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]

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[edit] Formulation

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

Let ρ be an absolutely irreducible, continuous, and odd[2] two-dimensional representation of G_\mathbb{Q} over a finite field

F = \mathbb{F}_{l^r}

of characteristic l,

\rho: G_\mathbb{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[3]. 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

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.[4]

[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. ^ [1]
  4. ^ These proofs can be found on Khare's web page.

[edit] External links