Isogenous group
Isogenous group refers to a concept in the field of algebraic geometry, particularly in the study of abelian varieties. An isogenous relationship between two abelian varieties indicates a certain kind of equivalence, defined by the existence of a morphism with finite kernel between them. This concept is crucial for understanding the structure and classification of abelian varieties, which are higher-dimensional generalizations of elliptic curves.
Definition[edit | edit source]
Two abelian varieties A and B over a field k are said to be isogenous if there exists a non-constant morphism f: A → B that is a finite morphism, meaning that the kernel of f, denoted as ker(f), is a finite group. This relationship is symmetric, meaning if A is isogenous to B, then B is isogenous to A through a dual morphism. The existence of an isogeny implies that A and B have the same dimension as algebraic varieties.
Properties[edit | edit source]
Isogenies share several important properties:
- Transitivity: If A is isogenous to B, and B is isogenous to C, then A is isogenous to C.
- Equivalence Relation: Isogeny is an equivalence relation among abelian varieties, partitioning them into isogeny classes within which every pair of varieties is isogenous.
- Preservation of Rational Points: Over finite fields, isogenies between abelian varieties can be used to study the distribution and number of rational points, which has implications for cryptography and number theory.
Applications[edit | edit source]
Isogenies have applications in various areas of mathematics and computer science:
- In cryptography, isogenies between elliptic curves are used in the construction of isogeny-based cryptographic protocols, which are considered promising for post-quantum cryptography.
- In number theory, isogenies are used to study the L-functions of abelian varieties, contributing to our understanding of the Birch and Swinnerton-Dyer conjecture and other deep problems.
- In algebraic geometry, the concept of isogeny is fundamental in the classification and study of abelian varieties, including the construction of moduli spaces.
Isogeny Classes[edit | edit source]
An isogeny class of abelian varieties over a field k is the set of all abelian varieties over k that are isogenous to each other. The structure of these classes can be quite complex, but they are central to understanding the relationships between different abelian varieties.
See Also[edit | edit source]
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