Proposition on Page 181 of Cornell, Silverman, Stevens, "Modular Forms and Fermat's Last Theorem" Does anyone have a reference for this proposition? The lecture seems to partially hinge on it, but the number theorists I have spoken with seem to doubt the result. We do know it is true if the Weil group is replaced by the Galois group, and if we require continuity of the representation. The problem is that the image of the Weil group could be, in theory, large, due to the real component. If there is indeed a mistake, then I am wondering how much of the chapter is in need of modification as a result. I am also wondering why "semisimple" is in quotes in the text.
Consider a $2$-dimensional semisimple representation of the Weil group
$$\rho: W_F \to \operatorname{GL}_2(\mathbb C)$$
where $F$ is a number field. The proposition claims that the image of $\rho$ in $\operatorname{PGL}_2(\mathbb C)$ is classified as one of the following three types:
Cyclic Type: $\rho$ is cyclic, hence is the direct sum of the two $1$-dimensional representations defined by characters $\mu$ and $\nu$.
Dihedral Type: The image of $\rho$ is dihedral, so that $\rho$ is induced from a 1-dimensional representation of an index 2 subgroup of $G_F$.
Exceptional Type: The image of $\rho$ in $\operatorname{PGL}_2(\mathbb C)$ is $A_4$, $S_4$, or $A_5$.
 A: To prove this proposition, we need some facts about Weil groups of number fields. I recommend John Tate's "Number Theoretic Background" for reference.
I will summarise some facts that we need: The Weil group $W_F$ of a number field $F$ is a topological group equipped with a continuous group homomorphism $r: W_F\to G_F$ with a dense image (where $G_F$ denotes the absolute Galois group of $F$). For any finite Galois extension $E/F$, define $W_E:=r^{-1}(G_E)$. Let $W_E^c=\overline{[W_E, W_E]}$ be the closure of the commutator subgroup of $W_E$. Let $W_{E/F}:=W_F/W_E^c$. Then we have an exact sequence
$$0\to W_E^{ab}:=W_E/W_E^c\to W_{E/F}\to G_{E/F}\to 0.$$
Also, the natural map $W_F\to \varprojlim W_{E/F}$  is an isomorphism of topological groups.
Let $\rho:W_F\to \text{GL}_2(\mathbb{C})$ be an semisimple continuous representation. If $\rho$ is reducible, then it is of cyclic type. So we may assume that $\rho$ is irreducible. By the no-small-subgroup property of $\text{GL}_2(\mathbb{C})$, $\rho$ factors thorough $W_{E/F}$. Since $W_E^{ab}$ is an abelian normal subgroup of $W_{E/F}$, and $\rho$ is irreducible, the image of $W_E^{ab}$ consists of constant matrices. Hence, the projection $\overline{\rho}:W_{E/F}\to \text{PGL}_2(\mathbb{C})$ factors through $G_{E/F}$, which is finite. Therefore, $\overline{\rho}$ has a finite image. Since $\rho$ is irreducible, the image is an exceptional or a dihedral group. If it is an exceptional group, then $\rho$ itself is of exceptional type.  If it is a dihedral group, then a standard representation-theoretic argument can show that $\rho$ itself is of dihedral type.
Note that this is no longer true if $\rho$ is not semisimple. For example, by the definition of the Weil group, there exist a morphism
$$f:W_{\mathbb{Q}}^{ab}\simeq \mathbb{Q}^\times\backslash\mathbb{A}_{\mathbb{Q}}^{\times}\simeq \mathbb{R}^{\times}_{>0}\times\prod_{p}\mathbb{Z}_p^\times\twoheadrightarrow\mathbb{R}$$
and
$$g\mapsto\begin{pmatrix} 1 & f(g) \\ 0 & 1 \end{pmatrix}$$
is not one of these types.
