# Langlands Program and Ramanujan Conjecture: a survey

## TL;DR

This survey introduces the Langlands Program and its connection to the Ramanujan Conjecture, covering both global and local aspects in characteristic zero and p, highlighting modern approaches and known cases.

## Contribution

It provides an accessible overview of the Langlands Program and Ramanujan Conjecture, emphasizing recent methods and known results to guide new researchers.

## Key findings

- Overview of automorphic L-functions via Langlands-Shahidi method
- Summary of known cases over function fields
- Discussion of open problems in number fields

## Abstract

We present topics in the Langlands Program to graduate students and a wider mathematically mature audience. We study both global and local aspects in characteristic zero as well as characteristic $p$. We look at modern approaches to the generalized Ramanujan conjecture, which is an open problem over number fields, and present known cases over function fields. In particular, we study automorphic $L$-functions via the Langlands-Shahidi method. Hopefully, our approach to the Langlands Program can help guide the interested reader into an exciting field of mathematical research.