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)

Comments

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.

References

[a1]	J. Neukirch, "Class field theory" , Springer (1986)