# Finite order Hecke characters

Let $\mathbb{A}_{\mathbb{Q}}^{\times}$ be the ring of ideles over the rational $\mathbb{Q}$. Let $\{ \chi : \mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^{\times} : \chi_\infty = 1, \chi^n =1 \}$ be the set of Hecke characters of order $n$ with trivial infinity part.

I want to whether they are finite in number corresponding bijectively to the Dirichlect characters $(\mathbb{Z}/n\mathbb{Z})^\times \to \mathbb{C}^\times$.

• You must have misunderstood something. The claim as stated isn't correct. – Fan Zheng Aug 1 '17 at 8:24

Just take $n=2$. Then there is only one homomorphism $(\mathbb{Z}/2\mathbb{Z})^\times \to \mathbb{C}^\times$, however there are many Hecke characters of order $2$. (e.g. take $$\widehat{\mathbb{Z}}^\times \to \mathbb{Z}_p^\times \to \{\pm 1\}, \quad x_p \mapsto \left(\frac{x_p \bmod p}{p}\right)$$ given by the Legendre symbol.)
• @user49908: Note also that $\chi_\infty$ is only trivial if the associated primitive Dirichlet character is even. If the associated primitive Dirichlet character is odd, then $\chi_\infty$ is the sign character $\mathbb{R}^\times\to\{\pm 1\}$. You can find a concise account over number fields here (see pages 7-8): arxiv.org/abs/1402.1332 – GH from MO Aug 1 '17 at 17:08