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

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

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

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

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

• Abelian variety
• Elliptic curve
• Morphism
• Finite morphism
• Cryptography
• Number theory
• Algebraic geometry