An abelian isogeny is a morphism between two abelian varieties that is surjective, has a finite kernel, and respects the group structure. This concept is significant in the study of the relationships between abelian varieties and provides insights into their structure, leading to deeper results in arithmetic geometry and the study of their endomorphisms.
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Abelian isogenies can be thought of as 'morphisms' between abelian varieties that preserve their group structures while potentially reducing their dimension.
The degree of an abelian isogeny measures how many points in the domain map to a single point in the codomain, giving insight into the relationship between the two varieties.
An important property of abelian isogenies is that they induce a bijection between points of finite order on the two varieties involved.
Isogenies can be composed; if you have two abelian isogenies, their composition is also an isogeny.
In many cases, an abelian variety can be expressed as a quotient of another via an isogeny, which connects different varieties in arithmetic geometry.
Review Questions
How do abelian isogenies reflect the structure and relationships between different abelian varieties?
Abelian isogenies serve as crucial morphisms that connect different abelian varieties while preserving their algebraic and group structures. By being surjective with a finite kernel, they reveal how one variety can map onto another, allowing us to analyze their properties and similarities. This understanding helps us grasp the fundamental relationships in arithmetic geometry and enables a better comprehension of endomorphisms among these varieties.
Discuss how the concept of kernel plays a role in defining an abelian isogeny and its significance in mapping between varieties.
The kernel of an abelian isogeny consists of points that are sent to the identity element in the codomain, which is crucial for ensuring that the morphism behaves like a group homomorphism. This finite kernel indicates how many elements in the domain correspond to each point in the target variety, making it essential for analyzing both the nature of the mapping and its implications for understanding group structures within abelian varieties. The size of this kernel can also provide insight into the geometric properties shared by these varieties.
Evaluate how understanding abelian isogenies enhances our knowledge of endomorphisms in arithmetic geometry.
Understanding abelian isogenies significantly enhances our knowledge of endomorphisms by demonstrating how these mappings preserve structural properties across different varieties. Isogenies allow for the exploration of various relationships between endomorphisms, illustrating how they can compose to form new morphisms. Furthermore, studying these connections leads to insights about automorphisms and symmetries present within abelian varieties, which are essential for advancing theories in arithmetic geometry.
Related terms
abelian variety: A complete algebraic variety that has a group structure, meaning it can be added together and has a zero element.
finite morphism: A morphism between varieties that is affine and satisfies the condition of having finite fibers over each point in the target variety.
kernel: In the context of group homomorphisms, the kernel is the set of elements that map to the identity element in the target group.