Open conditions for infinite multiplicity eigenvalues on elliptic curves

Abstract

Let E be an elliptic curve defined over a number field K. We show that for each root of unity zeta, the set Sigma(zeta) of sigma is an element of Gal((K) over bar /K) such that zeta is an eigenvalue of infinite multiplicity for or acting on E((K) over bar) circle times C has non-empty interior. For the eigenvalue -1, we can show more: for any sigma in Gal((K) over bar /K), the multiplicity of the eigenvalue - 1 is either 0 or infinity. It follows that Sigma(-1) is open. (c) 2006 Elsevier Inc. All rights reserved