Title:

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 $X_0(N)$ 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.
