Characters of the absolute Galois group and their restrictions

Question

I am reading this paper Cremona et. al. In section 3 (pages 5-6), they talk about characters more specifically about Dirichlet characters and characters of the absolute Galois group, $G_{\mathbb{Q}}$. I am a little lost probably because my background is not strong enough.

I have a couple of questions that I got stuck on were:

1. How to think of characters over $\mathbb{Q}$ as characters of $G_{\mathbb{Q}}$?
2. How does a character of $G_{\mathbb{Q}}$ restricts to a number field $K$ ?

Could someone recommend a resource to learn about these things? Should I be reading about Galois representations?

Thank you in advance!

Answer 1

• 1. For a finite image (equivalently continuous in the relevant topologies) character $$\rho : Gal(\Bbb{Q}(\zeta_\infty)/\Bbb{Q})\to \Bbb{C}^\times$$ Then the subfield fixed by $\ker(\rho)$ is a finite extension of $\Bbb{Q}$ (of degree $|Image(\rho)|$). Thus for some $k$, $\ker(\rho)$ contains $Gal(\Bbb{Q}(\zeta_\infty)/\Bbb{Q}(\zeta_k))$ and hence $\rho$ is a character $$Gal(\Bbb{Q}(\zeta_k)/\Bbb{Q})\cong \Bbb{Z}/k \Bbb{Z}^\times\to \Bbb{C}^\times$$ A character $$f:\Bbb{Z}/k \Bbb{Z}^\times\to \Bbb{C}^\times$$ corresponds to a Dirichlet character: $\chi(n)= f(n)$ if $\gcd(n,k)=1$, otherwise $\chi(n)=0$.

     Conversely any Dirichlet character gives a character $\Bbb{Z}/k \Bbb{Z}^\times\to \Bbb{C}^\times$ for some $k$ thus a character $Gal(\Bbb{Q}(\zeta_k)/\Bbb{Q})\to \Bbb{C}^\times$ and hence a continuous character $Gal(\overline{\Bbb{Q}}/\Bbb{Q})\to \Bbb{C}^\times$.

     The correspondence is not one-to-one, it is if we take the minimum possible $k$ and restrict to the primitive Dirichlet characters.

     Any character $Gal(\overline{\Bbb{Q}}/\Bbb{Q})\to \Bbb{C}^\times$ factorizes through the abelianization of $Gal(\overline{\Bbb{Q}}/\Bbb{Q})$ which is $Gal(\Bbb{Q}(\zeta_\infty)/\Bbb{Q})$ (Kronecker Weber theorem).

• 2. You probably meant restricting a character $Gal(\overline{\Bbb{Q}}/\Bbb{Q})\to \Bbb{C}^\times$ to a character $Gal(\overline{\Bbb{Q}}/K)\to \Bbb{C}^\times$.

     The continuous characters $Gal(\overline{\Bbb{Q}}/K)\to \Bbb{C}^\times$ correspond similarly to the finite image Hecke characters of $K$, but this is a much more complicated story, the theorems from class field theory.