isogenioiden

Isogenies are a fundamental concept in the study of elliptic curves and abelian varieties, which are algebraic structures that generalize elliptic curves. An isogeny is a morphism between two elliptic curves or abelian varieties that is also a group homomorphism. This means that it preserves the group structure of the curves or varieties.

Isogenies play a crucial role in the theory of elliptic curves and abelian varieties. They are used to study the arithmetic properties of these objects, such as their torsion points and their endomorphism rings. Isogenies also have applications in cryptography, particularly in the construction of elliptic curve cryptosystems.

One important property of isogenies is that they are surjective. This means that every point in the target curve or variety is the image of some point in the source curve or variety under the isogeny. Another important property is that isogenies are compatible with the group operation. This means that the image of the sum of two points under an isogeny is equal to the sum of the images of the two points under the isogeny.

Isogenies can be classified into two types: separable and inseparable. A separable isogeny is one that is also a separable morphism, meaning that its differential is non-zero. An inseparable isogeny is one that is not separable. In characteristic zero, all isogenies are separable, but in positive characteristic, inseparable isogenies can occur.

Isogenies are also used to study the moduli spaces of elliptic curves and abelian varieties. The moduli space of an elliptic curve or abelian variety is the set of all isomorphism classes of such objects, modulo the action of the automorphism group. Isogenies can be used to study the geometry of these moduli spaces, and to understand the relationships between different points in the moduli space.