In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre (1975, 1987), states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form. The conjecture in the level 1 case was proved by Chandrashekhar Khare in 2005,[1] and a proof of the full conjecture was completed jointly by Khare and Jean-Pierre Wintenberger in 2008.[2]
a theorem due to Shimura, Deligne, and Serre-Deligne attaches to a representation
where is the ring of integers in a finite extension of . This representation is characterized by the condition that for all prime numbers , coprime to we have
and
Reducing this representation modulo the maximal ideal of gives a mod representation of .
Serre's conjecture asserts that for any representation as above, there is a modular eigenform such that
.
The level and weight of the conjectural form are explicitly conjectured 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).
Optimal level and weight
The strong form of Serre's conjecture describes the level and weight of the modular form.
The optimal level is the Artin conductor of the representation, with the power of removed.
In 2005, Chandrashekhar Khare obtained a proof of the level 1 case of Serre conjecture,[5] and in 2008 a proof of the full conjecture in collaboration with Jean-Pierre Wintenberger.[6]
Notes
^Khare, Chandrashekhar (2006), "Serre's modularity conjecture: The level one case", Duke Mathematical Journal, 134 (3): 557–589, doi:10.1215/S0012-7094-06-13434-8.
^Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "On Serre's reciprocity conjecture for 2-dimensional mod p representations of Gal(Q/Q)", Annals of Mathematics, 169 (1): 229–253, doi:10.4007/annals.2009.169.229.
^Dieulefait, Luis (2007), "The level 1 weight 2 case of Serre's conjecture", Revista Matemática Iberoamericana, 23 (3): 1115–1124, arXiv:math/0412099, doi:10.4171/rmi/525.
^Khare, Chandrashekhar (2006), "Serre's modularity conjecture: The level one case", Duke Mathematical Journal, 134 (3): 557–589, doi:10.1215/S0012-7094-06-13434-8.
Stein, William A.; Ribet, Kenneth A. (2001), "Lectures on Serre's conjectures", in Conrad, Brian; Rubin, Karl (eds.), Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., vol. 9, Providence, R.I.: American Mathematical Society, pp. 143–232, ISBN978-0-8218-2173-2, MR 1860042
Serre's Modularity Conjecture 50 minute lecture by Ken Ribet given on October 25, 2007 ( slides PDF, PDF)
January 14, 2023
serre, modularity, conjecture, mathematics, introduced, jean, pierre, serre, 1975, 1987, states, that, irreducible, dimensional, galois, representation, over, finite, field, arises, from, modular, form, stronger, version, this, conjecture, specifies, weight, l. In mathematics Serre s modularity conjecture introduced by Jean Pierre Serre 1975 1987 states that an odd irreducible two dimensional Galois representation over a finite field arises from a modular form A stronger version of this conjecture specifies the weight and level of the modular form The conjecture in the level 1 case was proved by Chandrashekhar Khare in 2005 1 and a proof of the full conjecture was completed jointly by Khare and Jean Pierre Wintenberger in 2008 2 Serre s modularity conjectureFieldAlgebraic number theoryConjectured byJean Pierre SerreConjectured in1975First proof byChandrashekhar KhareJean Pierre WintenbergerFirst proof in2008 Contents 1 Formulation 2 Optimal level and weight 3 Proof 4 Notes 5 References 6 See also 7 External linksFormulation EditThe conjecture concerns the absolute Galois group G Q displaystyle G mathbb Q of the rational number field Q displaystyle mathbb Q Let r displaystyle rho be an absolutely irreducible continuous two dimensional representation of G Q displaystyle G mathbb Q over a finite field F F ℓ r displaystyle F mathbb F ell r r G Q G L 2 F displaystyle rho colon G mathbb Q rightarrow mathrm GL 2 F Additionally assume r displaystyle rho is odd meaning the image of complex conjugation has determinant 1 To any normalized modular eigenform f q a 2 q 2 a 3 q 3 displaystyle f q a 2 q 2 a 3 q 3 cdots of level N N r displaystyle N N rho weight k k r displaystyle k k rho and some Nebentype character x Z N Z F displaystyle chi colon mathbb Z N mathbb Z rightarrow F a theorem