# Conductor of a character

An integer associated to the character of a representation of the Galois group of a finite extension of a local field. Let $K$ be a field that is complete with respect to a discrete valuation, with residue class field $k$ of characteristic $p \geq 0$. Let $L/K$ be a Galois extension of degree $n$ with Galois group $G$ and suppose that the residue class field extension is separable. If $\chi$ is the character of some finite-dimensional complex representation of $G$, its conductor $f ( \chi )$ is defined by the formula:

$$f ( \chi ) = \ \sum _ {i = 0 } ^ \infty \frac{n _ {i} }{n _ {0} } ( \chi ( 1) - \chi ( G _ {i} )),$$

where

$$G _ {i} = \ \{ {g \in G } : { \nu _ {L} ( g ( x) - x) \geq i + 1 \textrm{ for } \ \textrm{ all } x \in L \ \textrm{ with } \ \nu _ {L} ( x) \geq 0 } \} ,$$

$$n _ {i} = | G _ {i} |,\ \chi ( G _ {i} ) = n _ {i} ^ {-1} \sum _ {g \in G _ {i} } \chi ( g) ,$$

where $\nu _ {L}$ is the corresponding valuation of $L$. If $p$ does not divide $n$, then $G _ {i} = \{ 1 \}$ for $i > 0$ and $f ( \chi ) = \chi ( 1) - \chi ( G _ {0} )$. If $\chi$ is the character of a rational representation $M$, then $\chi ( G _ {i} ) = \mathop{\rm dim} M ^ {G _ {i} }$. The conductor $f ( \chi )$ is a non-negative integer.

#### References

 [1] J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967) pp. Chapt. VI [2] E. Artin, J. Tate, "Class field theory" , Benjamin (1967) [3] J.-P. Serre, "Local fields" , Springer (1979) (Translated from French)

The ideal $\mathfrak p _ {k} ^ {f ( \chi ) }$, where $f ( \chi )$ is the conductor of a character $\chi$ of the Galois group of an extension of local fields, is also called the Artin conductor of $\chi$. There is a corresponding notion for extensions of global fields obtained by taking a suitable product over all finite primes, cf. [a1], p. 126. It plays an important role in the theory of Artin $L$- functions, cf. $L$- function.