Serre's modularity conjecture

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

Let $\backslash rho$ be an absolutely irreducible, continuous, two-dimensional representation of $G\_\backslash mathbb\{Q\}$ over a finite field $F\; =\; \backslash mathbb\{F\}\_\{\backslash ell^r\}$.

:$\backslash rho\; \backslash colon\; G\_\backslash mathbb\{Q\}\; \backslash rightarrow\; \backslash mathrm\{GL\}\_2(F).$

Additionally, assume $\backslash rho$ is odd, meaning the image of complex conjugation has determinant -1.

To any normalized modular eigenform

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

of level $N=N(\backslash rho)$, weight $k=k(\backslash rho)$, and some Nebentype character

:$\backslash chi\; \backslash colon\; \backslash mathbb\{Z\}/N\backslash mathbb\{Z\}\; \backslash rightarrow\; F^*$,

a theorem due to Shimura, Deligne, and Serre-Deligne attaches to $f$ a representation

:$\backslash rho\_f\backslash colon\; G\_\backslash mathbb\{Q\}\; \backslash rightarrow\; \backslash mathrm\{GL\}\_2(\backslash mathcal\{O\}),$

where $\backslash mathcal\{O\}$ is the ring of integers in a finite extension of $\backslash mathbb\{Q\}\_\backslash ell$. This representation is characterized by the condition that for all prime numbers $p$, coprime to $N\backslash ell$ we have

:$\backslash operatorname\{Trace\}(\backslash rho\_f(\backslash operatorname\{Frob\}\_p))=a\_p$

and

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

Reducing this representation modulo the maximal ideal of $\backslash mathcal\{O\}$ gives a mod $\backslash ell$ representation $\backslash overline\{\backslash rho\_f\}$ of $G\_\backslash mathbb\{Q\}$.

Serre's conjecture asserts that for any representation $\backslash rho$ as above, there is a modular eigenform $f$ such that

:$\backslash overline\{\backslash rho\_f\}\; \backslash cong\; \backslash rho$.

The level and weight of the conjectural form $f$ are explicitly conjectured 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).

The strong form of Serre's conjecture describes the level and weight of the modular form.

The optimal level is the Artin conductor of the representation, with the power of $l$ removed.

A proof of the level 1 and small weight cases of the conjecture was obtained in 2004 by Chandrashekhar Khare and Jean-Pierre Wintenberger,{{Citation |last=Khare |first=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=On Serre's reciprocity conjecture for 2-dimensional mod p representations of Gal(Q/Q) |journal=Annals of Mathematics |volume=169 |issue=1 |pages=229–253 |doi= 10.4007/annals.2009.169.229|doi-access=free }}. and by Luis Dieulefait,{{Citation |last=Dieulefait |first=Luis |year=2007 |title=The level 1 weight 2 case of Serre's conjecture |journal=Revista Matemática Iberoamericana |volume=23 |issue=3 |pages=1115–1124 |url=https://projecteuclid.org/euclid.rmi/1204128312 |doi=10.4171/rmi/525|arxiv=math/0412099 }}. independently.

In 2005, Chandrashekhar Khare obtained a proof of the level 1 case of Serre conjecture,{{Citation |last=Khare |first=Chandrashekhar |title=Serre's modularity conjecture: The level one case |year=2006 |journal=Duke Mathematical Journal |volume=134 |issue=3 |pages=557–589 |doi=10.1215/S0012-7094-06-13434-8 }}. and in 2008 a proof of the full conjecture in collaboration with Jean-Pierre Wintenberger.{{Citation |last=Khare |first=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre's modularity conjecture (I) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=485–504 |doi=10.1007/s00222-009-0205-7 |bibcode=2009InMat.178..485K |citeseerx=10.1.1.518.4611 }} and {{Citation |last=Khare |first=Chandrashekhar |last2=Wintenberger |first2=Jean-Pierre |year=2009 |title=Serre's modularity conjecture (II) |journal=Inventiones Mathematicae |volume=178 |issue=3 |pages=505–586 |doi=10.1007/s00222-009-0206-6 |bibcode=2009InMat.178..505K |citeseerx=10.1.1.228.8022 }}.

{{Reflist}}

• {{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | department=Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974) |mr=0382173 | year=1975 | journal=Astérisque | issn=0303-1179 | volume=24–25 | title=Valeurs propres des opérateurs de Hecke modulo l | pages=109–117}}
• {{Citation | last1=Serre | first1=Jean-Pierre | author1-link=Jean-Pierre Serre | title=Sur les représentations modulaires de degré 2 de Gal({{overline|Q}}/Q) | doi=10.1215/S0012-7094-87-05413-5 |mr=885783 | year=1987 | journal=Duke Mathematical Journal | issn=0012-7094 | volume=54 | issue=1 | pages=179–230}}
• {{Citation | last1=Stein | first1=William A. | last2=Ribet | first2=Kenneth A. | editor1-last=Conrad | editor1-first=Brian | editor2-last=Rubin | editor2-first=Karl | title=Arithmetic algebraic geometry (Park City, UT, 1999) | publisher=American Mathematical Society | location=Providence, R.I. | series=IAS/Park City Math. Ser. | isbn=978-0-8218-2173-2 |mr=1860042 | year=2001 | volume=9 | chapter=Lectures on Serre's conjectures | pages=143–232}}

• Wiles's proof of Fermat's Last Theorem

• {{usurped|1=[https://web.archive.org/web/20071205022659/http://fora.tv/2007/10/25/Kenneth_Ribet_Serre_s_Modularity_Conjecture Serre's Modularity Conjecture]}} 50 minute lecture by Ken Ribet given on October 25, 2007 ( [https://math.berkeley.edu/~ribet/cms.pdf slides] PDF, [https://web.archive.org/web/20110717033621/http://www.cirm.univ-mrs.fr/videos/2007/exposes/23/Ribet.pdf other version of slides] PDF)
• [https://web.archive.org/web/20011120190728/http://modular.fas.harvard.edu/papers/serre/ribet-stein.pdf Lectures on Serre's conjectures]

Category:Modular forms

Category:Theorems in number theory

Category:Conjectures that have been proved