Abelian varieties

Definition

Abelian varieties are a class of algebraic varieties that are also group varieties, meaning they have a group structure that is compatible with their algebraic structure. These varieties can be thought of as higher-dimensional generalizations of elliptic curves and play a central role in various areas of mathematics, particularly in number theory and algebraic geometry.

5 Must Know Facts For Your Next Test

1. Abelian varieties can be defined over any field, but their properties and structure are particularly interesting when studied over algebraically closed fields like the complex numbers.
2. They possess a rich geometric structure, including a well-defined notion of divisors and intersection theory, which is essential for the study of their cohomology.
3. The dual of an abelian variety is another abelian variety, and the correspondence between them plays a significant role in algebraic geometry and the theory of motives.
4. Abelian varieties have a connection to Galois representations, which helps bridge the gap between algebraic geometry and number theory.
5. They satisfy the property of being projective and complete, making them essential tools for understanding more complex geometric objects.

Review Questions

• How do abelian varieties generalize elliptic curves and what implications does this have for their study?
  • Abelian varieties generalize elliptic curves by extending the concept to higher dimensions while maintaining group structure. This generalization allows mathematicians to explore more complex interactions within number theory and algebraic geometry. The study of abelian varieties includes examining their points over various fields, their moduli spaces, and connections to other mathematical objects like Galois representations.
• Discuss the significance of the dual abelian variety in relation to an abelian variety's structure and properties.
  • The dual abelian variety provides crucial insights into the geometric structure of an abelian variety. It allows for a better understanding of divisors and cohomology on the original variety. Moreover, the interaction between an abelian variety and its dual can lead to deep results concerning rational points and can be pivotal in advancing research areas like arithmetic geometry.
• Evaluate how abelian varieties contribute to the development of modern number theory, particularly in relation to Galois representations.
  • Abelian varieties significantly enhance modern number theory through their intricate connection to Galois representations. They provide a framework for understanding how arithmetic properties relate to geometric structures. The interplay between these varieties and Galois representations has led to profound insights into rational points on varieties, modular forms, and ultimately contributes to the broader implications such as those seen in the proof of Fermat's Last Theorem.

Related terms

Elliptic Curve: A smooth, projective algebraic curve of genus one with a specified point, often studied for its rich structure and applications in number theory.

Tate Module: A module associated with an abelian variety that encodes information about its points over various fields, especially useful in the study of its properties and connections to Galois representations.

Riemann Surface: A one-dimensional complex manifold that can be used to study complex functions and their properties, serving as a foundational concept in complex analysis and algebraic geometry.