Serre's modularity conjecture

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In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre (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

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

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 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

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.