Automorphic L-functions and functoriality

TL;DR

This paper discusses the use of the Langlands-Shahidi method combined with converse theorems to establish new cases of Langlands functoriality over number fields, leading to progress on Ramanujan and Selberg conjectures.

Contribution

It demonstrates how the Langlands-Shahidi method and converse theorems can be used to prove new instances of Langlands functoriality and related conjectures.

Findings

Established new cases of Langlands functoriality over number fields.

Derived new estimates towards Ramanujan and Selberg conjectures.

Showcased the effectiveness of the Langlands-Shahidi method in automorphic forms.

Abstract

This is a report on the global aspects of the Langlands-Shahidi method which in conjunction with converse theorems of Cogdell and Piatetski-Shapiro has recently been instrumental in establishing a significant number of new and surprising cases of Langlands Functoriality Conjecture over number fields. They have led to striking new estimates towards Ramanujan and Selberg conjectures.