I already created [this post on Math Stack Exchange][1] 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][2]:

> **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!


  [1]: https://math.stackexchange.com/questions/2940155/local-factors-determine-weil-representations-proof-of-the-cyclic-case
  [2]: https://arxiv.org/pdf/1112.4889.pdf