Serre conjecture (number theory)

:"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. This was a significant step in number theory, though this was not realised for at least a decade. [The conjecture appeared in print in 1987, but probably dates to the late 1960s or early 1970s; Serre and Pierre Deligne apparently felt it was too much to ask. The Ribet-Stein reference states that Serre wrote down a conjecture of this type in a 1973 letter to John Tate.]

Formulation

The conjecture concerns the absolute Galois group $G\_mathbb\{Q\}$ of the rational number field $mathbb\{Q\}$.

Let $ho$ be an absolutely irreducible, continuous, and odd [Odd determines the sign of the determinant of ρ applied to complex conjugation "c", as being −1 rather than +1. Here "c" is one (or any) complex conjugation in "G", of order 2; given that "c" is of order 2, ρ can be classified by this sign.] two-dimensional representation of $G\_mathbb\{Q\}$ over a finite field

:$F\; =\; mathbb\{F\}\_\{l^r\}$

of characteristic $l$,

:$ho:\; G\_mathbb\{Q\}\; ightarrow\; GL\_2(F)$.

According to the conjecture, there exists a normalized modular eigenform

:$f\; =\; q+a\_2q^2+a\_3q^3+cdots$

of level $N=N(\; ho)$, weight $k=k(\; ho)$, and some Nebentype character

:$chi\; :\; mathbb\{Z\}/Nmathbb\{Z\}\; ightarrow\; F^*$

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

:$operatorname\{Trace\}(\; ho(operatorname\{Frob\}\_p))=a\_p$

and

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

The level and the weight of $ho$ are explicitly calculated in Serre's article [ [http://modular.ucsd.edu/scans/papers/serre/serre-sur_les_representations_modulaires_deg_degre_2_de_galqbar_over_q.pdf] ] . 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).

Proof

In 2005, Chandrashekhar Khare and Jean-Pierre Wintenberger published a proof of the level 1 Serre conjecture, and later a proof of the full conjecture. [These proofs can be found on [http://www.math.utah.edu/~shekhar/papers.html Khare's web page] .]

Notes

External links

* [http://modular.fas.harvard.edu/papers/serre/ribet-stein.pdf Ribet-Stein lectures (PDF)]
* [http://fora.tv/2007/10/25/Kenneth_Ribet_Serre_s_Modularity_Conjecture Ribet Lecture]