# Artin representations in Serre's book 'local fields'

Let $K$ be a complete local field with discrete valuation, and let $L/K$ be a finite Galois extension. Use $G=Gal(L/K)$ to denote the Galois group.

In Serre's book 'local fields', chapter 6, a linear representation of $G$, called Artin representation, is defined by identifying its character. Explicit description of such representations is discussed in the case of algebraic curves in section 4 of that chapter.

But I'm more interested in the case when $L,K$ are $p$-adic fields, and I wanna know the following questions:

(1) How to construct such representations explicitly in $p$-adic fields case?

(2) What's its role in ramification theory, or more general in (local) class field theory?

(3) What other properties do we know about such representations? Like when will it be irreducible?

Thank you very much for any answer or references for any of the above questions.

• I think the Artin character is always the sum of the Swan character and the unitary character, right? So, I have the feeling that the answer to the last question is: 'Never.' This decomposition corresponds to the measure of wild (resp. tame) ramification, if I'm not mistaken. – jmc Nov 16 '13 at 17:29
• Maybe,I'm not quite clear. But thanks for your comment. – user1832 Nov 18 '13 at 2:25

Serre (Annals, 1960) proves that the Artin representation corresponding to $L|K$ is rational over $\mathbf{Q}_l$ for every prime $l$ distinct from the residual characteristic $p$ and gives an example where it is not rational over $\mathbf{R}$.
Fontaine (Annales ENS, 1971)) gives a general criterion for the Artin representation to be rational over a given field of characteristic $0$, and proves that it is rational over the ring $W(k)$ of Witt vectors (if the residue field $k$ of $K$ is perfect). It might be a good idea to start with these classic papers to learn more about the Artin representation.