# Langlands reciprocity (general)

- kind: problem
- status: open
- year: 1967
- authors: Robert Langlands

**Statement.** Galois representations and automorphic representations correspond, compatibly with L-functions.

The general correspondence is known for GL(1) (class field theory) and in many cases for GL(2), plus pieces beyond. Each new case has taken years and has usually settled famous problems along the way. The geometric version was proved in 2024, and the arithmetic version remains the central open problem of the field.

Posed 1967, Robert Langlands.

## Related problems

- The original curation links this problem to [Geometric Langlands conjecture](/wiki/Problem:geometric-langlands.md); the argument for the link has not been written yet.
- The original curation links this problem to [Modularity theorem](/wiki/Problem:modularity-theorem.md); the argument for the link has not been written yet.

## Sources

- [Wikipedia: Langlands program](https://en.wikipedia.org/wiki/Langlands_program)

[The Langlands program](/wiki/Space:langlands-program.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations

- related to [geometric-langlands](/wiki/Problem:geometric-langlands.md) — The original curation links this problem to Geometric Langlands conjecture; the argument for the link has not been written yet.
- related to [modularity-theorem](/wiki/Problem:modularity-theorem.md) — The original curation links this problem to Modularity theorem; the argument for the link has not been written yet.

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