Curves of genus 2 and quadratic forms
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Authors
Kir, Harun
Date
2024-09-05
Type
thesis
Language
eng
Keyword
curves of genus 2 , quadratic forms , refined Humbert invariant , Humbert surface
Alternative Title
Abstract
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.
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ProQuest PhD and Master's Theses International Dissemination Agreement
Intellectual Property Guidelines at Queen's University
Copying and Preserving Your Thesis
This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.