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

TL;DR

This paper characterizes rational genus 2 curves with real multiplication by the ring of integers of Q(√5), providing explicit models and criteria for rational points on associated Hilbert modular surfaces.

Contribution

It offers a simple generic description and explicit Weierstrass models for such curves, advancing understanding of their moduli and rational points.

Findings

Identifies which rational moduli points correspond to rational curves

Provides explicit Weierstrass models for these curves

Develops techniques for reducing quadratic forms over polynomial rings

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.