Serre conjecture

The title of this article is ambiguous. The Quillen-Suslin theorem is also called the Serre conjecture.

The Serre Conjecture is a mathematical theorem about Galois representations and modular forms , which was proven in 2006 by Chandrashekhar Khare , Jean-Pierre Wintenberger and Mark Kisin . The Serre conjecture implies the modularity theorem and thus also the large theorem of Fermat . The Serre conjecture is based on a conjecture by Jean-Pierre Serre .

Independently of Khare and Wintenberger, Luis Dieulefait also proved special cases of the Serre conjecture in 2004, which are sufficient for the proof of Fermat's large theorem.

formulation

The conjecture concerns representations ( Galois representations ) of the absolute Galois group of rational numbers . The absolute Galois group contains all Galois groups of automorphisms of algebraic number fields, which are given as finite Galois extensions of the field of rational numbers. ${\displaystyle G_{\mathbb {Q} }}$ ${\displaystyle \mathbb {Q} }$

Be an absolutely irreducible, continuous and odd two-dimensional representation of over a finite body ${\displaystyle \rho }$ ${\displaystyle G_{\mathbb {Q} }}$

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

the characteristic , ${\displaystyle l}$

${\displaystyle \rho :G_{\mathbb {Q} }\rightarrow GL_{2}(F).\ }$

According to Serre's conjecture, each such representation becomes the congruence subgroup of the grade , weight , and secondary type through a representation in the space of the tip shapes ${\displaystyle \Gamma _{0}(N)}$ ${\displaystyle N=N(\rho )}$ ${\displaystyle k=k(\rho )}$

${\displaystyle \chi :\mathbb {Z} /N\mathbb {Z} \rightarrow F^{*}\ }$

defined, whereby module forms in characteristics with coefficients of the Fourier expansion in F are considered. The effect of the absolute Galois group in this representation is described by the Hecke operators , linear mappings in the space of the tip shapes of this type. There is a standardized Hecke eigenform , it is a simultaneous eigenfunction of all Hecke operators, with the Fourier expansion ${\displaystyle l}$

${\displaystyle f=q+a_{2}q^{2}+a_{3}q^{3}+\cdots \ }$

Special elements of the absolute Galois group , the Frobenius elements for the prime number p, contain essential information on the arithmetic of the number fields. After the conjecture of Serre further applies to all prime numbers , prime to : ${\displaystyle G_{\mathbb {Q} }}$ ${\displaystyle \operatorname {Frob} _{p}}$ ${\displaystyle p}$ ${\displaystyle N\,l}$

${\displaystyle \operatorname {Trace} (\rho (\operatorname {Frob} _{p}))=a_{p}\ }$

and

${\displaystyle \det(\rho (\operatorname {Frob} _{p}))=p^{k-1}\chi (p).\ }$

This means that the trace and the determinant - and thus essentially the effect of the Frobenius image in the representation under consideration - are determined by the Hecke's eigenform. Serre even suspected (and showed this explicitly with examples) that the parameters of the representation (level, weight, secondary type) can be calculated explicitly (strong Serre conjecture). ${\displaystyle \rho }$

It has been known for a very long time through the deep sentences of Gorō Shimura , Pierre Deligne , Barry Mazur and Robert Langlands that one can assign a representation (as required above) to each Hecke eigenform . The Serre conjecture claims the opposite: every irreducible, continuous and odd representation comes from a modular form. ${\displaystyle f\in S_{k}(N,\chi )}$

literature

Original works of evidence:

• Chandrashekhar Khare: Serre's modularity conjecture: The level one case , Duke Mathematical Journal, Volume 134, 2006, pp. 557-589
• Chandrashekhar Khare, Jean-Pierre Wintenberger: Serre's Modularity Conjecture , Part 1,2, Inventiones Mathematicae, Volume 158, 2009, pp. 485–504, 505–586, Part I (PDF file; 344 kB), Part II (PDF File; 974 kB)
• Khare, Wintenberger: On Serre's reciprocity conjecture for 2-dimensional mod p representations of Gal ( ) ${\displaystyle {\bar {\mathbb {Q} }}/\mathbb {Q} }$ , Annals of Mathematics, Volume 169, 2009, pp. 229-253
• Luis Dieulefait: The level 1 weight 2 case of Serre's conjecture , Revista Matemática Iberoamericana, Volume 23, 2007, pp. 1115-1124.
• Mark Kisin: Modularity of 2-adic Barsotti-Tate representations , Inventiones Mathematicae, Volume 178, 2009, pp. 587-634, Preprints, Kisin

To the Serre conjecture:

• William A. Stein , Ken Ribet : Lectures on Serre's conjecture , in: Brian Conrad, Karl Rubin (Eds.), Arithmetical algebraic geometry (Park City 1999), IAS / Park City Lectures 9, American Mathematical Society, 2001, p. 143 -232, pdf
• Gabor Wiese : The connection between modular forms and number fields , Essener Unikate No. 33, 2007, pdf
• LJP Kilford: Modular forms , Imperial College Press 2008, Chapter 6.2 (Galois representations attached to mod p modular forms), p. 152ff

Individual evidence

1. ↑ Serre, Valeurs propres de opérateurs de Hecke modulo I, Astérisque, Volume 24/25, 1975, pp. 109-117
2. ↑ Serre, Sur les représentations modulaires de degré 2 de , Duke Mathematical Journal, Volume 54, 1987, pp. 179-230 ${\displaystyle Gal({\overline {\mathbb {Q} }}/\mathbb {Q} )}$
3. ↑ Odd means that the element of the Galois group that corresponds to the complex conjugation is represented in the representation by the matrix -1
4. ↑ For the definition of the terms see modular form
5. ↑ See Theorem 3.26 in Haruzo Hida: Modular Forms and Galois cohomology . Cambridge University Press, Cambridge 2000.

Web links

• Serre's Modularity Conjecture 50-minute lecture by Ken Ribet on October 25, 2007 ( slides PDF, other versions of slides PDF)
• Gabor Wiese, The Serre Assumption of Modularity and Computer Algebra, 2010, pdf