Open conditions for infinite multiplicity eigenvalues on elliptic curves

Let E be an elliptic curve defined over a number field K. We show that for each root of unity ζ, the set Σζ of such that ζ is an eigenvalue of infinite multiplicity for σ acting on has non-empty interior. For the eigenvalue −1, we can show more: for any σ in , the multiplicity of the eigenvalue −1 is either 0 or ∞. It follows that Σ−1 is open.