due to Shimura Deligne and Serre Deligne attaches to f displaystyle f a representation r f G Q G L 2 O displaystyle rho f colon G mathbb Q rightarrow mathrm GL 2 mathcal O where O displaystyle mathcal O is the ring of integers in a finite extension of Q ℓ displaystyle mathbb Q ell This representation is characterized by the condition that for all prime numbers p displaystyle p coprime to N ℓ displaystyle N ell we have Trace r f Frob p a p displaystyle operatorname Trace rho f operatorname Frob p a p and det r f Frob p p k 1 x p displaystyle det rho f operatorname Frob p p k 1 chi p Reducing this representation modulo the maximal ideal of O displaystyle mathcal O gives a mod ℓ displaystyle ell representation r f displaystyle overline rho f of G Q displaystyle G mathbb Q Serre s conjecture asserts that for any representation r displaystyle rho as above there is a modular eigenform f displaystyle f such that r f r displaystyle overline rho f cong rho The level and weight of the conjectural form f displaystyle f are explicitly conjectured 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 Optimal level and weight EditThe strong form of Serre s conjecture describes the level and weight of the modular form The optimal level is the Artin conductor of the representation with the power of l displaystyle l removed Proof EditA proof of the level 1 and small weight cases of the conjecture was obtained in 2004 by Chandrashekhar Khare and Jean Pierre Wintenberger 3 and by Luis Dieulefait 4 independently In 2005 Chandrashekhar Khare obtained a proof of the level 1 case of Serre conjecture 5 and in 2008 a proof of the full conjecture in collaboration with Jean Pierre Wintenberger 6 Notes Edit Khare Chandrashekhar 2006 Serre s modularity conjecture The level one case Duke Mathematical Journal 134 3 557 589 doi 10 1215 S0012 7094 06 13434 8 Khare Chandrashekhar Wintenberger Jean Pierre 2009 Serre s modularity conjecture I Inventiones Mathematicae 178 3 485 504 Bibcode 2009InMat 178 485K CiteSeerX 10 1 1 518 4611 doi 10 1007 s00222 009 0205 7 and Khare Chandrashekhar Wintenberger Jean Pierre 2009 Serre s modularity conjecture II Inventiones Mathematicae 178 3 505 586 Bibcode 2009InMat 178 505K CiteSeerX 10 1 1 228 8022 doi 10 1007 s00222 009 0206 6 Khare Chandrashekhar Wintenberger Jean Pierre 2009 On Serre s reciprocity conjecture for 2 dimensional mod p representations of Gal Q Q Annals of Mathematics 169 1 229 253 doi 10 4007 annals 2009 169 229 Dieulefait Luis 2007 The level 1 weight 2 case of Serre s conjecture Revista Matematica Iberoamericana 23 3 1115 1124 arXiv math 0412099 doi 10 4171 rmi 525 Khare Chandrashekhar 2006 Serre s modularity conjecture The level one case Duke Mathematical Journal 134 3 557 589 doi 10 1215 S0012 7094 06 13434 8 Khare Chandrashekhar Wintenberger Jean Pierre 2009 Serre s modularity conjecture I Inventiones Mathematicae 178 3 485 504 Bibcode 2009InMat 178 485K CiteSeerX 10 1 1 518 4611 doi 10 1007 s00222 009 0205 7 and Khare Chandrashekhar Wintenberger Jean Pierre 2009 Serre s modularity conjecture II Inventiones Mathematicae 178 3 505 586 Bibcode 2009InMat 178 505K CiteSeerX 10 1 1 228 8022 doi 10 1007 s00222 009 0206 6 References EditSerre Jean Pierre 1975 Valeurs propres des operateurs de Hecke modulo l Journees Arithmetiques de Bordeaux Conf Univ Bordeaux 1974 Asterisque 24 25 109 117 ISSN 0303 1179 MR 0382173 Serre Jean Pierre 1987 Sur les representations modulaires de degre 2 de Gal Q Q Duke Mathematical Journal 54 1 179 230 doi 10 1215 S0012 7094 87 05413 5 ISSN 0012 7094 MR 0885783 Stein William A Ribet Kenneth A 2001 Lectures on Serre s conjectures in Conrad Brian Rubin Karl eds Arithmetic algebraic geometry Park City UT 1999 IAS Park City Math Ser vol 9 Providence R I American Mathematical Society pp 143 232 ISBN 978 0 8218 2173 2 MR 1860042See also EditWiles s proof of Fermat s Last TheoremExternal links EditSerre s Modularity Conjecture 50 minute lecture by Ken Ribet given on October 25 2007 slides PDF other version of slides PDF Lectures on Serre s conjectures Retrieved from https en wikipedia org w index php title Serre 27s modularity conjecture amp oldid 1094232517, wikipedia, wiki, book, books, library,