Equations for modular curves
<p>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, eq...
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| Format: | Thesis |
| Language: | English |
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1996
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| _version_ | 1826316593909465088 |
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| author | Galbraith, S Steven Galbraith |
| author2 | Birch, B |
| author_facet | Birch, B Galbraith, S Steven Galbraith |
| author_sort | Galbraith, S |
| collection | OXFORD |
| description | <p>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.</p><p>Heights of modular curves are studied and a discussion is given of the size of coefficients occurring in equations for $X_0(N)$.</p><p>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.</p> |
| first_indexed | 2024-03-06T21:51:49Z |
| format | Thesis |
| id | oxford-uuid:4b893bc3-f4fe-4877-872a-6a7dd4d5c76d |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-12-09T03:47:50Z |
| publishDate | 1996 |
| record_format | dspace |
| spelling | oxford-uuid:4b893bc3-f4fe-4877-872a-6a7dd4d5c76d2024-12-08T10:38:08ZEquations for modular curvesThesishttp://purl.org/coar/resource_type/c_db06uuid:4b893bc3-f4fe-4877-872a-6a7dd4d5c76dNumber theoryMathematicsEnglishOxford University Research Archive - Valet1996Galbraith, SSteven GalbraithBirch, B<p>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.</p><p>Heights of modular curves are studied and a discussion is given of the size of coefficients occurring in equations for $X_0(N)$.</p><p>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.</p> |
| spellingShingle | Number theory Mathematics Galbraith, S Steven Galbraith Equations for modular curves |
| title | Equations for modular curves |
| title_full | Equations for modular curves |
| title_fullStr | Equations for modular curves |
| title_full_unstemmed | Equations for modular curves |
| title_short | Equations for modular curves |
| title_sort | equations for modular curves |
| topic | Number theory Mathematics |
| work_keys_str_mv | AT galbraiths equationsformodularcurves AT stevengalbraith equationsformodularcurves |