Abelian varieties

Definition

Abelian varieties are a class of algebraic varieties that are complete, projective, and have a group structure. They generalize elliptic curves and can be viewed as higher-dimensional analogs of them, possessing rich geometric and arithmetic properties. These varieties play a significant role in various areas of mathematics, including number theory and algebraic geometry, making them crucial in the context of specific theorems and conjectures involving modular forms and more.

5 Must Know Facts For Your Next Test

1. Abelian varieties can be defined over any field and have applications in both algebraic geometry and number theory.
2. They are characterized by having a group structure that allows for the addition of points, making them important for studying rational points and their distributions.
3. The dimension of an abelian variety is referred to as its 'genus', which is a generalization of the genus concept found in elliptic curves.
4. Hasse's theorem applies to abelian varieties in relation to the number of rational points over finite fields, linking their arithmetic properties with geometry.
5. The Modularity theorem connects abelian varieties with modular forms, showing that every elliptic curve over the rationals can be associated with a modular form, establishing deep links between these areas.

Review Questions

• How do abelian varieties extend the concept of elliptic curves, and what are some key properties that they share?
  • Abelian varieties extend the concept of elliptic curves by generalizing the notion to higher dimensions. Like elliptic curves, they possess a group structure that allows for point addition, but abelian varieties can exist in multiple dimensions. Key properties shared include being complete and projective algebraic varieties, as well as having well-defined notions of dimension and genus, which help relate them to geometric structures and number theory.
• Discuss the importance of Hasse's theorem concerning abelian varieties and how it relates to their rational points over finite fields.
  • Hasse's theorem is significant for abelian varieties as it provides a way to estimate the number of rational points on these varieties over finite fields. The theorem establishes that the number of such points falls within a specific interval, known as the Hasse interval, which is dependent on the dimension of the variety. This result not only highlights the arithmetic structure of abelian varieties but also shows how their properties can be linked to broader questions in number theory.
• Evaluate the implications of the Modularity theorem on our understanding of abelian varieties and their connection to modular forms.
  • The Modularity theorem has profound implications for understanding abelian varieties, particularly in how it reveals connections between these geometric objects and modular forms. By demonstrating that every elliptic curve over the rationals can be associated with a modular form, it opens pathways for exploring deeper relationships in number theory. This connection enriches our comprehension of both fields and leads to insights about rational points on abelian varieties, contributing to ongoing research into their properties and behaviors in various mathematical contexts.

Related terms

Elliptic Curves: Elliptic curves are a special case of abelian varieties, defined by cubic equations in two variables, and are foundational in number theory and cryptography.

Tate Module: The Tate module is an important construction related to abelian varieties that captures their p-adic properties and is used in the study of their rational points.

Jacobian Variety: A Jacobian variety is a specific type of abelian variety associated with a non-singular projective curve, providing a way to study the curve's algebraic properties.