Shimura Varieties

Definition

Shimura varieties are a class of complex algebraic varieties that generalize the concept of modular curves and arise in the study of arithmetic geometry, particularly in connection with complex multiplication. They are defined as quotients of symmetric spaces by arithmetic groups, playing a crucial role in the Langlands program and connections to number theory, especially in terms of understanding abelian varieties with additional structure.

5 Must Know Facts For Your Next Test

1. Shimura varieties are often equipped with a Hodge structure, which allows for a rich interplay between algebraic geometry and Hodge theory.
2. The study of Shimura varieties has deep implications for number theory, including connections to L-functions and the Birch and Swinnerton-Dyer conjecture.
3. They can be realized as moduli spaces for abelian varieties with additional structures, such as level structures or polarizations.
4. The definition of Shimura varieties is closely tied to the theory of automorphic forms, making them key players in the Langlands program.
5. Their rational points have been the subject of extensive research due to their connections to problems in arithmetic geometry, such as determining when these points correspond to solutions of Diophantine equations.

Review Questions

• How do Shimura varieties relate to modular forms and what role do they play in understanding abelian varieties?
  • Shimura varieties generalize modular forms by serving as moduli spaces for abelian varieties with additional structure. Modular forms can be seen as functions associated with points on these varieties. The deep connections between Shimura varieties and modular forms enhance our understanding of the properties of abelian varieties, particularly through their Hodge structures and rational points, linking complex analysis with algebraic geometry.
• Discuss the significance of the Langlands program in relation to Shimura varieties and their role in number theory.
  • The Langlands program establishes profound connections between number theory and representation theory, with Shimura varieties serving as crucial objects within this framework. These varieties facilitate the study of automorphic forms, allowing mathematicians to relate Galois representations to geometric objects. This interplay is pivotal for understanding L-functions and forms an essential part of ongoing research into deep conjectures in number theory.
• Evaluate the impact of Shimura varieties on modern arithmetic geometry and how they influence current mathematical research.
  • Shimura varieties significantly impact modern arithmetic geometry by providing a rich landscape for exploring fundamental questions about rational points and moduli problems. Their relationships with other mathematical structures, like L-functions and Hodge theory, fuel current research directions that seek to resolve longstanding conjectures. As such, studying Shimura varieties not only deepens our understanding of classical results but also opens new avenues for exploration within the field.

Related terms

Modular Forms: Functions on the upper half-plane that are invariant under the action of a modular group, serving as a fundamental tool in number theory and the study of elliptic curves.

Symmetric Spaces: Spaces that exhibit symmetry properties, often studied in differential geometry, which serve as the foundation for defining Shimura varieties through their quotient by discrete groups.

Langlands Program: A set of conjectures connecting number theory and representation theory, providing deep insights into the relationships between Galois groups and automorphic forms.