Difference between revisions of "Conductor of a character"
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| + | 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 | 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 | + | 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==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1967) pp. Chapt. VI</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E. Artin, J. Tate, "Class field theory" , Benjamin (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.-P. Serre, "Local fields" , Springer (1979) (Translated from French)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1967) pp. Chapt. VI</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E. Artin, J. Tate, "Class field theory" , Benjamin (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.-P. Serre, "Local fields" , Springer (1979) (Translated from French)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | The ideal | + | 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. [[#References|[a1]]], p. 126. It plays an important role in the theory of Artin $ L $- | ||
| + | functions, cf. [[L-function| $ L $- | ||
| + | function]]. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Neukirch, "Class field theory" , Springer (1986)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Neukirch, "Class field theory" , Springer (1986)</TD></TR></table> | ||
Latest revision as of 18:32, 14 June 2020
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) |
How to Cite This Entry:
Conductor of a character. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conductor_of_a_character&oldid=19015
Conductor of a character. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Conductor_of_a_character&oldid=19015
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article