Equations for modular curves

Thesis

The primary topic of this thesis is the construction of explicit projective equations for the modular curves $X_0(N)$. The techniques may also be used to obtain equations for $X_0^+(p)$ and, more generally, $X_0(N) / W_n$. The thesis contains a number of tables of results. In particular, equations are given for all curves $X_0(N)$ having genus $2 le g le 5$. Equations are also given for all $X_0^+(p)$ having genus 2 or 3, and for the genus 4 and 5 curves $X_0^+(p)$ when $p le 251$. The most successful tool used to obtain these equations is the canonical embedding, combined with the fact that the differentials on a modular curve correspond to the weight 2 cusp forms. A second method, designed specifically for hyperelliptic curves, is given. A method for obtaining equations using weight 1 theta series is also described.

Heights of modular curves are studied and a discussion is given of the size of coefficients occurring in equations for $X_0(N)$.

Finally, the explicit equations are used to study the rational points on $X_0^+(p)$. Exceptional rational points on $X_0^+(p)$ are exhibited for $p = 73,103,137$ and 191.

Authors: Galbraith, S; Steven Galbraith

• Publication date: 1996
• DOI: 10.5287/ora-goekeeq44
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Modular curves
• Subjects: Number theory; Mathematics