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In mathematics, the '''Artin conductor''' is a number or [[ideal (ring theory)|ideal]] associated to a character of a [[Galois group]] of a [[local field|local]] or [[global field|global]] [[field (mathematics)|field]], introduced by {{harvs|txt|last=Artin|first=Emil|authorlink=Emil Artin|year1=1930|year2=1931}} as an expression appearing in the [[functional equation]] of an [[Artin L-function]].


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==Local Artin conductors==
 
Suppose that ''L'' is a finite [[Galois extension]] of the local field ''K'', with Galois group ''G''. If χ is a character of ''G'', then the Artin conductor of χ is the number
:<math>f(\chi)=\sum_{i\ge 0}\frac{g_i}{g_0}(\chi(1)-\chi(G_i))</math>
where ''G''<sub>''i''</sub> is the ''i''-th [[ramification group]] (in [[lower numbering]]), of order ''g''<sub>''i''</sub>, and χ(''G''<sub>''i''</sub>) is the average value of χ on ''G''<sub>''i''</sub>.<ref name=S67158>Serre (1967) p.158</ref>  By a result of Artin, the local conductor is an integer.<ref name=S67159>Serre (1967) p.159</ref><ref name=MP329>{{cite book | first1=Yu. I. | last1=Manin | authorlink1= | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | page=329 }}</ref> If χ is unramified, then its Artin conductor is zero. If ''L'' is unramified over ''K'', then the Artin conductors of all χ are zero.
 
The ''wild invariant''<ref name=MP329/> or ''Swan conductor''<ref name=Sn249>Snaith (1994) p.249</ref> of the character is
 
:<math> f(\chi) - (\chi(1)-\chi(G_0)), </math>
 
in other words, the sum of the higher order terms with ''i'' > 0.
 
==Global Artin conductors==
 
The '''global Artin conductor''' of a representation χ of the Galois group ''G'' of a finite extension ''L''/''K'' of global fields is an ideal of ''K'', defined to be
 
:<math>\mathfrak{f}(\chi) = \prod_p p^{f(\chi,p)}</math>
 
where the product is over the primes ''p'' of ''K'', and ''f''(χ,''p'') is the local Artin conductor of the restriction of χ to the decomposition group of some prime of ''L'' lying over ''p''.<ref name=S67159/> Since the local Artin conductor is zero at unramified primes, the above product only need be taken over primes that ramify in ''L''/''K''.
 
==Artin representation and Artin character==
 
Suppose that ''L'' is a finite Galois extension of the local field ''K'', with Galois group&nbsp;''G''. The '''Artin character''' ''a''<sub>''G''</sub> of ''G'' is the character
 
:<math>a_G=\sum_\chi f(\chi)\chi</math>
 
and the '''Artin representation''' ''A''<sub>''G''</sub> is the complex linear representation of ''G'' with this character. {{harvtxt|Weil|1946}} asked for a direct construction of the Artin representation. {{harvs|txt|last=Serre|year=1960}} showed that the Artin representation can be realized over the local field '''Q'''<sub>''l''</sub>, for any prime ''l'' not equal to the residue characteristic ''p''. {{harvtxt|Fontaine|1971}} showed that it can be realized over the corresponding ring of Witt vectors. It cannot in general be realized over the rationals or over the local field '''Q'''<sub>''p''</sub>, suggesting that there is no easy way to construct the Artin representation explicitly.<ref name=S67160>Serre (1967) p.160</ref>
 
==Swan representation==
 
The Swan character ''sw''<sub>''G''</sub> is given by
 
:<math>sw_G= a_G -r_G+1 \, </math>
 
where ''r''<sub>''g''</sub> is the character of the regular representation and 1 is the character of the trivial representation.<ref name=Sn248>Snaith (1994) p.248</ref> The Swan character is the character of a representation of ''G''. {{harvs|txt|authorlink=Richard Swan|last=Swan|year=1963}} showed that there is a unique projective representation of ''G'' over the ''l''-adic integers with character the Swan character.
 
==Applications==
 
The Artin conductor appears in the [[conductor-discriminant formula]] for the discriminant of a global field.<ref name=S67160>Serre (1967) p.160</ref>
 
The optimal level in the [[Serre modularity conjecture]] is expressed in terms of the Artin conductor.  
 
The Artin conductor appears in the functional equation of the [[Artin L-function]].
 
