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

Let K be a number field, K¯ an algebraic closure of K and E/K an elliptic curve defined over K. Let GK be the absolute Galois group Gal(K¯/K) of K¯ over K. This paper proves that there is a subset Σ⊆GK of Haar measure 1 such that for every σ∈Σ, the spectrum of σ in the natural representation E(K¯)⊗C of GK 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¯)⊗C of GK.