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.

Principle

Reciprocity and functoriality: spectral/analytic data arising from automorphic forms correspond to arithmetic representations of Galois groups, and transfers between groups correspond to transfers of L‑functions and local factors, forming a unifying principle across number theory and harmonic analysis.

Demonstration

Class field theory realizes the Langlands correspondence for GL(1): characters of the idele class group correspond to one-dimensional Galois representations. The modularity theorem (a GL(2) instance) links elliptic curves over Q to modular forms and their automorphic representations.

Misapplication

Treating the general Langlands conjectures as proven in full generality, or misidentifying local components and ramification behavior when attempting explicit matches, leads to incorrect conclusions; also conflating geometric and number-theoretic variants without noting distinctions causes errors.

Consequence

Where established, the correspondence yields powerful translations: automorphic methods produce arithmetic results (e.g., information about Galois representations, L-values, and reciprocity laws), and arithmetic input informs analytic properties like functional equations and spectral decompositions.

Reversal

One can view the correspondence either as assigning arithmetic (Galois) parameters to automorphic representations or as constructing automorphic objects predicted by arithmetic properties; reversing perspective emphasizes either the spectral or arithmetic primacy.

Boundary

The full Langlands program is largely conjectural in many settings; proven cases include global and local correspondences for GL_n over number and function fields and many instances for classical groups, but general functoriality and reciprocity statements remain open in full generality.

Semantic Tension

Tensions arise between the classical (number-theoretic) Langlands correspondence, the geometric Langlands program (algebro-geometric and categorical in nature), and specialized reciprocity laws; the term 'Langlands' thus covers several related but distinct frameworks.

Synthesis

The Langlands correspondence postulates a grand duality translating analytic automorphic data into arithmetic Galois data and vice versa, organized by functoriality and reciprocity: it is a unifying conjectural architecture connecting harmonic analysis, representation theory and number theory.