I am reading about the Langlands program (mostly for fun). I am mostly self-taught in algebraic number theory. I have read that one recovers abelian class field theory from Langlands by setting $n=1$ so to speak, and looking at the group $G = GL_1$. However, my question has to do with the details of this statement.
More specifically, given an abelian Galois finite extension $F$ of $\mathbb{Q}$, with Galois group $G = \operatorname{Gal}(F/\mathbb{Q})$, consider a $1$-dimensional representation:
$$ \sigma: G \to GL_1(\mathbb{C}). $$
By Artin's abelian class field theory (please correct me if I make some wrong statements), there is a positive integer $N_{\sigma}$, and a Dirichlet character:
$$ \chi_{\sigma}: (\mathbb{Z}/N_{\sigma}\mathbb{Z})^{\times} \to \mathbb{C}^*, $$
such that, for any unramified prime $p$, we have:
$$ \sigma(\{\operatorname{Fr}_p\}) = \chi_{\sigma}([p]), $$
where $\operatorname{Fr}_p$ denotes the Frobenius element associated to $p$ (and $\{-\}$ denotes the conjugacy class) and $[p]$ denotes the class of $p$ modulo $N_{\sigma}$.
My question is, why is a Dirichlet character (such as the one above) a special case of an automorphic representation of $GL_1$ please? For some reason, this is not so obvious to me.
First of all, should one consider $GL_1$ over the adeles over $\mathbb{Q}$ or over $F$? I suspect it is the adeles over $\mathbb{Q}$. But then, given an unramified prime $p$, how does one define the "local" group homomorphism from $\mathbb{Q}^{\times}_p \to \mathbb{C}^*$ associated to $\chi_{\sigma}$?
So my question is really, how are the Dirichlet characters a special case of Hecke's Größencharacters (I love writing the 'esset' ß)?
