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] The conjecture concerns the absolute Galois group
be an absolutely irreducible, continuous, two-dimensional representation of
is odd, meaning the image of complex conjugation has determinant -1.
To any normalized modular eigenform of level
, and some Nebentype character a theorem due to Shimura, Deligne, and Serre-Deligne attaches to
is the ring of integers in a finite extension of
This representation is characterized by the condition that for all prime numbers
we have and Reducing this representation modulo the maximal ideal of
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).
The optimal level is the Artin conductor of the representation, with the power of
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.