Group scheme

Definition

A group scheme is a mathematical structure that generalizes the notion of groups to the category of schemes, allowing the concepts of algebraic groups to be studied in a more flexible setting. This concept facilitates the understanding of isogenies and Néron models by providing a framework to analyze group actions and morphisms within the realm of algebraic geometry. Group schemes can be viewed as schemes that also possess the structure of a group, equipped with both multiplication and inverse operations defined in a compatible way.

5 Must Know Facts For Your Next Test

1. Group schemes can be defined over any base scheme, making them versatile in various geometric contexts.
2. They encompass both finite group schemes (which are discrete) and algebraic group schemes (which can have continuous structures).
3. The morphisms between group schemes preserve their group structures, allowing one to study their interactions systematically.
4. Every abelian variety can be viewed as a group scheme, emphasizing the role of group structures in algebraic geometry.
5. Group schemes play a crucial role in defining isogenies between abelian varieties, allowing us to classify them based on their morphisms.

Review Questions

• How do group schemes enhance the understanding of isogenies between abelian varieties?
  • Group schemes provide a structured framework to analyze isogenies since they ensure that both the source and target abelian varieties maintain their group properties. By considering isogenies as morphisms between group schemes, one can utilize the rich geometric structure inherent in these schemes to study properties like kernel and image, which are crucial for understanding the relationships between abelian varieties.
• Discuss the importance of Néron models in relation to group schemes and their applications in arithmetic geometry.
  • Néron models serve as a way to study abelian varieties over different bases by constructing a suitable model that retains essential properties across various local and global fields. In relation to group schemes, Néron models help in understanding how these varieties behave when reduced modulo primes, particularly in terms of their group structure. This connection allows for deeper insights into the arithmetic properties and degeneration of abelian varieties.
• Evaluate the role of group schemes in modern arithmetic geometry, particularly in relation to their impact on modular forms and Galois representations.
  • Group schemes are pivotal in modern arithmetic geometry as they bridge the gap between algebraic geometry and number theory. Their role extends to modular forms where one studies congruences and relationships through isogenies defined by these schemes. Furthermore, the connection between group schemes and Galois representations provides essential insights into how symmetries in algebraic structures relate to field extensions, thereby enriching the theory behind arithmetic geometry.

Related terms

abelian variety: A complete algebraic variety that has a group structure, meaning it is both a geometrical object and a group, allowing for rich interaction between geometry and algebra.

Néron model: An extension of an abelian variety over a local or global base that allows for the study of its reduction and properties over different fields, providing insight into its behavior at different places.

isogeny: A morphism between abelian varieties that preserves the group structure and has finite degree, providing important information about the relationship between different abelian varieties.