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:
- 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.
- 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!
