Es wird eine alternative Normalform für Elliptische Kurven gegeben und deren Modulraum studiert. Weiterhin wird die Konfiguration der Wendepunkte eingehend studiert. Für weitere Details verweisen wir auf die Einleitung.
Inhaltsverzeichnis (Table of Contents)
- Einleitung
- Introduction
- Hesse pencil and Hesse configuration
- A geometrical approach
- The Hesse configuration
- The Hesse group
- An algebraic approach.
- The Weierstrass normal form and the j-invariant.
- The Cayleyan curve
- Polar hypersurfaces
- Polar quadrics and the Hessian
- The dual Hesse pencil and the Cayleyan of a plane cubic
- References
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
The bachelor thesis focuses on exploring the Hesse configuration, a geometric structure formed by the inflection points of an elliptic curve. The work aims to provide a detailed understanding of this configuration through two distinct approaches: a geometric approach using the Hesse pencil and an algebraic approach employing the group structure of elliptic curves.
- The Hesse configuration of nine points and twelve lines, its uniqueness and its realizations in different planes.
- The Hesse pencil, a linear system of cubic curves in the projective plane, and its role in defining the Hesse configuration.
- The group structure of elliptic curves and its connection to the Hesse configuration.
- The Cayleyan curve, a dual object to the Hesse configuration, and its construction through the theory of polar hypersurfaces.
- The Hessian group and its action on the Hesse pencil.
Zusammenfassung der Kapitel (Chapter Summaries)
The first part of the thesis delves into the Hesse configuration, presenting a geometric approach through the Hesse pencil. It examines the properties of the Hesse pencil and its relationship to the configuration, including the determination of a normal form for elliptic curves and the computation of the j-invariant. The second part explores the Hesse configuration from an algebraic perspective, leveraging the group structure of elliptic curves to understand the configuration's properties. This section also introduces the theory of polar hypersurfaces and its application to defining the Cayleyan curve, a dual object to the Hesse configuration.
Schlüsselwörter (Keywords)
The key terms and concepts explored in this thesis include: Hesse configuration, elliptic curves, inflection points, Hesse pencil, Cayleyan curve, polar hypersurfaces, Hessian group, j-invariant, group structure, projective geometry, and algebraic geometry.
Frequently Asked Questions
What is the Hesse configuration?
The Hesse configuration is a geometric structure consisting of nine inflection points and twelve lines, formed by the intersection of cubic curves in a projective plane.
What is a Hesse pencil?
A Hesse pencil is a linear system of cubic curves in the projective plane that is used to define and study the properties of the Hesse configuration.
How are elliptic curves related to the Hesse configuration?
The nine points of the Hesse configuration are exactly the inflection points of an elliptic curve, and their arrangement is linked to the curve's group structure.
What is the Cayleyan curve?
The Cayleyan curve is a dual object to the Hesse configuration, constructed through the theory of polar hypersurfaces and Hessian quadrics.
What is the j-invariant in this context?
The j-invariant is an algebraic value used to characterize elliptic curves and is computed as part of the normal form analysis within the Hesse pencil.
- Quote paper
- Jakob Bongartz (Author), 2012, The Hesse pencil and its Cayleyan, Munich, GRIN Verlag, https://www.grin.com/document/335107
