Serre's modularity conjecture

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

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

${\displaystyle \rho \colon G_{\mathbb {Q} }\rightarrow \mathrm {GL} _{2}(F).}$

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

To any normalized modular eigenform

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

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

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

a theorem due to Shimura, Deligne, and Serre-Deligne attaches to ${\displaystyle f}$ a representation

${\displaystyle \rho _{f}\colon G_{\mathbb {Q} }\rightarrow \mathrm {GL} _{2}({\mathcal {O}}),}$

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

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

and

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

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

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

${\displaystyle {\overline {\rho _{f}}}\cong \rho }$.

The level and weight of the conjectural form ${\displaystyle f}$ 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 ${\displaystyle l}$ 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,[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 2008 a proof of the full conjecture in collaboration with Jean-Pierre Wintenberger.[6]

• Serre, Jean-Pierre (1975), "Valeurs propres des opérateurs de Hecke modulo l", Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974), Astérisque, 24–25: 109–117, ISSN 0303-1179, MR 0382173
• Serre, Jean-Pierre (1987), "Sur les représentations modulaires de degré 2 de Gal(Q/Q)", Duke Mathematical Journal, 54 (1): 179–230, doi:10.1215/S0012-7094-87-05413-5, ISSN 0012-7094, MR 0885783
• Stein, William A.; Ribet, Kenneth A. (2001), "Lectures on Serre's conjectures", in Conrad, Brian; Rubin, Karl (eds.), Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., 9, Providence, R.I.: American Mathematical Society, pp. 143–232, ISBN 978-0-8218-2173-2, MR 1860042

• 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