Serre's modularity conjecture

Formulation

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

Let ρ be an absolutely irreducible, continuous, two-dimensional representation of G Q over a finite field F = F ℓ r .

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

To any normalized modular eigenform

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

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

where O is the ring of integers in a finite extension of Q ℓ . This representation is characterized by the condition that for all prime numbers p , coprime to N ℓ we have

and

Reducing this representation modulo the maximal ideal of O gives a mod ℓ representation ρ f ¯ of G Q .

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

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 during 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.