Galois properties of elliptic curves with an isogeny

giving the action of GQ on Tp(E), the p-adic Tate module for E and for a prime p. If E doesn’t have complex multiplication, then a famous theorem of Serre [Ser2] asserts that the image of ρE,p has finite index in AutZp ( Tp(E) ) for all p and that the index is 1 for all but finitely many p. This paper concerns some of the exceptional cases where the index is not 1. If E has a cyclic isogeny of degree p defined over Q, then Tp(E)/pTp(E) is isomorphic to E[p] and has a 1-dimensional Fp-subspace which is invariant under the action of GQ. Hence ρE,p can’t be surjective if such an isogeny exists. Our primary objective in this paper is to show that, under various assumptions, the image of ρE,p is as large as allowed by the p-power isogenies defined over Q. Assume that E has a Q-isogeny of degree p and let Φ denote its kernel. Let Ψ = E[p] / Φ. The actions of GQ on Φ and Ψ are given by two characters φ, ψ : GQ → F×p , respectively. Our main result is the following.