Horizontal Isogeny

Definition

A horizontal isogeny is a morphism between elliptic curves that preserves the structure of the curve while mapping points from one curve to another in a way that maintains their respective j-invariants. This type of isogeny reflects the relationship between curves that are defined over the same field, enabling a deeper understanding of their geometric properties and how they interact within the broader context of elliptic curves.

5 Must Know Facts For Your Next Test

1. Horizontal isogenies are particularly significant in the study of endomorphism rings of elliptic curves, as they often correspond to special types of endomorphisms.
2. These isogenies can be constructed using points on the elliptic curves, where the kernel consists of specific points that satisfy certain properties.
3. The degree of a horizontal isogeny indicates how many points on one curve correspond to a point on another curve.
4. Horizontal isogenies are relevant in number theory, particularly in understanding the arithmetic of elliptic curves and their applications to problems like the Birch and Swinnerton-Dyer conjecture.
5. They play a crucial role in defining modular forms and understanding how different elliptic curves can relate to one another through these morphisms.

Review Questions

• How does a horizontal isogeny differ from other types of isogenies, and what implications does this have for the study of elliptic curves?
  • A horizontal isogeny specifically relates to morphisms that preserve j-invariants between elliptic curves defined over the same field, distinguishing it from vertical or other types of isogenies. This preservation means that horizontal isogenies maintain essential properties of the elliptic curves, impacting their classification and relationships in arithmetic geometry. Understanding these differences helps mathematicians analyze how elliptic curves interact within their various structures.
• Discuss the significance of the degree of a horizontal isogeny in relation to its geometric interpretation on elliptic curves.
  • The degree of a horizontal isogeny reflects how many points on one elliptic curve map to a single point on another curve. Geometrically, this degree can be visualized through the idea of counting pre-images under the morphism defined by the isogeny. A higher degree indicates more intricate mappings and relationships between the curves, revealing valuable information about their underlying structure and potential applications in number theory.
• Evaluate the role horizontal isogenies play in linking the concepts of modular forms and elliptic curves within number theory.
  • Horizontal isogenies serve as a bridge between modular forms and elliptic curves by illustrating how these mathematical entities can relate through specific morphisms. The correspondence between isogenies and modular forms allows mathematicians to explore deep connections in number theory, particularly through results like the Taniyama-Shimura-Weil conjecture. By evaluating these relationships, we gain insights into how complex analysis and algebraic geometry interconnect, impacting fields such as cryptography and arithmetic geometry.

Related terms

Elliptic Curve: An elliptic curve is a smooth, projective algebraic curve of genus one, equipped with a specified point, known as the identity element.

Isogeny: An isogeny is a morphism between elliptic curves that has finite degree and preserves the group structure of the curves.

j-invariant: The j-invariant is a value associated with an elliptic curve that classifies curves up to isomorphism over the complex numbers, reflecting their complex structure.