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| In mathematics, '''Serre's modularity conjecture''', introduced by {{harvs|txt|last=Serre|authorlink=Jean-Pierre Serre|year1=1975|year2=1987}} based on some 1973–1974 correspondence with [[John Tate]], states that an odd irreducible two-dimensional [[Galois representation]] over a finite field arises from a modular form, and a stronger version of his conjecture specifies the weight and level of the modular form. It was proved by [[Chandrashekhar Khare]] in the level 1 case<ref>{{Citation |last=Khare |first=Chandrashekhar |title=Serre's modularity conjecture: The level one case |year=2006 |journal=Duke Mathematical Journal |volume=134 |issue=3 |pages=557–589 |doi=10.1215/S0012-7094-06-13434-8 }}.</ref> in 2005 and later in 2008 a proof of the full conjecture was worked out jointly by [[Chandrashekhar Khare]] and [[Jean-Pierre Wintenberger]].<ref>{{Citation |last=Khare |first=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre’s modularity conjecture (I) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=485–504 |doi=10.1007/s00222-009-0205-7 }} and {{Citation |last=Khare |first=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre’s modularity conjecture (II) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=505–586 |doi=10.1007/s00222-009-0206-6 }}.</ref>
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| ==Formulation==
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| The conjecture concerns the [[absolute Galois group]] <math>G_\mathbb{Q}</math> of the [[rational number field]] <math>\mathbb{Q}</math>.
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| Let <math>\rho</math> be an [[absolutely irreducible]], continuous, two-dimensional representation of <math>G_\mathbb{Q}</math> over a finite field that is odd (meaning that complex conjugation has determinant -1)
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| :<math>F = \mathbb{F}_{\ell^r}</math>
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| of [[characteristic (field theory)|characteristic]] <math>\ell</math>,
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| :<math> \rho: G_\mathbb{Q} \rightarrow \mathrm{GL}_2(F).\ </math>
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| To any normalized [[modular eigenform]]
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| :<math> f = q+a_2q^2+a_3q^3+\cdots\ </math>
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| of [[level of a modular form|level]] <math> N=N(\rho) </math>, [[weight of a modular form|weight]] <math> k=k(\rho) </math>, and some [[Nebentype character]]
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| :<math> \chi : \mathbb{Z}/N\mathbb{Z} \rightarrow F^*\ </math>,
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| a theorem due to Shimura, Deligne, and Serre-Deligne attaches to <math> f </math> a representation
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| :<math> \rho_f: G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\mathcal{O}),\ </math>
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| where <math> \mathcal{O} </math> is the ring of integers in a finite extension of <math> \mathbb{Q}_\ell </math>. This representation is characterized by the condition that for all prime numbers <math>p</math>, [[coprime]] to <math>N\ell</math> we have | |
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| :<math> \operatorname{Trace}(\rho_f(\operatorname{Frob}_p))=a_p\ </math>
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| and | |
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| :<math> \det(\rho_f(\operatorname{Frob}_p))=p^{k-1} \chi(p).\ </math>
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| Reducing this representation modulo the maximal ideal of <math> \mathcal{O} </math> gives a mod <math> \ell </math> representation <math> \overline{\rho_f} </math> of <math> G_\mathbb{Q} </math>.
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| Serre's conjecture asserts that for any <math> \rho </math> as above, there is a modular eigenform <math> f </math> such that
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| :<math> \overline{\rho_f} \cong \rho </math>.
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| The level and weight of the conjectural form <math> f </math> are explicitly calculated in Serre's article. In addition, he derives a number of results from this conjecture, among them [[Fermat's Last Theorem]] and the now-proven Taniyama–Weil (or Taniyama–Shimura) conjecture, now known as the [[modularity theorem]] (although this implies Fermat's Last Theorem, Serre proves it directly from his conjecture).
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| ==Optimal level and weight==
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| The strong form of Serre's conjecture describes the level and weight of the modular form.
