Moduli for rational genus 2 curves with real multiplication for discriminant 5

TL;DR

The paper provides a precise, practical moduli description for genus 2 curves with RM by $\mathcal{O}_5$, showing generic RM-5 curves over $\mathbb{Q}$ correspond to rational points on the rational surface $Y_-(5)$, parameterized by $(m,n)\in \mathbb{Q}^2$ with $m^2-5n^2-5$ a norm from $\mathbb{Q}(\sqrt{5})$. It develops a concrete reduction strategy for Mestre conics using two birational models of $Y_-(5)$ and explicit reductions over polynomial rings, yielding a simple reduced conic $x_1^2-5x_2^2+(m^2-5n^2-5)x_3^2=0$ and explicit Weierstrass equations in terms of $(m,n,u,v)$. The work also clarifies the relationship to known RM-5 families (Mestre and Brumer), describes the rational moduli set of RM-5 curves, and discusses extensions to other discriminants $D$ where $Y_-(D)$ is rational, including numerical evidence for simple obstruction patterns at $D=17$. Overall, the results provide explicit, implementable criteria and formulas for constructing rational RM-5 genus 2 curves and pave the way for analogous descriptions for additional discriminants.

Abstract

Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rational points on the Hilbert modular surfaces via their Jacobians, but the converse is not true. We give a simple generic description of which rational moduli points correspond to rational curves, as well as give associated Weierstrass models, in the case of RM by the ring of integers of $\mathbb{Q}(\sqrt{5})$. To prove this, we provide some techniques for reducing quadratic forms over polynomial rings.

Theorems & Definitions (18)

• Theorem 1.1
• Proposition 2.1
• proof
• Proposition 2.2: wilson
• Proposition 2.3
• Remark 2.4
• Example 3.1
• Theorem 4.1: mestre
• Theorem 4.2: mestre
• Theorem 5.1