I have a couple of questions concerning the cyclotomic character.
For the moment I know very little about the mod $\ell$ cyclotomic character, namely that $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts on $\mu_\ell$, the group of $\ell$th roots of unity in $\overline{\mathbb{Q}}$, and this action gives rise to a homomorphism
$$
\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \operatorname{Aut}(\mu_\ell)
$$
which gives the map in question $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathbb{F}_\ell^*$. Now,
1) does anyone have any textbook reference for either the $\ell$-adic or mod $\ell$ cyclotomic character?
2) assuming that the above is correct, how to rigorously define the $\ell$-adic cyclotomic character?
Thank you very much !
Best Answer
The mod $\ell$ cyclotomic character is defined by considering the group $μ_{\ell}$ of $\ell$-th roots of unity in $\overline{\mathbb{Q}}$; the action of the Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ on the cyclic group $μ_{\ell}$ gives rise, as you wrote, to a continuous homomorphism $$ \chi_{\ell}: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow Aut(μ_{\ell}). $$ Since $μ_{\ell}$ is a cyclic group of order $\ell$, its group of automorphisms is the group $\mathbb{F}_{\ell}^*$. We obtain a map $Gal(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \mathbb{F}_{\ell}^*$ , which is the character we want.