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

Abstract

Our aim in this work is to produce equations for curves of genus 2 whose Jacobians have real multiplication (RM) by , and to examine the conjecture that any abelian surface with RM by is isogenous to a simple factor of the Jacobian of a modular curve for some .

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 which parametrizes such curves which have a rational Weierstrass point.

We proceed to give a calculation of the -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 . We examine when a given polarized abelian surface defined over a number field with an action of an order in a real field , also defined over , can be made principally polarized after -isogeny, and prove, in particular, that this is possible when the conductor of 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 . From this description, we are able to generate a fund of equations for these curves, employing a method due to Mestre.