Abelian Variety

Definition

An abelian variety is a complete algebraic variety that is also a group, meaning it has a well-defined addition operation that satisfies the group axioms. Abelian varieties arise naturally in algebraic geometry and are fundamentally important in the study of algebraic curves and number theory due to their rich structure and symmetry properties.

5 Must Know Facts For Your Next Test

1. Abelian varieties can be defined over any field, but they have particularly interesting properties when considered over fields of characteristic zero.
2. They have a rich structure, including a group structure that allows for addition and scalar multiplication, leading to many connections with number theory.
3. Every abelian variety can be expressed as a product of elliptic curves, which are the simplest types of abelian varieties.
4. The notion of a polarization is crucial in understanding the geometry of abelian varieties, allowing us to define notions like divisors and line bundles.
5. Abelian varieties play a central role in modern number theory, particularly in the context of the Langlands program and the proof of Fermat's Last Theorem.

Review Questions

• How does the group structure of an abelian variety enhance its applications in algebraic geometry?
  • The group structure of an abelian variety provides a framework for defining various geometric operations such as addition and scalar multiplication. This structure allows mathematicians to study properties like divisors, line bundles, and cohomology groups, which are essential in understanding the geometry of algebraic curves. Moreover, the symmetry inherent in these varieties makes them important in categorizing solutions to geometric problems and providing insight into their underlying properties.
• Discuss the significance of the Jacobian in relation to algebraic curves and abelian varieties.
  • The Jacobian serves as an abelian variety associated with an algebraic curve, encapsulating crucial information about the curve's geometry. It parametrizes line bundles on the curve, allowing one to transition between geometric properties and algebraic characteristics. The relationship between a curve and its Jacobian is foundational; through tools like the Torelli Theorem, we see how understanding one can illuminate aspects of the other, bridging topics within algebraic geometry.
• Evaluate how the connection between abelian varieties and number theory impacts contemporary mathematical research.
  • The relationship between abelian varieties and number theory is profound, influencing various contemporary mathematical theories such as the Langlands program. This program seeks to connect Galois representations with automorphic forms through abelian varieties, suggesting a deep interplay between number theory and geometry. Additionally, understanding these varieties helps solve classic problems like Fermat's Last Theorem, demonstrating how geometric insights can yield breakthroughs in number theory. Thus, abelian varieties serve as a critical nexus for ongoing research across multiple mathematical disciplines.

Related terms

Jacobian: The Jacobian of a curve is an abelian variety that parametrizes line bundles on the curve and plays a key role in the study of algebraic curves.

Riemann Surfaces: Riemann surfaces are one-dimensional complex manifolds that can be associated with algebraic curves and are often used to study abelian varieties.

Torelli Theorem: The Torelli Theorem states that an algebraic curve is determined by its Jacobian, linking the geometry of curves with the theory of abelian varieties.