I already created this post on Math Stack Exchange but I was not so sure if this question fits better here. If it is not, I want to apologize in advance and feel free to delete my post.

I want to understand the proof of the following Theorem from "Euler Factors determine Weil Representations" by Tim and Vladimir Dokchitser:

Theorem 1 Every Frobenius-semisimple Weil representation $\rho$ is uniquely determined by its local polynomials $P(\rho/F,T)$ over finite separable extensions $F/K$.

Before we talk about the cyclic case of the proof, let us recall some definitions first:

• Let $K$ be a local field and $G_K = \operatorname{Gal}(\bar{K}/K)$ be the absolute Galois group of $K$. An (arithmetic) Frobenius element is any element $\operatorname{Frob}_K \in G_K$ that acts as $x \mapsto x^{|k|}$ on $\bar{k}$, the algebraic closure of the residue field $k$ of $K$.
• The Weil group $W_K$ is the subgroup of $G_K$ of all automorphisms that act as an integral power of Frobenius on the residue field.
• A Weil representation is a representation $\rho: W_K \to \operatorname{GL}_n(\mathbb{C})$ such that $\rho(I_K)$ is finite. It is called Frobenius-semisimple if the image of some (equivalently, any) Frobenius element is diagonalizable.
• The local polynomial $P(\rho,T)$ is the inverse characteristic polynomial of $\operatorname{Frob}_K^{-1}$ on the inertia invariants of $\rho$, i.e. $$P(\rho,T) = \det(1-T \cdot \operatorname{Frob_K^{-1}}).$$ Similarly, for a finite extension $F/K$, we write $P(\rho/F)$ for the local polynomial of the restriction of $\rho$ to $W_F$, i.e. $$P(\rho/F,T) = P(\rho|_{W_F},T).$$

Now I would like to talk about the proof of the cyclic case which is given in the paper:

Step 1: Cyclic. Suppose $\rho$ factors through a finite cyclic group $G = \operatorname{Gal}(F/K) \simeq C_n$ and $F/K$ has ramification degree $e$. By Lemma 2 (cf. below), there is a cyclic totally ramified extension $K/K$ of degree $e$ such that $FL/L$ is unramified of degree $n$. The restriction map $\operatorname{Gal}(FL/L) \to \operatorname{Gal}(F/K)$ is an isomorphism, as it is clearly injective and both groups have order $n$. So $\rho/L$ determines $\rho$, and $\rho/L$ can be recovered from its local polynomial $P(\rho/L,T)$.

In our proof we used the following Lemma which we shall take for granted in this post:

Lemma 2 Let $F/K$ be a cyclic extension of degree $n$ and ramification degree $e$. Then there exists a cyclic totally ramified extension $L/K$ of degree $e$ such that $FL/L$ is unramified of degree $n$.

Now I have the following specific questions about the proof above:

• Could you give me a good argument why the restriction map $\operatorname{Gal}(FL/L) \to \operatorname{Gal}(F/K)$ is injective? Let us say we have $\sigma, \sigma' \in \operatorname{Gal}(FL/L)$ which are not equal. Then there exists an $x \in FL \setminus L$ such that $\sigma(x) \neq \sigma'(x)$. If $x \in F$ then the restrictions $\bar{\sigma}, \bar{\sigma}$ are obviously not equal. But what happens if $x$ is not in $F$?
• Why does $\rho/L$ determine $\rho$ then? And how can we recover $\rho/L$ from its local polynomial $P(\rho/L,T)$? What do "determining" and "recovering" even mean in these cases?

Additional Remark:

I would also like to add a diagram which my professor drew to explain me something relating to this Theorem (or Lemma?):

$$\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} & & & & \operatorname{Gal}(LF/L) & \ra{\chi|_L, \text{ unramified}} & \operatorname{Gal}(K^{ur}/K) & & & & \\ & & & & \da{\simeq} & & \da{=} & & & & \\ 1 & \ra{} & \operatorname{Gal}(F/K^{ur}) & \ra{} & \operatorname{Gal}(F/K) & \ra{} & \operatorname{Gal}(K^{ur}/K) & \ra{} & 1\\ & & & & \da{\chi} & & & & & & \\ & & & & \mathbb{C}^\times & & & & & & \\ \end{array}$$ (Sorry, did not know how to do a long equal for $\operatorname{Gal}(K^{ur}/K)$.)

And next to this diagram, he also wrote $P(\chi|_L,T) = 1 - \operatorname{Frob}_L \cdot T$.

To this remark, I have the following questions (if it is really related to the proof):

• What is $\chi$ supposed to mean?
• Is the diagram commutative?
• Why all of the sudden does the maximal unramified extension (I think?) $K^{ur}$ of $K$ appear here?
• If or how is this related to the proof of Theorem 1 (resp. Lemma 2)?

Could you please help me answering my questions? I feel like I lack a lot of background knowledge which is required here, so it would be really nice if you could explain them to me carefully. Thank you in advance!