---
tag: RWXXNZNQYJ
kind: statement
role: theorem
name: Serre's modularity conjecture
msc: [11F80, 11F11, 11F33]
uses: [JQEYEQJY2G, GWKN5VG78A, A50Y012RBP, S4SV3QZ8B1, H5145REFMW, 0ZXZ82FTX2]
about: [JQEYEQJY2G, GWKN5VG78A]
sources:
  - url: https://en.wikipedia.org/wiki/Serre%27s_modularity_conjecture
same_as:
  wikidata: Q681147
  wikipedia: Serre's modularity conjecture
---

## Statement

Let $p$ be a prime, and let $\overline{\F}_p$ be an algebraic closure of the [[finite field|0ZXZ82FTX2]] $\F_p$.
Let $\bar\rho \colon \Gal(\overline{\Q}/\Q) \to \mathrm{GL}_2(\overline{\F}_p)$ be a continuous [[Galois representation|JQEYEQJY2G]] that is irreducible and odd, that is, $\det \bar\rho(c) = -1$ for a complex conjugation $c$.
Then $\bar\rho$ is modular, in the following sense.
There is a normalised [[cusp form|A50Y012RBP]] $f = \sum_{n \ge 1} a_n q^n$ of some weight $k \ge 2$ and level $N \ge 1$ that is an eigenform of all [[Hecke operators|S4SV3QZ8B1]].
Let $\mathrm{Frob}_\ell$ denote a [[Frobenius element|H5145REFMW]] at a prime $\ell$.
There is also a ring homomorphism $a \mapsto \bar{a}$ from the ring generated by the $a_n$ to $\overline{\F}_p$ such that, for every prime $\ell \nmid pN$,
$$\tr \bar\rho(\mathrm{Frob}_\ell) = \bar{a}_\ell.$$

## Remarks

Serre stated the conjecture in 1987 in a strong form.
In it, $N$ is the Artin conductor of $\bar\rho$ away from $p$, and $k$ is an explicit weight $k(\bar\rho)$ read off from the restriction of $\bar\rho$ to inertia at $p$.
Ribet's level-lowering theorem and later work showed that the weak form above implies the strong form.
Khare and Wintenberger proved the conjecture, with a step by Kisin, in work published in 2009.
Their proof is an induction on the prime $p$ and the conductor, using modularity lifting theorems and the existence of compatible families of Galois representations.
Oddness is necessary: for every modular eigenform the attached representation is odd.
Example: for an elliptic curve $E$ over $\Q$, the action on the $p$-torsion points $E[p]$ gives an odd $\bar\rho$, since $\det \bar\rho$ is the cyclotomic character modulo $p$.
When this $\bar\rho$ is irreducible, its modularity also follows from the [[modularity theorem|CS1NVJBZMH]].

## Explanations

### L1 human

Serre's conjecture says that every two-dimensional Galois representation modulo $p$ of the right kind comes from a modular form.
The "right kind" is just irreducible and odd, two conditions that are easy to check.
So a large class of arithmetic objects is matched with analytic ones.

### L2 human

The strong form gives a route to Fermat's Last Theorem.
A putative solution would give a representation of level $2$ and weight $2$, and there are no nonzero cusp forms of that level and weight.
The theorem is a mod $p$ case of the [[Langlands program|J9T2M084W5]] for $\mathrm{GL}_2$ over $\Q$.
It also implies that the Artin conjecture holds for odd two-dimensional complex Galois representations of $\Q$.