Automorphic representations

Definition

Automorphic representations are sophisticated mathematical constructs that arise in the context of number theory, particularly in the study of forms on algebraic groups over global fields. They generalize the concept of classical modular forms and are key in understanding the connections between number theory and harmonic analysis, especially as they relate to L-functions and Galois representations.

5 Must Know Facts For Your Next Test

1. Automorphic representations can be viewed as generalizations of both modular forms and characters, encapsulating more complex symmetries and properties.
2. They are classified according to their properties under the action of a certain group, often linked to the Galois group in number theory.
3. Automorphic representations can be constructed from either algebraic varieties or from the representation theory of p-adic groups, showing their versatility.
4. The connection between automorphic representations and L-functions is crucial for understanding many deep results in number theory, including the proof of the Taniyama-Shimura-Weil conjecture.
5. These representations play a significant role in modern research areas such as the Langlands program, providing insights into the relationships between different fields of mathematics.

Review Questions

• How do automorphic representations relate to modular forms and why is this connection significant?
  • Automorphic representations generalize modular forms by allowing for a broader class of symmetries and interactions with algebraic groups. This connection is significant because it extends the rich theory of modular forms to higher-dimensional contexts, enabling new insights into both number theory and harmonic analysis. Through these connections, automorphic representations help illuminate deep relationships between different areas of mathematics.
• Discuss the role of automorphic representations in the Langlands program and their implications for number theory.
  • Automorphic representations play a central role in the Langlands program, which seeks to establish profound connections between Galois groups and automorphic forms. The program proposes that there is a correspondence between these representations and Galois representations, leading to deep insights about L-functions. This has implications for many areas in number theory, including the study of Diophantine equations and the understanding of rational points on algebraic varieties.
• Evaluate how the classification of automorphic representations enhances our understanding of L-functions within number theory.
  • The classification of automorphic representations enhances our understanding of L-functions by providing a structured framework through which to analyze their properties. By studying how these representations transform under various groups, mathematicians can derive important results about L-functions, such as their analytic properties and functional equations. This relationship is essential for proving key conjectures in number theory, as it links abstract representation theory with concrete arithmetic properties of numbers.

Related terms

Modular forms: Modular forms are complex functions that are holomorphic and exhibit certain symmetry properties, playing a crucial role in number theory and the theory of automorphic forms.

L-functions: L-functions are complex analytic functions that encode important number-theoretic information, often associated with Dirichlet characters or modular forms.

Langlands program: The Langlands program is a set of conjectures connecting number theory and representation theory, proposing deep links between Galois groups and automorphic forms.