Langlands program

A network of conjectures linking representation theory and harmonic analysis to number theory, especially by relating automorphic representations to Galois representations.

Galois representation: A homomorphism from a Galois group into a group of invertible matrices. It encodes arithmetic symmetries on the number-theoretic side of Langlands correspondences.
Local Langlands correspondence: A correspondence between representations of a reductive group over a local field and representations of its Weil–Deligne group. It gives the program’s local form, relating representation theory to local Galois data.
Robert Langlands: A Canadian mathematician who formulated the conjectures now known as the Langlands program. His 1967 letter to André Weil set out the program’s foundational ideas.
Local Langlands correspondence for GL(n): The theorem matching irreducible admissible representations of GL(n) over a local field with n-dimensional Weil–Deligne representations. It is a fundamental proven instance of local Langlands reciprocity.
Langlands functoriality conjecture: The conjecture that homomorphisms of L-groups induce transfers between corresponding automorphic representations. Its general form remains a central unresolved prediction of the program.
Automorphic representation: An irreducible representation arising in spaces of automorphic forms on a reductive group. It is a principal object on the analytic side of the program.
Global Langlands correspondence: A conjectural correspondence between automorphic representations and Galois representations over global fields. It expresses the program’s central global relationship.
Langlands letter: A 1967 letter in which Robert Langlands outlined conjectures connecting automorphic forms and number theory. It introduced the broad reciprocity and functoriality vision behind the program.
Modularity theorem: The theorem that every elliptic curve over the rationals corresponds to a modular form. It realizes a major arithmetic-to-automorphic correspondence predicted by Langlands ideas.
Arthur conjectures: Conjectures describing automorphic representations through parameters and multiplicity formulas for reductive groups. They extend Langlands principles to the structure and classification of automorphic spectra.

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