Infinite multiplicity of roots of unity of the Galois group in the representation on elliptic curves

Abstract

Let K be a number field, (K) over bar an algebraic closure of K and E/K an elliptic curve defined over K. Let G(K) be the absolute Galois group Gal((K) over bar /K) of K over K. This paper proves that there is a subset Sigma subset of G(K) of Haar measure 1 such that for every sigma is an element of Sigma, the spectrum of a in the natural representation E((K) over bar) circle times C of G(K) consists of all roots of unity, each of infinite multiplicity. Also, this paper proves that any complex conjugation automorphism in GK has the eigenvalue -1 with infinite multiplicity in the representation space E((K) over bar) circle times C of G(K). (c) 2005 Elsevier Inc. All rights reserved