Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.337773 
Title:  Equations for modular curves  
Author:  Galbraith, Steven D. 
ISNI:
0000 0001 3486 6708


Awarding Body:  University of Oxford  
Current Institution:  University of Oxford  
Date of Award:  1996  
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Abstract:  
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.


Supervisor:  Birch, Bryan  Sponsor:  Commonwealth Scholarship  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.337773  DOI:  Not available  
Keywords:  Number theory ; Mathematics ; Modular curves  
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