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Serre's modularity conjecture
Field	Algebraic number theory
Conjectured by	Jean-Pierre Serre
Conjectured in	1975
First proof by	Chandrashekhar Khare; Jean-Pierre Wintenberger
First proof in	2008

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]

Formulation

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

\rho \colon G_{\mathbb {Q} }\rightarrow \mathrm {GL} _{2}(F).

Additionally, assume $\rho$ 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}+\cdots

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

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

,

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

\rho _{f}\colon 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 representation $\rho$ 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 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 $l$ removed.

Proof

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

^ 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 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.
^ 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

Serre, Jean-Pierre (1975), "Valeurs propres des opérateurs de Hecke modulo l", Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), Astérisque, 24–25: 109–117, ISSN 0303-1179, MR 0382173
Serre, Jean-Pierre (1987), "Sur les représentations modulaires de degré 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 1860042