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I'm trying to decipher Lang's Algebraic Number Theory when it comes to L-functions. Let $K/\mathbb{Q}$ be an abelian extension. Then we are supposed to have a factorization

$$\zeta_K(s)=\zeta_\mathbb{Q}(s)\prod_{\chi\neq 1} L(s,\chi)$$

I'm trying to understand this factorization in more concrete terms. First of all Lang gives the definition of $\chi$ as a group on ideles. I was wondering if there is a simple more concrete and computable definition in terms of ray classes?

Let's say we take some simple example like $\mathbb{Q}(\sqrt{-3})$. In what group are these $\chi$ defined and how would I compute them? Is there some algorithm that works in general? I understand that this has something to do with the conductor, which would be $(3)p_\infty$ in the above case.

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2 Answers 2

Completely explicitly, let $G$ be the Galois group of the extension $K/\mathbb{Q}$. So $G$ is a quotient of the absolute Galois group $G_{\mathbb{Q}}$ of $\mathbb{Q}$. Then the product runs over the non-trivial irreducible complex representations of the absolute Galois group of $\mathbb{Q}$ that factor through $G$, i.e. over the non-trivial characters of $G$, thought of as characters of $G_{\mathbb{Q}}$. The zeta function of $\mathbb{Q}$ corresponds to the trivial character. Thus, in the example $K=\mathbb{Q}(\sqrt{3})$ the product only consists of one term, corresponding to the non-trivial character of $G\cong\mathbb{Z}/2\mathbb{Z}$, the so-called quadratic character attached to $K$.

This works in much greater generality: for any base field instead of $\mathbb{Q}$ and for arbitrary Galois groups, not necessarily abelian. The Dirichlet $L$-functions are then replaced by Artin $L$-functions.


Edit:

Here is an explanation of why the character is what I claim it is: first, embed $K$ into a cyclotomic field. In the particular example $K=\mathbb{Q}(\sqrt{3})$, we have $K\subset F=\mathbb{Q}(\mu_{12})$. The Galois group of $F/\mathbb{Q}$ is isomorphic to $(\mathbb{Z}/12\mathbb{Z})^\times\cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. Once we fix a primitive 12-th root of unity $\zeta$, the isomorphism is given by $\sigma\mapsto a$ where $\sigma(\zeta) = \zeta^a$. You can explicitly write down the 4 Dirichlet characters of $(\mathbb{Z}/12\mathbb{Z})^\times$ and I will leave this to you. Now, you should identify the subgroup of $(\mathbb{Z}/12\mathbb{Z})^\times$ that corresponds to the field $K$ (notice that $F$ has three quadratic subfields: $\mathbb{Q}(\sqrt{\pm3})$ and $\mathbb{Q}(\sqrt{-1})$) and you will find that out of the three non-trivial Dirichlet characters you just write down, exactly one will factor through this subgroup. That's the character you are looking for. No serious class field theory comes into this, really, nothing beyond Kronecker-Weber.

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Can't we go a little lowbrow here and eliminate any mention of representations and absolute Galois groups and just say (in the abelian case) that $\chi$ runs over the non-trivial homomorphisms from the Galois group of $K/{\bf Q}$ to the non-zero complex numbers? –  Gerry Myerson May 2 '11 at 4:23
    
I'm still puzzled about why we get that particular character. We have functions from $G_\mathbb{Q}$ to $\mathbb{C}^\times$ and $K$ is the fixed field of some $H$ which has index two. So we need to look at the characters $H^\bot = \widehat{G/H}$? How do you deduce what the character should be? –  dstt May 2 '11 at 4:26
    
Is there any way to express this through the ray class formulation of global CFT? It's usually much easier to understand the idelic formulation once you understand the ray class formulation. –  dstt May 2 '11 at 4:28

In my mind, the concrete example of this factorization is for the Gaussian integers. In this case we have that $$\chi(n) = \begin{cases} 1 \;\;\;\;\;\; \text{if } n \equiv 1 \mod{4} \\ -1 \;\;\;\; \text{if } n \equiv 3 \mod{4}\\ 0 \; \; \; \; \; \text{otherwise} \end{cases}$$ is a Dirichlet character $\chi$ of $\mathbb{Z}[i]$, $L(s,\chi)$ is the Dirichlet L-function evaluated at $\chi$, and $\zeta_{\mathbb{Q}(i)}(s)$ is the Dedekind zeta function over the Gaussian integers. Then, $$\zeta_{\mathbb{Q}(i)}(s) = \zeta(s)L(s,\chi)$$ is the factorization of the Dedekind zeta function as a product of the Riemann zeta function and a Dirichlet L-function over $\mathbb{Z}[i]$.

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