# Inflation property of Artin $L$-series

I have been looking at several different proofs that Artin $$L$$-series of Abelian extensions coincide with Hecke $$L$$-series.

In Serge Lang's $$\textit{Algebraic Number Fields}$$ (XII §2) and Jürgen Neukirch's $$\textit{Algebraische Zahlentheorie}$$ (VII §10) they use a property which I paraphrase in the following manner:

Let $$E/K$$ be an Abelian extension, $$G:=\textrm{Gal}(E/K)$$, and let $$\chi$$ be a simple character of $$G$$. We then have $$G/\textrm{Ker}(\chi) \cong \textrm{Gal}(E_{\chi}/K)$$, where $$E_{\chi}$$ is the fixed subfield corresponding to $$\textrm{Ker}(\chi) \vartriangleleft G$$. By inflation, we may view $$\chi$$ is a character of $$\textrm{Gal}(E_{\chi}/K)$$, and we have: $$L(E_{\chi}/K,\chi,s) = L(E/K,\chi,s)$$

I have some scruples with this argument.

I understand the property of inflation to mean the following:

Let $$E/K$$ be a Galois extension, $$G:=\textrm{Gal}(E/K)$$, and let $$E'/K$$ be a bigger Galois extension ($$E \subset E'$$), $$G':=\textrm{Gal}(E'/K)$$, and let $$\chi$$ be a simple character of $$G$$. We then have: $$L(E'/K,\chi',s) = L(E/K,\chi,s)$$ $$\chi' = \chi \circ \pi$$, where $$\pi: G' \to G$$ is the canonical projection.

That is, inflation allows us to pass $$\textit{from a smaller Galois extension to a bigger one}$$.

But Lang and Neukirch (et al.) seem to be going the other way: They take a character of a bigger Galois extension and pass to a smaller one.

But this is manifestly impossible. Take $$\textit{e.g.}$$ $$\mathbb{Q} / \mathbb{Q}$$ for the smaller extension and $$\mathbb{Q}(i) / \mathbb{Q}$$ for the bigger one. The above would imply that: $$L(\mathbb{Q}(i) / \mathbb{Q}, \chi, s) = L( \mathbb{Q} / \mathbb{Q} , \chi , s) = \zeta(s)$$ On the left, we could let $$\chi$$ be the non-trivial sign character. On the right, we must necessarily have the trivial character.

How is this to be understood?

The converse to the inflation is that, if $$\rho$$ is a representation of $$G=Gal(E/K)$$ then $$H=\ker(\rho)$$ is normal and $$\tilde{\rho}(gH)=\rho(g)$$ is a (faithful) representation of $$G/H=Gal(E/K)/Gal(E/E^H)=Gal(E^H/K)$$ and $$L(E/K,\rho,s)=L(E^H/K,\tilde{\rho},s)$$