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| The optimal level is the [[Artin conductor]] of the representation, with the power of ''l'' removed.
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| ==Proof==
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| A proof of the level 1 and small weight cases of the conjecture was obtained during 2004 by [[Chandrashekhar Khare]] and [[Jean-Pierre Wintenberger]],<ref>{{Citation |last=Khare |first=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=On Serre's reciprocity conjecture for 2-dimensional mod p representations of Gal(Q/Q) |journal=[[Annals of Mathematics]] |volume=169 |issue=1 |pages=229–253 |doi= |url=http://annals.princeton.edu/annals/2009/169-1/p05.xhtml }}.</ref> and by [[Luis Dieulefait]],<ref>{{Citation |last=Dieulefait |first=Luis |year=2007 |title=The level 1 weight 2 case of Serre's conjecture |journal=Revista Matemática Iberoamericana |volume=23 |issue=3 |pages=1115–1124 |url=http://projecteuclid.org/euclid.rmi/1204128312 }}.</ref> independently.
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| In 2005, Chandrashekhar Khare obtained a proof of the level 1 case of Serre conjecture,<ref>{{Citation |last=Khare |first=Chandrashekhar |title=Serre's modularity conjecture: The level one case |year=2006 |journal=Duke Mathematical Journal |volume=134 |issue=3 |pages=557–589 |doi=10.1215/S0012-7094-06-13434-8 }}.</ref> and in 2008 a proof of the full conjecture in collaboration with Jean-Pierre Wintenberger.<ref>{{Citation |last=Khare |first=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre’s modularity conjecture (I) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=485–504 |doi=10.1007/s00222-009-0205-7 }} and {{Citation |last=Khare |first=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre’s modularity conjecture (II) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=505–586 |doi=10.1007/s00222-009-0206-6 }}.</ref>
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974) | publisher=[[Société Mathématique de France]] | location=Paris | id={{MR|0382173}} | year=1975 | journal=Astérisque | issn=0303-1179 | volume=24–25 | chapter=Valeurs propres des opérateurs de Hecke modulo l | pages=109–117}}
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| *{{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Sur les représentations modulaires de degré 2 de Gal({{overline|Q}}/Q) | url=http://dx.doi.org/10.1215/S0012-7094-87-05413-5 | doi=10.1215/S0012-7094-87-05413-5 | id={{MR|885783}} | year=1987 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=54 | issue=1 | pages=179–230}}
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| *{{Citation | last1=Stein | first1=William A. | last2=Ribet | first2=Kenneth A. | editor1-last=Conrad | editor1-first=Brian | editor2-last=Rubin | editor2-first=Karl | title=Arithmetic algebraic geometry (Park City, UT, 1999) | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=IAS/Park City Math. Ser. | isbn=978-0-8218-2173-2 | id={{MR|1860042}} | year=2001 | volume=9 | chapter=Lectures on Serre's conjectures | pages=143–232}}
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| ==External links==
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| *[http://fora.tv/2007/10/25/Kenneth_Ribet_Serre_s_Modularity_Conjecture Serre's Modularity Conjecture] 50 minute lecture by [[Ken Ribet]] given on October 25, 2007 ( [http://math.berkeley.edu/~ribet/cms.pdf slides] PDF, [http://www.cirm.univ-mrs.fr/videos/2007/exposes/23/Ribet.pdf other version of slides] PDF)
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| *[http://modular.fas.harvard.edu/papers/serre/ribet-stein.pdf Lectures on Serre's conjectures]
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| [[Category:Modular forms]]
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| [[Category:Theorems in number theory]]
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Ed is what individuals call me and my wife doesn't like it at all. My husband doesn't like it the way I do but what I truly like performing is caving but I don't have the time recently. Ohio is where his home is and his family members loves it. My working day occupation is an invoicing officer but I've currently applied for an additional one.
My weblog: online reader