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This post can be seen as a continuation of this post I created on MathOverflow.

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 Artin representation 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 case of an Artin representation which is given in the paper:

Step 2: Artin to cyclic. Suppose $\rho$ factors through a finite quotient, equivalently it has local polynomial $P(\rho/F,T) = (1-T)^{\dim{\rho}}$ over some Galois extension $F/K$. By character theory for the finite group $G=\operatorname{Gal}(F/K)$, it is enough to recover the character of $\rho$. Thus it suffices to recover the restriction of $\rho$ to every cyclic subgroup $\langle g \rangle < G$ since this gives the value of the character on the conjugacy class of $g$. This is done in Step 1 (the cyclic case).

We shall take Step 1 for granted in this post. Anyone who is more interested in this step can take a look at the MathOverflow post.

Now I have the following specific questions regarding the second step:

  1. Is it important that the local polynomial of $\rho/F$ for some Galois extension is $(1-T)^{\dim{\rho}}$? I don't really see how this is used in the proof but it bugged me since the authors mentioned it.
  2. How exactly do we get the value of the character if we restrict $\rho$ to every cyclic subgroup? It seems to come from character theory but my background knowledge about it is not really good.

What did I understand?

  • Restricting our representation $\rho$ to any cyclic subgroup $\langle g \rangle < G$ yields an intermediate subextension $M_g$ of $F/K$ such that $\langle g \rangle = \operatorname{Gal}(F/M_g)$ (follows from the Fundamental Theorem of Galois Theory). Then one applies this restriction to Step 1 of the proof.

Could you please help me with this question? If there is something unclear with my explanation, please let me know. Thank you!

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