Modular Forms and $L$-functions in the Langlands Program

The Langlands program predicts deep connections between automorphic forms and Galois representations, mediated by $L$-functions. We begin with the classical theory of modular forms for $\text{SL}_2(\mathbb{Z})$, define Hecke eigenforms, and construct the associated $L$-function $L(f, s) = \sum_{n=1}^{\infty} a_n n^{-s}$. We then outline how these objects fit into the broader framework of automorphic representations of $\text{GL}_n(\mathbb{A})$, culminating \in a discussion of functoriality and recent progress on the Ramanujan conjecture.