Abelian Varieties

Definition

Abelian varieties are complete algebraic varieties that have a group structure, meaning they allow for addition of points in a way that satisfies the properties of a group. These varieties can be thought of as higher-dimensional generalizations of elliptic curves and play a critical role in the study of algebraic geometry and number theory, especially in relation to complex tori and their connections to Hodge structures.

5 Must Know Facts For Your Next Test

1. Abelian varieties can be defined over any field and have important connections to number theory, especially through their points defined over finite fields.
2. The dimension of an abelian variety corresponds to the number of independent parameters needed to define it, making them essential in higher-dimensional algebraic geometry.
3. The group structure on abelian varieties allows for well-defined operations such as addition and scalar multiplication, which are foundational in the study of their properties.
4. An abelian variety can be embedded into projective space, meaning that they can be studied using tools from both algebraic geometry and complex analysis.
5. The study of abelian varieties leads to significant results in arithmetic geometry, including the proof of major conjectures such as the Mordell-Weil theorem regarding rational points.

Review Questions

• How do abelian varieties generalize the concept of elliptic curves, and what implications does this have for their study?
  • Abelian varieties generalize elliptic curves by allowing for higher dimensions while maintaining a group structure. Just like elliptic curves, abelian varieties can be used to study rational points and modular forms. The transition from elliptic curves to higher-dimensional abelian varieties opens up new avenues in number theory, particularly in understanding the solutions to polynomial equations in several variables.
• Discuss the role of Hodge structures in understanding abelian varieties and their cohomological properties.
  • Hodge structures provide a framework for analyzing the cohomology groups associated with abelian varieties. By using Hodge decomposition, one can extract geometric information from these cohomological groups, leading to insights about their topological and algebraic properties. This connection is crucial in studying both the deformation theory of abelian varieties and their relations with complex tori.
• Evaluate the significance of the Mordell-Weil theorem in the context of abelian varieties and its impact on arithmetic geometry.
  • The Mordell-Weil theorem states that the group of rational points on an abelian variety over a number field is finitely generated. This result is pivotal because it allows mathematicians to classify and understand rational solutions to equations defining these varieties. Its impact on arithmetic geometry is profound as it connects deep aspects of number theory with geometric structures, opening doors to further research into more complicated varieties and their properties.

Related terms

Elliptic Curve: A smooth projective algebraic curve of genus one, equipped with a specified point, serving as the simplest example of an abelian variety.

Complex Tori: A complex manifold that is locally modeled on complex Euclidean space and can be viewed as a quotient of complex space by a lattice.

Hodge Structure: A decomposition of the cohomology of a smooth projective variety that reveals deep relationships between algebraic geometry and topology.