Serre conjecture (number theory)

The Quillen–Suslin theorem was also conjectured by Serre; and may also be called Serre's Conjecture.

Given a representation

$\rho: G_Q \rightarrow GL_2(F)\$ ,

where F is a finite field of characteristic l, $ρ$ is an absolutely irreducible, continuous, and odd representation of the absolute Galois group of the rationals Q, then there exists a normalized modular eigenform

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

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

$\chi : Z/NZ \rightarrow F^*\$

such that for all prime numbers p, coprime to Nl we have

$\operatorname{Trace}(\rho(\operatorname{Frob}_p))=a_p\$

and

$\det(\rho(\operatorname{Frob}_p))=p^{k-1} \chi(p).\$

The level and the weight of $ρ$ are explicitly calculated in Serre's article [1].

One of the corollaries of Serre's conjecture is the proof of Fermat's Last Theorem.