Automorphic representations

Definition

Automorphic representations are a type of mathematical structure arising in the context of number theory and representation theory, where they generalize classical representations of groups over local and global fields. These representations play a crucial role in connecting different areas such as the theory of L-functions, harmonic analysis, and number theory, particularly in understanding solutions to equations and their symmetries.

5 Must Know Facts For Your Next Test

1. Automorphic representations can be thought of as a bridge between number theory and harmonic analysis, as they help understand the distribution of prime numbers through their connections to L-functions.
2. They are classified into different types based on properties like the type of group they represent, which can be algebraic groups over various fields.
3. The Langlands program, a set of conjectures connecting number theory and representation theory, heavily relies on the study of automorphic representations.
4. Automorphic representations can be realized via automorphic forms, which are functions defined on spaces associated with groups that satisfy certain invariance properties.
5. Understanding automorphic representations is essential for proving many deep results in number theory, such as the proof of Fermat's Last Theorem.

Review Questions

• How do automorphic representations relate to L-functions and what significance do they have in number theory?
  • Automorphic representations are deeply interconnected with L-functions, as they provide a way to construct and analyze these functions. L-functions encapsulate information about prime distributions and other number-theoretic phenomena. Automorphic representations help facilitate this connection by allowing for the study of symmetries and invariants within these functions, leading to significant insights into properties of numbers.
• Discuss the role of automorphic forms in defining automorphic representations and their impact on harmonic analysis.
  • Automorphic forms serve as a critical tool in defining automorphic representations by providing the necessary functions that exhibit invariance under group actions. These forms encapsulate complex transformations that relate to symmetry properties in number theory. Their study enhances harmonic analysis by enabling mathematicians to decompose functions into simpler components, ultimately leading to deeper understanding of both representation theory and number theoretic results.
• Evaluate how the Langlands program utilizes automorphic representations to bridge various areas of mathematics and its implications for modern research.
  • The Langlands program is a vast collection of conjectures that connects number theory, representation theory, and geometry through the lens of automorphic representations. By postulating relationships between these fields, it allows mathematicians to leverage results from one area to gain insights into another. This cross-pollination has led to groundbreaking discoveries in modern mathematics, such as advancements in our understanding of prime numbers and the structure of Galois representations. The implications for research are profound, as they guide investigations into unproven conjectures and inspire new mathematical theories.

Related terms

L-functions: Analytic functions that encode number-theoretic information and are used to study properties of numbers, including prime distribution.

Hecke algebras: Algebras that arise in the representation theory of groups, particularly related to automorphic forms and L-functions.

modular forms: Complex functions that are invariant under certain transformations and are used extensively in number theory and automorphic representations.