Hecke Characters

Let \(K\) be a number field. A Hecke Grossencharacter, or simply known as Hecke character, is a continuous complex-valued multiplicative homomorphism from the idele class group of \(K\) to \(\mathbb{C}^{\times}\).

The idele class group is defined as \[\begin{equation} \mathop{\mathrm{\mathfrak{C}}}(K) = \mathbb{A}_K^{\times} / K^{\times}. \end{equation}\]

See Hecke \(L\)-functions for some exciting application.