$L$-function of character in terms of other character

Question

Let $F/K$ be a finite Galois field extension and $\varphi : \mathfrak{I}_K \rightarrow \mathbb{C}^{\times}$ a Hecke character of $K$.Define $\psi = \varphi \circ N_{F/K}$ as a Hecke character of $F$.

The $L$-function of a Hecke character was given by $$L(\varphi,s) = \prod_{\mathfrak{p}} \frac{1}{1 - \varphi(\mathfrak{p}) \mathfrak{N}_{K/\mathbb{Q}}(\mathfrak{p})^s},$$ the product being over all prime ideals.

I am looking for a connection between $L(\psi,s)$ and $L(\varphi,s)$. On page 121 of de Shalit - Iwasawa theory of elliptic curves with CM, I found the equation $$L(\psi,s) = \prod_{\chi \in \mathrm{Gal}(F/K)} L(\chi \varphi,s),$$ at least in the case that $F/K$ is abelian, and this sort of result is what I want. However, I don't understand it; I don't even know how to interpret $\chi \varphi$. I appreciate any help.

Answer 1

This is a brief sketch, but hopefully it is helpful in some way.

For simplicity, I will only look at unramified finite places of $F$ and $K$. Let $p$ be a nonarchimedean place of $K$, and $q$ be a place in $F$ over $p$. We want to compare the local $L$-factor at $p$: $$L_p(\varphi, s) = (1 - \varphi(p) N_{K/\mathbb{Q}}(p)^{-s})^{-1}$$ with the product of all $L$-factors of places above $p$: $$\prod_{q | p} L_q(\psi, s) = \prod (1 - \psi(q) N_{F/\mathbb{Q}}(q)^{-s})^{-1} = \prod (1 - \varphi(p^{f_q}) N_{K/\mathbb{Q}}(N_{F/K}(q))^{-s})^{-1} = \prod (1 - \left(\varphi(p) N_{K/\mathbb{Q}}(p)^{-s}\right)^{f})^{-1} = (1 - \left(\varphi(p) N_{K/\mathbb{Q}}(p)^{-s}\right)^{f})^{-g}$$ where $f,g$ are the residue field degree and number of primes above $p$ respectively.

1. $\phi$ unramified at $p$ implies that $\psi$ is unramified at all $q$ above $p$.

2. By global class field theory, if $F/K$ is finite abelian Galois, we have $$Gal(F/K) \cong J_K/N J_F$$ This allows us to interpret a character of the Galois group as a Hecke character.

3. Given a Galois character $\chi:Gal(F/K) \to \mathbb{C}$, it is unramified at $p$, i.e. the restriction to any inertia group $I(q|p)$ is trivial, if and only if the corresponding Hecke character $\varphi_{\chi}$ is unramified at $p$. The unramified characters are all we care about, since the ramified ones has 1 as $L$-factor which can be ignored. Moreover, $$\varphi_{\chi}(p) = \chi(Frob_p)$$ So there are $f$ possible values for $\varphi_{\chi}(p)$, each occuring $g$ times, corresponding to the $g$ possible lifts of the character from the decomposition group.

4. Note the factorization $1-T^f = \prod_{\mu} (1 - \mu f)$ where $\mu$ is a $f$-th root of unity. Note also that the Galois group of residue field extension has order $f$, so we may as well write $$(1 - \left(\varphi(p) N_{K/\mathbb{Q}}(p)^{-s}\right)^{f})^{-g} = \prod_{\chi_1 \in Gal(k_q/k_p)^{\vee}} (1 - \left(\chi_1(Frob_p)\varphi(p) N_{K/\mathbb{Q}}(p)^{-s}\right)^{-g}$$ where $k_p,k_q$ are residue fields for $p$ and $q$ respectively, and $\vee$ means the character group. But each $\chi_1$ can be lifted uniquely to a character $\chi_1'$ on the decomposition group $D(q|p)$ by imposing that the value on $I(q|p)$ is 1. Each $\chi_1'$ has $g$ lifts to a character $\chi$ on $Gal(F/K)$, so $$(1 - \left(\varphi(p) N_{K/\mathbb{Q}}(p)^{-s}\right)^{f})^{-g} = \prod_{\chi_1 \in Gal(k_q/k_p)^{\vee}} (1 - \left(\chi_1(Frob_p)\varphi(p) N_{K/\mathbb{Q}}(p)^{-s}\right)^{-g} = \prod_{\chi \in Gal(F/K)^{\vee}} (1 - \left(\chi(Frob_p)\varphi(p) N_{K/\mathbb{Q}}(p)^{-s}\right)^{-1} = \prod_{\chi \in Gal(F/K)^{\vee}} (1 - \left(\varphi_{\chi}(p)\varphi(p) N_{K/\mathbb{Q}}(p)^{-s}\right)^{-1}$$ (Here in the last product, we implicitly consider only the characters unramified at $p$, so that $\chi(Frob_p)$ makes sense) If we declare $$L_p(\varphi \chi, s) = (1 - \left(\varphi_{\chi}(p)\varphi(p) N_{K/\mathbb{Q}}(p)^{-s}\right)^{-1}$$ then $$\prod_{q | p} L_q(\psi, s) = \prod_{\chi \in Gal(F/K)^{\vee}} L_p (\varphi \chi,s)$$ Multiplying over all places, we are done.