---
tag: J9T2M084W5
kind: conjecture
name: Langlands program
msc: [11R39, 11F70, "22E55"]
uses: [JQEYEQJY2G, 3C73HND6XA, SEH68BA1QQ, MWBDFS2KJ5, H5145REFMW, BT5V3TSNY6]
about: [JQEYEQJY2G, 3C73HND6XA]
sources:
  - url: https://en.wikipedia.org/wiki/Langlands_program
same_as:
  wikidata: Q1393253
  wikipedia: Langlands program
history:
  posed_by: "Robert Langlands"
  first_stated: "1967"
---

## Statement

The following is the global reciprocity conjecture for $\mathrm{GL}_n$, one central part of the program.
Let $F$ be an [[algebraic number field|MWBDFS2KJ5]], let $G_F$ be its absolute Galois group, let $n \ge 1$, and let $\ell$ be a prime.
Fix a field isomorphism $\iota \colon \overline{\Q}_\ell \to \C$.
Let $\rho \colon G_F \to \mathrm{GL}_n(\overline{\Q}_\ell)$ be an irreducible [[Galois representation|JQEYEQJY2G]].
Assume that $\rho$ is unramified at all but finitely many [[places|BT5V3TSNY6]] of $F$ and de Rham at the places above $\ell$.
Then there is a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb{A}_F)$, where $\mathbb{A}_F$ is the [[adele ring|SEH68BA1QQ]] of $F$, with the following property.
For all but finitely many finite places $v$ of $F$, both $\rho$ and $\pi_v$ are unramified at $v$.
Let $\mathrm{Frob}_v$ be a [[Frobenius element|H5145REFMW]] at $v$.
Then the Satake parameters of $\pi_v$ are, up to the standard normalising twist, the images under $\iota$ of the eigenvalues of $\rho(\mathrm{Frob}_v)$.

## Remarks

The Langlands program is a web of conjectures proposed by Robert Langlands from 1967 on.
It relates Galois representations, [[automorphic forms|3C73HND6XA]] and representations of reductive groups over local and global fields.
Besides reciprocity, its main parts are the principle of functoriality, which predicts transfers of automorphic representations between different reductive groups, and the local Langlands correspondence.
For $n = 1$, reciprocity is a reformulation of [[class field theory|BEKCEYHCHX]] in terms of [[Hecke characters|R65XKHN4D1]].
The [[modularity theorem|CS1NVJBZMH]] for elliptic curves over $\Q$ and [[Serre's modularity conjecture|RWXXNZNQYJ]] are proved cases for $n = 2$ and $F = \Q$.
The local Langlands correspondence for $\mathrm{GL}_n$ over a $p$-adic field was proved by Harris and Taylor and, independently, by Henniart.
In general, both reciprocity and functoriality remain open.

## Explanations

### L1 human

The Langlands program predicts a dictionary between two worlds.
On one side are symmetries of the solutions of polynomial equations, recorded by Galois groups and their representations.
On the other side are highly symmetric functions from analysis, the automorphic forms.
The dictionary says that the counts of solutions modulo primes, on the first side, match the coefficients of functions on the second side.
Proved pieces of this dictionary, such as the modularity of elliptic curves, led to the proof of Fermat's Last Theorem.