Homomorphism of Abelian Varieties

Definition

A homomorphism of abelian varieties is a morphism between two abelian varieties that respects the group structure, meaning it preserves the addition operation defined on these varieties. This concept is vital as it allows for the comparison and interaction between different abelian varieties, which include elliptic curves as a special case. These homomorphisms reveal much about the underlying algebraic and geometric properties of the varieties involved.

5 Must Know Facts For Your Next Test

1. Homomorphisms of abelian varieties can be classified into several types based on properties such as injectivity, surjectivity, and whether they are isogenies.
2. The kernel of a homomorphism between two abelian varieties can provide important information about their structure, often leading to insights into the relationship between them.
3. Homomorphisms can induce interesting geometric and arithmetic properties, such as producing new points on one variety from points on another through the homomorphism.
4. In the context of elliptic curves, every non-constant morphism between elliptic curves can be viewed as a homomorphism that preserves their group structure.
5. The study of homomorphisms between abelian varieties plays a crucial role in many areas of algebraic geometry, including the understanding of their endomorphism rings.

Review Questions

• How does a homomorphism of abelian varieties preserve the group structure during its operation?
  • A homomorphism of abelian varieties maintains the group structure by ensuring that the addition operation on points in one variety corresponds to the addition operation on points in another. Specifically, if you take two points in one abelian variety and apply the homomorphism, the image will equal the sum of the images of those two points in the other variety. This preservation is fundamental as it ensures that the mapping respects the underlying algebraic framework.
• Discuss how the concept of kernels in homomorphisms can shed light on the relationship between different abelian varieties.
  • The kernel of a homomorphism between abelian varieties contains points that map to the identity element in the target variety. Analyzing this kernel can reveal valuable information about how two varieties are related. For example, if the kernel is finite, it may indicate an isogeny between the varieties, suggesting they have similar structures. Understanding kernels also helps identify potential morphisms between varieties by highlighting constraints imposed by their group structures.
• Evaluate how homomorphisms play a role in establishing connections between elliptic curves and other types of abelian varieties.
  • Homomorphisms are crucial in linking elliptic curves to broader classes of abelian varieties. Since every elliptic curve can be viewed as an abelian variety, studying homomorphisms provides insights into their interactions with more complex varieties. These mappings can facilitate discussions about properties such as torsion points and rational points, linking back to fundamental concepts in number theory. This interconnection allows mathematicians to apply results from one area to solve problems in another, illustrating the versatility and richness of algebraic geometry.

Related terms

Abelian Variety: An abelian variety is a complete algebraic variety that has a group structure, meaning that it allows for a well-defined addition operation on its points.

Elliptic Curve:

An elliptic curve is a specific type of abelian variety defined over a field, characterized by having a genus of one and containing a specified point known as the identity.

Isogeny: An isogeny is a special type of homomorphism between abelian varieties that is also surjective and has a finite kernel, allowing for a deeper understanding of their structure.