==Notes==
{{reflist}}
 
==References==
*{{Citation | last1=Artin | first1=Emil | author1-link=Emil Artin | title=Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren. | language=German | doi=10.1007/BF02941010 | jfm=56.0173.02  | year=1930 | journal=Abhandlungen Hamburg | volume=8 | pages=292–306}}
*{{Citation | last1=Artin | first1=Emil | author1-link=Emil Artin | title=Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper. | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002171422 | language=German | doi=10.1515/crll.1931.164.1 | zbl=
0001.00801  | year=1931 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | volume=164 | pages=1–11}}
*{{Citation | last1=Fontaine | first1=Jean-Marc | title=Colloque de Théorie des Nombres (Univ. Bordeaux, Bordeaux, 1969) | url=http://www.numdam.org/item?id=MSMF_1971__25__71_0 | publisher=[[Société Mathématique de France]] | location=Paris | series=Mémoires de la Société Mathématique de France | mr=0374106  | year=1971 | volume=25 | chapter=Sur les représentations d'Artin | pages=71–81}}
*{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Sur la rationalité des représentations d'Artin | jstor=1970142 | mr=0171775  | year=1960 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=72 | pages=405–420}}
* {{citation | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | chapter=VI. Local class field theory | pages=128–161 | editor1-last=Cassels | editor1-first=J.W.S. | editor1-link=J. W. S. Cassels | editor2-last=Fröhlich | editor2-first=A. | editor2-link=Albrecht Fröhlich | title=Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union | location=London | publisher=Academic Press | year=1967 | zbl=0153.07403 }}
* {{citation | title=Explicit Brauer Induction: With Applications to Algebra and Number Theory | volume=40 | series=Cambridge Studies in Advanced Mathematics | first=V. P. | last=Snaith | authorlink= | publisher=[[Cambridge University Press]] | year=1994 | isbn=0-521-46015-8 | zbl=0991.20005 }}
*{{Citation | last1=Swan | first1=Richard G. | title=The Grothendieck ring of a finite group | doi=10.1016/0040-9383(63)90025-9 | mr=0153722  | year=1963 | journal=[[Topology (journal)|Topology. An International Journal of Mathematics]] | issn=0040-9383 | volume=2 | pages=85–110}}
*{{Citation | last1=Weil | first1=André | author1-link=André Weil | title=L'avenir des mathématiques | mr=0020961  | year=1946 | journal=Bol. Soc. Mat. São Paulo | volume=1 | pages=55–68}}
 
[[Category:Number theory]]
[[Category:Representation theory]]
[[Category:Zeta and L-functions]]

Latest revision as of 22:37, 4 December 2013

In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by Template:Harvs as an expression appearing in the functional equation of an Artin L-function.

Local Artin conductors

Suppose that L is a finite Galois extension of the local field K, with Galois group G. If χ is a character of G, then the Artin conductor of χ is the number

f(χ)=i0gig0(χ(1)χ(Gi))

where Gi is the i-th ramification group (in lower numbering), of order gi, and χ(Gi) is the average value of χ on Gi.[1] By a result of Artin, the local conductor is an integer.[2][3] If χ is unramified, then its Artin conductor is zero. If L is unramified over K, then the Artin conductors of all χ are zero.

The wild invariant[3] or Swan conductor[4] of the character is

f(χ)(χ(1)χ(G0)),

in other words, the sum of the higher order terms with i > 0.

Global Artin conductors

The global Artin conductor of a representation χ of the Galois group G of a finite extension L/K of global fields is an ideal of K, defined to be

f(χ)=ppf(χ,p)

where the product is over the primes p of K, and f(χ,p) is the local Artin conductor of the restriction of χ to the decomposition group of some prime of L lying over p.[2] Since the local Artin conductor is zero at unramified primes, the above product only need be taken over primes that ramify in L/K.

Artin representation and Artin character

Suppose that L is a finite Galois extension of the local field K, with Galois group G. The Artin character aG of G is the character

aG=χf(χ)χ

and the Artin representation AG is the complex linear representation of G with this character. Template:Harvtxt asked for a direct construction of the Artin representation. Template:Harvs showed that the Artin representation can be realized over the local field Ql, for any prime l not equal to the residue characteristic p. Template:Harvtxt showed that it can be realized over the corresponding ring of Witt vectors. It cannot in general be realized over the rationals or over the local field Qp, suggesting that there is no easy way to construct the Artin representation explicitly.[5]

Swan representation

The Swan character swG is given by

swG=aGrG+1

where rg is the character of the regular representation and 1 is the character of the trivial representation.[6] The Swan character is the character of a representation of G. Template:Harvs showed that there is a unique projective representation of G over the l-adic integers with character the Swan character.

Applications

The Artin conductor appears in the conductor-discriminant formula for the discriminant of a global field.[5]

The optimal level in the Serre modularity conjecture is expressed in terms of the Artin conductor.

The Artin conductor appears in the functional equation of the Artin L-function.

Notes

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    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  1. Serre (1967) p.158
  2. 2.0 2.1 Serre (1967) p.159
  3. 3.0 3.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  4. Snaith (1994) p.249
  5. 5.0 5.1 Serre (1967) p.160
  6. Snaith (1994) p.248