Langlands reciprocity principle for GL(n)

Abstract

Langlands program is one of the most significant areas of research in modern mathematics as it asserts a connection between number theory, representation the ory and automorphic forms. Langlands program is basically a study to associate the representations of Galois groups over local and global fields with the automorphic rep resentations of algebraic groups over local fields and adeles, respectively. In this thesis, we particularly focus on the Langlands reciprocity principles and related conjectures for the general linear group GLn which are discussed in two parts: local Langlands correspondence and global Langlands reciprocity conjecture for GLn. We divide the local Langlands correspondence for GLn into two cases whether GLn is over a non archimedean or an archimedean local field. The non-archimedean case is proven thanks to Harris and Taylor [1] for the non-archimedean local fields of zero characteristic in 2001 and thanks to Laumon, Stuhler, Rapoport [2] for those of finite characteristic in 1993. The proof of the archimedean case is given thanks to Langlands [3]. Local Langlands correspondence states a well-defined one to one correspondence of the set of equivalence classes of n-dimensional (Frobenius) semisimple complex Weil-Deligne representations of the Weil group WF of F with the set of equivalence classes of irre ducible admissible representations of GLn(F) where F is a local field. The local L and "-factors attached to these certain representations of WF and of GLn(F) are preserved under this correspondence and the preservations are called the naturality properties. In the local part, our objective is to grasp this correspondence at best. For the global part, we respectively discuss the adele ring AK of a global field K, global class field the ory of K, automorphic and cuspidal representations of GLn(AK), global automorphic L and "-factors and hypothetical Langlands group of K in order to state the global Langlands reciprocity conjecture.