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

Let K be a number field, image an algebraic closure of K and E/K an elliptic curve defined over K. Let GK be the absolute Galois group image of image over K. This paper proves that there is a subset Σsubset of or equal toGK of Haar measure 1 such that for every σset membership, variantΣ, the spectrum of σ in the natural representation image 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 image of GK.