Isogenies

An isogeny is a concept in abstract algebra, particularly in the study of groups and elliptic curves. Broadly defined, an isogeny is a special type of homomorphism between algebraic groups that is surjective and has finite kernel. In the context of abelian varieties, such as elliptic curves, an isogeny is a surjective regular map with finite kernel. This means that the "fibers" of the map consist of finitely many points. The existence of an isogeny between two elliptic curves implies a strong connection between them, suggesting they share many properties.

Isogenies play a crucial role in number theory and cryptography. For instance, they are fundamental to the theory of complex multiplication, which provides a way to construct elliptic curves with specific properties. In cryptography, isogenies between elliptic curves are used in protocols like SIDH (Supersingular Isogeny Diffie-Hellman), which offers potential advantages in terms of key size compared to traditional elliptic curve cryptography. The study of isogenies involves understanding their degree, which is the size of the kernel, and how they behave under composition.