Serre conjecture (number theory)

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

In mathematics, Jean-Pierre Serre conjectured the following result regarding two-dimensional Galois representations (of the absolute Galois group G_\mathbb{Q} of the rational number field \mathbb{Q}).

Let ρ be an absolutely irreducible, continuous, and odd two-dimensional representation of G_\mathbb{Q} over a finite field F = \mathbb{F}_{l^r} of characteristic l,

\rho: G_Q \rightarrow GL_2(F)\.

According to the conjecture, 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 : \mathbb{Z}/N\mathbb{Z} \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]. 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).

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