# Understanding Hecke Characters as Extension of Dirichlet Characters

I understand the concept of a Dirichlet character, and am interested in its generalizations to arbitrary number fields. I have heard that this generalization is called a Hecke character. However, I am not familiar with adeles or ideles, so I don't understand that definition. I know there is a "classical" definition of Hecke characters, but I am having trouble finding an easy to understand source about this definition. Could anyone provide a reference on the classical definition of Hecke characters that covers some of the preliminary knowledge needed to understand this definition?

• Hecke's original papers are quite good! But I'm suspecting you don't want to read papers in German? Commented Nov 11, 2020 at 1:23
• Commented May 26, 2021 at 19:08

The group of fractional ideals (coprime with some ideal $$J$$..) is a free abelian group generated by the prime ideals $$I_{K,J}\cong \prod_{P\not\ni J}' P^\Bbb{Z}$$

So it is easy to construct all the homomorphisms $$\psi: I_{K,J}\to \Bbb{C}^\times$$.

The Hecke characters are those whose restriction to the principal ideals is defined in term of reduction $$\bmod J$$ and complex embeddings $$\sigma_j$$, ie. $$\psi(aO_K)=\chi(a) = \phi(a)\prod_j \sigma_j(a)^{r_j}|\sigma_j(a)|^{s_j}$$ where $$\phi$$ is an homomorphism $$O_K/J^\times\to \Bbb{C}^\times$$

It is easy to generate all the possible $$\chi$$, the big restriction is that we need it to be trivial on $$O_K^\times$$ so that $$\psi(aO_K)=\chi(a)$$ is well-defined.

Finally we extend $$\psi$$ by defining $$\psi(I_l)$$ for the finitely generators of the class group (of fractional ideals coprime with $$J$$..) :

$$Cl_J(K)\cong \prod C_{n_l}$$,

$$I_l$$ is a generator of $$C_{n_l}$$, so $$I_l^{n_l}=(a_l)$$ is principal, we choose $$\psi(I_l)$$ such that $$\psi(I_l)^{n_l}=\psi(a_l O_K)$$.

Try with $$K$$ a quadratic imaginary field, $$O_K^\times$$ is finite thus it is much easier to construct all the possible $$\chi$$.

Try also with $$\psi(I)=\varphi(I\cap O)$$ where $$\varphi$$ is a character of the class group of $$O=\Bbb{Z}[mi]$$ with $$m\ne 1$$.

The point of this construction is that $$\sum \psi(I) N(I)^{-s}$$ (in addition to its Euler product) is the Mellin transform of some kind of theta function similar to $$\sum_n e^{-\pi n^2 x}$$ which will give the analytic continuation and functional equation.

• I appreciate the answer, and apparently I need to learn more abstract algebra before studying Hecke characters! Commented Nov 11, 2020 at 1:11

The original papers, in German, are quite well written, and certainly do not use ideles or adeles. Yes, identical titles:

E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen der Verteilung der Primzahlen}, Math. Z. 1 no. 4 (1918), 357-376.

E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen der Verteilung der Primzahlen, Math. Z. 6 no. 1-2 (1920), 11-51.