Terminology

In this section, we define twisted powers and rigid primes for an elliptic curve, and recall the definition of Tamagawa numbers of an abelian variety.

Definition 1.1 (Twisted Powers)   A twisted power of an elliptic curve $E$ over a field $K$ is an abelian variety $A$ over $K$ that is isomorphic over $\overline{K}$ to $E^{\times n}$ for some positive integer $n$.

We recall the standard notion of Tamagawa number of an abelian variety $A$, and introduce the notation $\overline{c}_{A,p}$ for the order of the group of components of $A$ over $\overline{\mathbb{F}}_p$.

Definition 1.2 (Tamagawa Numbers)   Let $A$ be an abelian variety over  $\mathbb{Q}$ with Néron model $\mathcal{A}$ over  $\mathbb{Z}$, and let $p$ be a prime. The component group of $A$ at $p$ is the finite group scheme $\Phi_{A,p} = \mathcal{A}_{\mathbb{F}_p}/\mathcal{A}_{\mathbb{F}_p}^0$, where $\mathcal{A}_{\mathbb{F}_p}^0$ is the identity component of $\mathcal{A}_{\mathbb{F}_p}$. The Tamagawa number of $A$ at $p$ is $c_{A,p} = \char93 \Phi_{A,p}(\mathbb{F}_p)$. We also set $\overline{c}_{A,p} = \char93 \Phi_{A,p}(\overline{\mathbb{F}}_p)$.

Definition 1.3 (Rigid Primes)   Let $E$ be an elliptic curve over  $\mathbb{Q}$. A prime $p$ is rigid for $E$ if $p$ does not divide $2\cdot N_E \cdot \prod_{\ell\mid N_E} \overline{c}_{E,\ell}$ and the representation $\rho_{E,p}:\Gal(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \Aut(E[p])$ is irreducible. Here $N_E$ is the conductor of $E$ and $\overline{c}_{E,p} = \char93 \Phi_{E,p}(\overline{\mathbb{F}}_p)$ is the order of the component group of $E$ at $p$.