Langlands reciprocity conjecture for $\mathrm{GL}(n)$ over number fields

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LanglandsReciprocity.reciprocity_conjecture

by Lucas · Oct 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

automorphic-formsgalois-representationsl-functionslanglands-programnumber-theory

Throughout, $K$ is a number field, $G_K=\mathrm{Gal}(\bar K/K)$ its absolute Galois group with the Krull topology, and $\mathbb A_K$ its adele ring. An Artin representation is a continuous homomorphism $\rho: G_K\to\mathrm{GL}_n(\mathbb C)$ . We say $\rho$ is automorphic if there are a nonzero bounded cusp form $\varphi$ on $\mathrm{GL}_n(\mathbb A_K)$ with unitary central character and a finite set $S$ of primes of $K$ such that for every prime $\mathfrak p\notin S$ : $\rho$ is unramified at $\mathfrak p$ , $\varphi$ is right invariant under $\mathrm{GL}_n(\mathcal O_{\mathfrak p})$ , and

T_{\mathfrak p,i}\,\varphi \;=\; N\mathfrak p^{\,i(n-i)/2}\,e_i\big(\rho(\mathrm{Frob}_{\mathfrak p})\big)\,\varphi\qquad(1\le i\le n),

where $T_{\mathfrak p,i}$ is the Hecke operator of the double coset of $\mathrm{diag}(\varpi,\dots,\varpi,1,\dots,1)$ ( $i$ entries $\varpi$ ) and $e_i(\rho(\mathrm{Frob}_{\mathfrak p}))$ is the $i$ -th elementary symmetric function of the eigenvalues of an arithmetic Frobenius. Equivalently, the cuspidal automorphic representation generated by $\varphi$ has the same local $L$ -factors as $\rho$ at all $\mathfrak p\notin S$ .

Conjecture. Let $\rho:G_K\to\mathrm{GL}_n(\mathbb C)$ be a continuous irreducible representation. Then $\rho$ is automorphic: there is a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n(\mathbb A_K)$ with

L_{\mathfrak p}(s,\pi)=L_{\mathfrak p}(s,\rho)=\det\big(1-\rho(\mathrm{Frob}_{\mathfrak p})N\mathfrak p^{-s}\big)^{-1}\quad\text{for almost all }\mathfrak p.

This is Langlands' reciprocity conjecture: every Artin $L$ -function of an irreducible representation is the $L$ -function of a cuspidal automorphic representation of $\mathrm{GL}(n)$ . It implies Artin's conjecture that $L(s,\rho)$ is entire for nontrivial irreducible $\rho$ .

Formalization Note Irreducibility is required: for reducible $\rho$ the matching automorphic representation is an isobaric sum, not cuspidal. The conjecture is stated in its standard form, with matching of local factors at almost all primes.

Preamble

import Definitions.Def_LanglandsReciprocity_defs
import Mathlib

Formal statement

namespace LanglandsReciprocity

open NumberField

/-- **Goal (Langlands reciprocity conjecture for `GL(n)`).** Every irreducible
`n`-dimensional Artin representation of a number field `K` is automorphic: some cuspidal
automorphic form on `GL(n, 𝔸_K)` has the same local `L`-factors as `ρ` at almost all primes. -/
theorem reciprocity_conjecture (K : Type*) [Field K] [NumberField K] (n : ℕ)
    (ρ : Gal(AlgebraicClosure K/K) →* GL (Fin n) ℂ) (hρ : Continuous ρ)
    (hirr : IsIrreducible ρ) :
    IsAutomorphic ρ := by sorry

end LanglandsReciprocity

Source

Langlands program (Wikipedia article, uploaded PDF), section Reciprocity and section Status; https://en.wikipedia.org/wiki/Langlands_program; R. P. Langlands, Problems in the theory of automorphic forms, LNM 170 (1970), https://doi.org/10.1007/BFb0079065

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Langlands reciprocity conjecture for GL(n)\mathrm{GL}(n)GL(n) over number fields

Langlands reciprocity conjecture for $\mathrm{GL}(n)$ over number fields