In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands (1967, 1970), it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles.
A grand unified theory of mathematics which includes the search for a generalization of Artin reciprocity (known as Langlands reciprocity) to non-Abelian Galois extensions of number fields. In a January 1967 letter to André Weil, Langlands proposed that the mathematics of algebra (Galois representations) and analysis (automorphic forms) are intimately related, and that congruences over finite fields are related to infinite-dimensional representation theory. In particular, Langlands conjectured that the transformations behind general reciprocity laws could be represented by means of matrices.
The Langlands program gives a very broad picture connecting automorphic forms and L-functions. It roughly states that, among other things, any L-function defined number-theoretically is the same as the one which can be defined as the automorphic L-function of some GL(n). In this loose way, every L-function is (conjecturally) viewed as one and the same object.
