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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:337773 |
| Date | January 1996 |
| Creators | Galbraith, Steven D. |
| Contributors | Birch, Bryan |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:4b893bc3-f4fe-4877-872a-6a7dd4d5c76d : http://www.math.auckland.ac.nz/~sgal018/thesis.pdf |
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