# Dirichlet Character, Galois Representation and Motives

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

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

• 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 $p \mid N$. – Joël Oct 8 '17 at 23:55
• Second question: yes, every Artin representation is attached to a pure motive, called an Artin motive. – Joël Oct 8 '17 at 23:56
• @Joël Thank you – Wenzhe Oct 9 '17 at 8:41