Langlands Functoriality Conjecture

TL;DR

This paper surveys the Langlands functoriality conjecture, discussing key methods like the Langlands-Shahidi approach, local and global conjectures, and converse theorems, highlighting progress and results in establishing functoriality.

Contribution

It provides an overview of the techniques and recent results related to the Langlands functoriality conjecture, emphasizing the role of the Langlands-Shahidi method and converse theorems.

Findings

Survey of key techniques for functoriality

Summary of important cases where functoriality is established

Discussion of recent progress and results

Abstract

Functoriality conjecture is one of the central subjects of the present-day mathematics. Functoriality is the profound problem formulated by Robert Langlands in the late 1960s in order to establish nonabelian class field theory. In this expository paper, we describe the Langlands-Shahidi method, the local and global Langlands conjectures and the converse theorems which are powerful tools for the establishment of functoriality of some important cases, and survey the interesting results related to functoriality conjecture.