Serre's modularity conjecture

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 conjecture
FieldAlgebraic number theory
Conjectured byJean-Pierre Serre
Conjectured in1975
First proof byChandrashekhar Khare
Jean-Pierre Wintenberger
First proof in2008

Formulation

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The conjecture concerns the absolute Galois group   of the rational number field  .

Let   be an absolutely irreducible, continuous, two-dimensional representation of   over a finite field  .

 

Additionally, assume   is odd, meaning the image of complex conjugation has determinant -1.

To any normalized modular eigenform

 

of level  , weight  , and some Nebentype character

 ,

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

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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.

Proof

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A 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

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  1. ^ 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.
  2. ^ 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.
  3. ^ 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.
  4. ^ 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.
  5. ^ 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.
  6. ^ 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

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See also

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