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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 !

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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.

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  • $\begingroup$ Thank you very much for your reply ! I had a look at the wikipedia page, and I am still not sure how to get the $\ell$-adic cyclotomic character from the construction of $\chi_\ell$ that you mention (which I also sketched). Is it more complicated than "taking inverse limits", where of course by that I mean describe the inverse system, the transition maps, etc. ? $\endgroup$ – user62423 Mar 15 '14 at 10:09
  • $\begingroup$ See planetmath.org/padiccyclotomiccharacter. $\endgroup$ – Dietrich Burde Mar 16 '14 at 20:05

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