Open problem, Arithmetic and number theory, Algebra, posed 1967

Langlands program

Partially solved

For any reductive algebraic group $G$ over a global field $F$ , establish a canonical correspondence between automorphic representations of $G(\mathbb{A}_F)$ and Galois representations of $\mathrm{Gal}(\overline{F}/F)$ into the Langlands dual group ${}^L G$ (reciprocity), such that homomorphisms between dual groups induce compatible transfers of automorphic representations (functoriality) preserving $L$ -functions $L(s, \pi, r) = L(s, \rho, r)$ for every representation $r$ .

Research frontier as of 2026

As of 2026, the global Langlands correspondence is proved for $\mathrm{GL}_n$ over function fields (Drinfeld, Lafforgue) and parameterized in the arithmetic direction for general reductive groups $G$ over function fields (Vincent Lafforgue, 2018). In May 2024, Arinkin, Beraldo, Campbell, Chen, Faergeman, Gaitsgory, Lin, Raskin, and Rozenblyum released a five-paper proof of the unramified categorical geometric Langlands conjecture. Over number fields such as $\mathbb{Q}$ , automorphy is known for many classical and Shimura-type Galois representations via Taylor–Wiles patching and Arthur's endoscopic classification, but general reciprocity and functoriality remain wide open.

Best known results

Global Langlands correspondence for $\mathrm{GL}_n$ over function fields (Laurent Lafforgue, 2002) and local Langlands correspondence for $\mathrm{GL}_n$ over $p$ -adic fields (Harris–Taylor, Henniart, 2000).
Complete proof of the unramified categorical geometric Langlands conjecture for reductive groups $G$ over complex algebraic curves (Gaitsgory, Raskin, et al., 2024).
Geometrization of the local Langlands correspondence on the Fargues–Fontaine curve for general reductive groups $G$ over $p$ -adic fields (Fargues–Scholze, 2021).

Tools and where they stop

Tool	Achieved	Where it stops
Arthur–Selberg trace formula and endoscopy	Combined with Ngô Bảo Châu's proof of the Fundamental Lemma, establishes functorial transfer from classical groups (orthogonal, symplectic, unitary) to $\mathrm{GL}_n$ (Arthur, 2013).	General functoriality (Beyond Endoscopy) requires isolating arbitrary $L$ -poles in the trace formula without relying on endoscopic group comparisons.
Taylor–Wiles modularity lifting and Calegari–Geraghty patching	Proves modularity of elliptic curves over $\mathbb{Q}$ and totally real fields, the Sato–Tate conjecture, and potential automorphy for symmetric powers of $\mathrm{GL}_2$ representations.	Requires Galois representations to appear in the cohomology of Shimura varieties or satisfy restrictive regularity and self-duality conditions over $\mathbb{Q}$ .

Open questions

Does every motivic Galois representation over a number field correspond to an automorphic representation of a reductive group $G$ ?
Can Langlands functoriality be proved for all symmetric power transfers $\mathrm{Sym}^n$ and general homomorphisms of Langlands dual groups over number fields?

References

Ngô Bảo Châu (2010). Le lemme fondamental pour les algèbres de Lie · DOI:10.1007/s10240-010-0026-7
Stephen Gelbart (1984). An elementary introduction to the Langlands program · DOI:10.1090/S0273-0979-1984-15237-6
Dennis Gaitsgory, Sam Raskin (2024). Proof of the geometric Langlands conjecture I: construction of the functor · arXiv:2405.03599 [preprint, not peer-reviewed]