---
tag: R65XKHN4D1
kind: concept
name: Hecke character
msc: [11R42, 11R56]
sources:
  - url: https://en.wikipedia.org/wiki/Hecke_character
same_as:
  wikidata: Q1948965
  wikipedia: Hecke character
---

## Summary

A Hecke character of a [[number field|MWBDFS2KJ5]] $K$ is a continuous homomorphism from the [[idele class group|9GHD9MXNYV]] $C_K$ to the nonzero complex numbers.
Hecke characters generalize [[Dirichlet characters|7XB5VJKDPG]], which are the Hecke characters of finite order of $\Q$.
Each Hecke character has an L-function with an Euler product, a meromorphic continuation and a functional equation, proved by Hecke and again by [[Tate's thesis|QYW79XKJTP]].

## Explanations

### L1 human

A Dirichlet character is a pattern of complex numbers attached to the residues modulo $m$, compatible with multiplication.
A Hecke character does the same for a number field, but it may also depend on the real and complex embeddings in a continuous way.
In the language of ideles it is just a continuous character of the idele group that ignores the elements of $K$.
A simple example is the idele norm $x \mapsto \prod_v \abs{x_v}_v$, which is $1$ on every element of $K$ by the product formula.