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. This was a significant step in number theory, though this was not realised for at least a decade.^[1]

Contents 1 Formulation 2 Proof 3 Notes 4 External links

[edit] Formulation

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

Let ρ be an absolutely irreducible, continuous, and odd^[2] two-dimensional representation of over a finite field

of characteristic l,

.

According to the conjecture, there exists a normalized modular eigenform

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

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

and

The level and the weight of ρ 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).

[edit] Proof

A proof of the level 1 and small weight cases of the conjecture was obtained during 2004 by Chandrashekhar Khare and Jean-Pierre Wintenberger^[3] and by Luis Dieulefait^[4], independently. In 2005, Chandrashekhar Khare obtained a proof of the level 1 case of Serre conjecture^[5], and in 2006 a proof of the full conjecture in collaboration with Jean-Pierre Wintenberger.^[6]

[edit] Notes

1. ^ 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.
2. ^ 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.
3. ^ http://annals.princeton.edu/annals/2009/169-1/index.xhtml
4. ^ http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.rmi/1204128312
5. ^ http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1156771903
6. ^ This proof can be found on http://www.math.ucla.edu/~shekhar/ [ Khare's web page].

[edit] External links

• Serre's Modularity Conjecture 50 minute lecture by Ken Ribet given on October 25, 2007 ( slides PDF , other version of slides PDF)
• Lectures on Serre's conjectures