Serre's modularity conjecture

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In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre based on some 1973–1974 correspondence with John Tate (Serre 1975, 1987), 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,[1] in 2005 and later in 2008 a proof of the full conjecture was worked out jointly by Khare and Jean-Pierre Wintenberger.[2]

Formulation[edit]

The conjecture concerns the absolute Galois group G_\mathbb{Q} of the rational number field \mathbb{Q}.

Let \rho be an absolutely irreducible, continuous, two-dimensional representation of G_\mathbb{Q} over a finite field that is odd (meaning that complex conjugation has determinant -1)

F = \mathbb{F}_{\ell^r}

of characteristic \ell,

 \rho: G_\mathbb{Q} \rightarrow \mathrm{GL}_2(F).\

To any normalized modular eigenform

 f = q+a_2q^2+a_3q^3+\cdots\

of level  N=N(\rho) , weight  k=k(\rho) , and some Nebentype character

 \chi : \mathbb{Z}/N\mathbb{Z} \rightarrow F^*\ ,

a theorem due to Shimura, Deligne, and Serre-Deligne attaches to  f a representation

 \rho_f: G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\mathcal{O}),\

where  \mathcal{O} is the ring of integers in a finite extension of  \mathbb{Q}_\ell . This representation is characterized by the condition that for all prime numbers p, coprime to N\ell we have

 \operatorname{Trace}(\rho_f(\operatorname{Frob}_p))=a_p\

and

 \det(\rho_f(\operatorname{Frob}_p))=p^{k-1} \chi(p).\

Reducing this representation modulo the maximal ideal of  \mathcal{O} gives a mod  \ell representation  \overline{\rho_f} of  G_\mathbb{Q} .

Serre's conjecture asserts that for any  \rho as above, there is a modular eigenform  f such that

 \overline{\rho_f} \cong \rho .

The level and weight of the conjectural form  f 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).

Optimal level and weight[edit]

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 l removed.

Proof[edit]

A proof of the level 1 and small weight cases of the conjecture was obtained during 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]

  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, 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, 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, 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, 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, doi:10.1007/s00222-009-0206-6 .

References[edit]

External links[edit]