An automorphic representation is a concept in mathematics, particularly in the field of number theory and representation theory. It refers to a type of representation of a group that is associated with automorphic forms, which are complex functions that exhibit certain symmetry properties. These representations help in understanding the relationships between different mathematical objects, such as L-functions and modular forms. In essence, automorphic representations allow mathematicians to study the properties of numbers and shapes through a unified framework. They play a crucial role in various areas, including the Langlands program, which seeks to connect number theory with geometry and representation theory. This connection has profound implications for understanding the structure of Galois groups and arithmetic geometry.