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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245601.png" /> be a field that is complete with respect to a discrete valuation, with residue class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245602.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245603.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245604.png" /> be a Galois extension of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245605.png" /> with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245606.png" /> and suppose that the residue class field extension is separable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245607.png" /> is the character of some finite-dimensional complex representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245608.png" />, its conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c0245609.png" /> is defined by the formula:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456010.png" /></td> </tr></table>
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456011.png" /></td> </tr></table>
+
$$
 +
G _ {i}  = \
 +
\{ {g \in G } : {
 +
\nu _ {L} ( g ( x) - x) \geq  i + 1  \textrm{ for } \
 +
\textrm{ all }  x \in L \
 +
\textrm{ with } \
 +
\nu _ {L} ( x) \geq  0 } \}
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456012.png" /></td> </tr></table>
+
$$
 +
n _ {i}  = | G _ {i} |,\  \chi ( G _ {i} )
 +
= n _ {i}  ^ {-} 1 \sum _ {g \in G _ {i} } \chi ( g) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456013.png" /> is the corresponding valuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456015.png" /> does not divide <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456016.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456017.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456020.png" /> is the character of a rational representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456022.png" />. The conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456023.png" /> is a non-negative integer.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456025.png" /> is the conductor of a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456026.png" /> of the Galois group of an extension of local fields, is also called the Artin conductor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456027.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456028.png" />-functions, cf. [[L-function|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024560/c02456029.png" />-function]].
+
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>

Revision as of 17:46, 4 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=46446
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article