Curves of genus 2 and quadratic forms

Our approach focuses on a special associated integral quadratic form qC, which is intrinsically attached to a curve C of genus 2. This form, known as the refined Humbert invariant, was introduced by Kani (1994). Our first main result is the classification of those imprimitive ternary quadratic forms which can occur as one of the forms qC, for some curve C of genus 2. We use this classification to classify the qC’s for curves with a prescribed group of automorphisms, and/or with some additional structure such as having an elliptic subcover of prescribed degree. This study, along with other results, has concrete applications on understanding the nature of intersections of Humbert surfaces. For instance, we prove that the intersection of two Humbert surfaces is not empty, and we determine the intersection of infinitely many prescribed Humbert surfaces in terms of ternary qC’s. To obtain these results, and many others, it is useful to study generalized Humbert sets. A generalized Humbert set H(q), associated with a given quadratic form q, was introduced by Kani in order to understand the nature of curves of genus 2. Regarding the structure of these sets, we find all ternary quadratic forms q for which H(q) is irreducible by relying on SAGE. Additionally, in our joint work with Kani, we give formulas for the cardinality of H(q) for any ternary form q. As an application of this formula, we establish formulas for the number of isomorphism classes of genus 2 curves with certain specified properties, such as curves with a prescribed automorphism group and having an elliptic subcover of a prescribed degree. We examine Shimura curves following the approaches of Lin and Yang, who studied these curves by associating positive binary quadratic forms with them. We clarify the connection between these forms and the refined Humbert invariant. By doing so, we obtain interesting results, such as improving a criterion for the existence of CM points on Shimura curves given by Lin and Yang.

Department dc:contributor.department

Mathematics and Statistics

Year dc:date.issued

2024

Author dc:creator

• Kir, Harun

Advisor dc:contributor.supervisor

• Kani, Ernst

• curves of genus 2
• quadratic forms
• refined Humbert invariant
• Humbert surface

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1974/33353

OAI identifier oai:identifier

oai:queensu.scholaris.ca:1974/33353