The Mathematical Institute, University of Oxford, Eprints Archive

Curves of genus 2 with real multiplication by a square root of 5

Wilson, J. (1998) Curves of genus 2 with real multiplication by a square root of 5. PhD thesis, University of Oxford.

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Abstract

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.

Item Type:Thesis (PhD)
Subjects:H - N > Number theory
Research Groups:Number Theory Group
ID Code:32
Deposited By:Eprints Administrator
Deposited On:10 Mar 2004
Last Modified:20 Jul 2009 14:18

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