- 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-dimensionalGalois representation s. This was a significant step innumber 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 andPierre 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 toJohn Tate .]Formulation
The conjecture concerns the
absolute Galois group G_mathbb{Q} of therational number field mathbb{Q}.Let ho be an
absolutely irreducible , continuous, and odd [Odd determines the sign of the determinant of ρ applied tocomplex 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 theModularity Theorem (although this implies Fermat's Last Theorem, Serre proves it directly from his conjecture).Proof
In 2005,
Chandrashekhar Khare andJean-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]
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