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Suppose $\chi$ is a real Drichlet character of modulus $N$, \begin{equation} \chi: (\mathbb{Z}/N\mathbb{Z})^* \rightarrow \mathbb{C} \end{equation} which induces an Artin representation \begin{equation} \rho: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}^1(\mathbb{C}) \end{equation} Since $\chi$'s value is 1 or -1, both of which are in $\mathbb{Q}_{\ell}$, so there is a representation defined similarly as $\rho$, \begin{equation} \rho_{\ell}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}^1(\mathbb{Q}_{\ell}) \end{equation} Is the characteristic polynomial of the geometric Frobenius at $p$ (for almost all $\ell$) of the form? \begin{equation} 1-\chi(p)\, T \end{equation} which is just 1 for some $p$, a polynomial of degree 0.

Is there a (mixed) motive, whose $\ell$-adic realisation is the representation $\rho_{\ell}$?

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    $\begingroup$ The first question is not well-defined. The image by $\rho_\ell$ of the geometric Frobenius at $p$ is well-defined only if $\rho_\ell$ is unramified at $p$, which means that $\rho_l(I_p)=1$ where $I_p$ is the inertia subgroup. So in your case it is only defined when $p$ does not divide $N\ell$. However, when $p$ is unramified, one consider, for certain purposes such as defining the $L$-function the characteristic polynomial of the action of $\rho_\ell(Frob_p)$ on the subspace fixed by $\rho_\ell(I_p)$, which is always defined. With this definition, you're right, you get 1 when $p \mid N$. $\endgroup$
    – Joël
    Commented Oct 8, 2017 at 23:55
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    $\begingroup$ Second question: yes, every Artin representation is attached to a pure motive, called an Artin motive. $\endgroup$
    – Joël
    Commented Oct 8, 2017 at 23:56
  • $\begingroup$ @Joël Thank you $\endgroup$
    – Wenzhe
    Commented Oct 9, 2017 at 8:41

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