Automorphic forms

Definition

Automorphic forms are complex analytic functions that arise in the study of symmetries of algebraic objects, particularly in the context of number theory and representation theory. They can be thought of as generalizations of modular forms, exhibiting invariance under the action of a discrete group, which plays a crucial role in connecting different areas of mathematics, including L-functions and their properties.

5 Must Know Facts For Your Next Test

1. Automorphic forms generalize the concept of modular forms and can be defined over different types of groups beyond just the modular group.
2. The study of automorphic forms has significant implications in various branches of mathematics, including number theory, representation theory, and arithmetic geometry.
3. Automorphic forms are related to L-functions through the Langlands program, which seeks to connect number theory with representation theory.
4. The space of automorphic forms can be equipped with a rich structure that allows for the application of techniques from harmonic analysis.
5. Automorphic forms play a critical role in understanding special values of L-functions, which can relate to deep conjectures like the Birch and Swinnerton-Dyer conjecture.

Review Questions

• How do automorphic forms relate to modular forms and what implications does this relationship have for number theory?
  • Automorphic forms generalize modular forms by allowing for invariance under actions of broader groups than just the modular group. This relationship is significant for number theory as it opens up connections between different types of functions and symmetries, enabling deeper insights into the distribution of prime numbers and properties of L-functions.
• Discuss the role of automorphic forms in the Langlands program and how this relates to L-functions.
  • Automorphic forms serve as a central component in the Langlands program, which aims to establish deep connections between number theory and representation theory. They are connected to L-functions by providing an avenue to understand special values and functional equations associated with these functions. This interplay allows mathematicians to explore conjectures linking Galois representations to automorphic representations.
• Evaluate the significance of automorphic forms in modern mathematics, particularly in relation to their applications in various mathematical fields.
  • Automorphic forms hold a significant position in modern mathematics due to their versatility and application across multiple fields such as number theory, representation theory, and even mathematical physics. Their rich structures not only provide insight into classical problems but also facilitate advancements in contemporary research areas. The intricate connections between automorphic forms and other mathematical concepts lead to breakthroughs in understanding special values of L-functions and contribute to ongoing research efforts like the Langlands program.

Related terms

Modular Forms: Modular forms are a specific type of automorphic form that are invariant under the action of the modular group, often used in number theory and related fields.

L-functions: L-functions are complex functions that encode important arithmetic information about algebraic objects, such as number fields or automorphic forms, and have properties linked to their analytic continuation and functional equations.

Hecke Algebras: Hecke algebras are commutative algebras associated with the action of Hecke operators on automorphic forms, providing a framework for studying their arithmetic properties.