Explicit moduli for curves of genus 2 with real multiplication by ℚ(√5)

1. Motivation. Let J₀(N) denote the Jacobian of the modular curve X₀(N) parametrizing pairs of N-isogenous elliptic curves. The simple factors of J₀(N) have real multiplication, that is to say that the endomorphism ring of a simple factor A contains an order in a totally real number field of degree dim A. We shall sometimes abbreviate "real multiplication" to "RM" and say that A has maximal RM by the totally real field F if A has an action of the full ring of integers of F. We say that a curve C has RM (or maximal RM) by F when the Jacobian Jac(C) does. Let us call an abelian variety modular if it is isogenous to a simple factor of J₀(N) for some N. Save for some technical restrictions, it is now known that all elliptic curves (that is, RM abelian varieties of dimension 1) are modular. It is also conjectured that all RM abelian varieties are modular [14]. In a recent paper, Taylor and Shepherd-Barron [15] have shown that many abelian surfaces with maximal real multiplication by ℚ(√5) are modular. (Again, there are some technical conditions to be met.) It is well known that principally polarized abelian surfaces are either Jacobians, or products. Thus principally polarized abelian surfaces with maximal RM are amenable to a fairly explicit description, if one can determine which curves give these surfaces as Jacobians. Our aim, then, is to attempt to give a description of those curves of genus 2 with maximal RM by ℚ(√5) both in terms of their moduli and by giving equations for the curves. (We also note that it follows from other work of ours [17, Chapter 4] that an abelian surface with RM is almost always isogenous over the ground field to a principally polarized abelian surface with maximal RM.)