Néron model

Definition

The Néron model is a way to extend an abelian variety defined over a field to a more general setting, allowing for a smooth and proper family of varieties over a base scheme. This concept is crucial for studying the reduction of abelian varieties and understanding how they behave over various fields, especially in the context of arithmetic geometry and algebraic groups.

5 Must Know Facts For Your Next Test

1. The Néron model provides a way to handle abelian varieties over local or global fields by giving them a canonical way to behave when reduced modulo primes.
2. It allows for the examination of rational points on abelian varieties in relation to their Néron models, offering insights into the structure of their group of rational points.
3. When dealing with Néron models, one often considers the concept of 'good reduction', which refers to conditions under which the Néron model retains desirable properties compared to the original abelian variety.
4. The Néron model can be constructed for every abelian variety over a number field, providing a systematic approach to studying their arithmetic properties.
5. In the context of the Mordell-Weil theorem, Néron models play a crucial role by helping to understand the rank and torsion structure of groups formed by rational points on abelian varieties.

Review Questions

• How does the Néron model facilitate the understanding of the reduction behavior of abelian varieties?
  • The Néron model allows mathematicians to study how an abelian variety defined over a field behaves when reduced modulo primes. By extending the variety to a smooth and proper family over a base scheme, one can analyze its structure at different levels of abstraction, including its rational points. This framework helps to understand phenomena like good and bad reduction, making it easier to apply tools from arithmetic geometry.
• Discuss the significance of good reduction in relation to Néron models and their implications on rational points.
  • Good reduction refers to conditions where the Néron model of an abelian variety retains similar geometric and arithmetic properties as the original variety when reduced. This is significant because it ensures that the study of rational points on the original variety can be effectively translated to the study of points on its Néron model. When good reduction occurs, results about torsion points and ranks from the Néron model can provide important insights into the original abelian variety's structure and properties.
• Evaluate how Néron models contribute to proving the Mordell-Weil theorem and what insights this provides into torsion points.
  • Néron models are instrumental in proving the Mordell-Weil theorem, which states that the group of rational points on an abelian variety over a number field is finitely generated. By using these models, mathematicians can study how torsion points behave under reduction and gain insights into the overall structure of the group. The construction of Néron models allows for a clearer understanding of both rank and torsion, showing how these components interact and giving rise to powerful results in arithmetic geometry.

Related terms

Abelian variety:

An abelian variety is a complete algebraic variety that has a group structure, allowing for the definition of addition on its points and serving as a key object of study in both algebraic geometry and number theory.

Reduction modulo p: Reduction modulo p involves taking an algebraic variety defined over a field of characteristic zero and considering its behavior over a field of positive characteristic, often revealing essential properties about the original variety.

Smooth morphism: A smooth morphism is a type of morphism between algebraic varieties that behaves nicely in terms of dimensionality and singularities, ensuring that fibers vary smoothly across the base space.