Langlands program and Ramanujan Conjecture: A survey

Langlands Functoriality Conjecture

Functoriality conjecture is one of the central and influential subjects of the present day mathematics. Functoriality is the profound lifting problem formulated by Robert Langlands in the late 1960s in order to establish nonabelian class field theory. In this expository article, I describe the Langlands-Shahidi method, the local and global Langlands conjectures and the converse theorems which a...

Lecture on Langlands Functoriality Conjecture

On the Geometric Langlands Conjecture

0.1. Background. Let X be a smooth, complete, geometrically connected curve over the finite field Fq. Denote by F the field of rational functions on X and by A the ring of adèles of F . The Langlands conjecture, recently proved by L. Lafforgue [Laf], establishes a correspondence between cuspidal automorphic forms on the group GLn(A) and irreducible, almost everywhere unramified, n–dimensional `...

On the Conjecture of Langlands and Rapoport

FORENOTE (2007): The remarkable conjecture of Langlands and Rapoport (1987) gives a purely group-theoretic description of the points on a Shimura variety modulo a prime of good reduction. In an article in the proceedings of the 1991 Motives conference (Milne 1994, §4), I gave a heuristic derivation of the conjecture assuming a sufficiently good theory of motives in mixed characteristic. I wrote...

Hilbert modular forms and the Ramanujan conjecture

Let F be a totally real field. In this paper we study the Ramanujan Conjecture for Hilbert modular forms and the Weight-Monodromy Conjecture for the Shimura varieties attached to quaternion algebras over F . As a consequence, we deduce, at all finite places of the field of definition, the full automorphic description conjectured by Langlands of the zeta functions of these varieties. Concerning ...