Curves of genus 2 with real multiplication by a square root of 5
Our aim in this work is to produce equations for curves of genus 2 whose Jacobians have real multiplication (RM) by $\mathbb{Q}(\sqrt{5})$, and to examine the conjecture that any abelian surface with RM by $\mathbb{Q}(\sqrt{5})$ is isogenous to a simple factor of the Jacobian of a modular curve $X_0...
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| Format: | Thesis |
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University of Oxford;Mathematical Institute
1998
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| _version_ | 1797112474516848640 |
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| author | Wilson, J |
| author_facet | Wilson, J |
| author_sort | Wilson, J |
| collection | OXFORD |
| description | Our aim in this work is to produce equations for curves of genus 2 whose Jacobians have real multiplication (RM) by $\mathbb{Q}(\sqrt{5})$, and to examine the conjecture that any abelian surface with RM by $\mathbb{Q}(\sqrt{5})$ is isogenous to a simple factor of the Jacobian of a modular curve $X_0(N)$ for some $N$. To this end, we review previous work in this area, and are able to use a criterion due to Humbert in the last century to produce a family of curves of genus 2 with RM by $\mathbb{Q}(\sqrt{5})$ which parametrizes such curves which have a rational Weierstrass point. We proceed to give a calculation of the $\mbox{\ell}$-adic representations arising from abelian surfaces with RM, and use a special case of this to determine a criterion for the field of definition of RM by $\mathbb{Q}(\sqrt{5})$. We examine when a given polarized abelian surface $A$ defined over a number field $k$ with an action of an order $R$ in a real field $F$, also defined over $k$, can be made principally polarized after $k$-isogeny, and prove, in particular, that this is possible when the conductor of $R$ is odd and coprime to the degree of the given polarization. We then give an explicit description of the moduli space of curves of genus 2 with real multiplication by $\mathbb{Q}(\sqrt{5})$. From this description, we are able to generate a fund of equations for these curves, employing a method due to Mestre. |
| first_indexed | 2024-03-07T08:24:46Z |
| format | Thesis |
| id | oxford-uuid:7dca2555-b555-4849-9640-3770ecbe3f94 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T08:24:46Z |
| publishDate | 1998 |
| publisher | University of Oxford;Mathematical Institute |
| record_format | dspace |
| spelling | oxford-uuid:7dca2555-b555-4849-9640-3770ecbe3f942024-02-12T11:32:47ZCurves of genus 2 with real multiplication by a square root of 5Thesishttp://purl.org/coar/resource_type/c_db06uuid:7dca2555-b555-4849-9640-3770ecbe3f94Mathematical Institute - ePrintsUniversity of Oxford;Mathematical Institute1998Wilson, JOur aim in this work is to produce equations for curves of genus 2 whose Jacobians have real multiplication (RM) by $\mathbb{Q}(\sqrt{5})$, and to examine the conjecture that any abelian surface with RM by $\mathbb{Q}(\sqrt{5})$ is isogenous to a simple factor of the Jacobian of a modular curve $X_0(N)$ for some $N$. To this end, we review previous work in this area, and are able to use a criterion due to Humbert in the last century to produce a family of curves of genus 2 with RM by $\mathbb{Q}(\sqrt{5})$ which parametrizes such curves which have a rational Weierstrass point. We proceed to give a calculation of the $\mbox{\ell}$-adic representations arising from abelian surfaces with RM, and use a special case of this to determine a criterion for the field of definition of RM by $\mathbb{Q}(\sqrt{5})$. We examine when a given polarized abelian surface $A$ defined over a number field $k$ with an action of an order $R$ in a real field $F$, also defined over $k$, can be made principally polarized after $k$-isogeny, and prove, in particular, that this is possible when the conductor of $R$ is odd and coprime to the degree of the given polarization. We then give an explicit description of the moduli space of curves of genus 2 with real multiplication by $\mathbb{Q}(\sqrt{5})$. From this description, we are able to generate a fund of equations for these curves, employing a method due to Mestre. |
| spellingShingle | Wilson, J Curves of genus 2 with real multiplication by a square root of 5 |
| title | Curves of genus 2 with real multiplication by a square root of 5 |
| title_full | Curves of genus 2 with real multiplication by a square root of 5 |
| title_fullStr | Curves of genus 2 with real multiplication by a square root of 5 |
| title_full_unstemmed | Curves of genus 2 with real multiplication by a square root of 5 |
| title_short | Curves of genus 2 with real multiplication by a square root of 5 |
| title_sort | curves of genus 2 with real multiplication by a square root of 5 |
| work_keys_str_mv | AT wilsonj curvesofgenus2withrealmultiplicationbyasquarerootof5 |