed points. We partition G by considering the possible group structures for the 1eigenspace 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 ℓadic representation and the Galois action on the torsion points of order a power of ℓ.


Acknowledgements
The second author gratefully acknowledges financial support from the SFBHigher Invariants at the University of Regensburg.


Author information
Davide Lombardo:
Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy
davide.lombardo@unipi.it
Antonella Perucca:
Universität Regensburg, Universitätsstrasse 31, 93053 Regensburg, Germany
mail@antonellaperucca.net

