The Langlands Program is a profound system of theorems, conjectures, and philosophies focused on the relationship between Galois representations and automorphic forms. Although this area is renowned for its esoteric nature, it originated from a very concrete question in basic number theory: solving polynomial equations over the integers. Starting from this point, we will explore the central concepts and statements of the Arithmetic Langlands Program, without assuming prior knowledge of algebraic number theory. Specifically, we will progress from polynomial equations to Frobenius elements, then to Galois representations, modular forms, Hecke operators, and automorphic forms, and finally discuss the relationships between Galois representations and automorphic forms as predicted by the Langlands Program.

Notes from Daebeom Choi.