Serre's modularity conjecture

In mathematics, Serre's modularity conjecture, introduced by, 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, and a proof of the full conjecture was completed jointly by Khare and Jean-Pierre Wintenberger in 2008.

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


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

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, and by Luis Dieulefait, independently.

In 2005, Chandrashekhar Khare obtained a proof of the level 1 case of Serre conjecture, and in 2008 a proof of the full conjecture in collaboration with Jean-Pierre Wintenberger.