The 1-eigenspace for matrices in GL(2)(Z(l))

作者：Lombardo Davide^*; Perucca Antonella

来源：The New York Journal of Mathematics, 2017, 23: 897-925.

• Haar measure
• general linear group
• Cartan subgroup
• l-adic representation
• elliptic curve

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 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.

• 出版日期2017