Serre's Modularity Conjecture - Formulation

Formulation

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

Let be an absolutely irreducible, continuous, two-dimensional representation of over a finite field that is odd (meaning that complex conjugation has determinant -1)

of characteristic ,

To any normalized modular eigenform

of level, weight, and some Nebentype character

,

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

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

and

Reducing this representation modulo the maximal ideal of gives a mod representation of .

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

.

The level and weight of the conjectural form 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).