Langlands Correspondence

A family of conjectural and proven links between representations of Galois groups or Weil groups and automorphic representations of reductive groups, organizing deep relationships between number theory, harmonic analysis, and arithmetic geometry through L-parameters and functoriality principles.

A broad network of conjectures and theorems positing deep relationships between automorphic representations of reductive groups over global or local fields and n-dimensional representations (or parameters) of absolute Galois groups or Weil–Deligne groups; often formulated as matching of L-functions and local-global compatibilities.

A broad network of conjectures and theorems proposing deep correspondences between automorphic representations (analytic objects on reductive groups over global or local fields) and arithmetic/Galois data (such as n-dimensional Galois representations), organized into local and global cases and guided by functoriality principles.