Generic models for genus 2 curves with real multiplication

TL;DR

This paper develops a new algorithm to construct explicit models of genus 2 curves with real multiplication for 12 new cases, expanding the known examples and simplifying their equations.

Contribution

It introduces an algorithm for minimising conic bundles over projective planes and applies it to find generic models for genus 2 curves with real multiplication.

Findings

Obtained models for 12 new genus 2 curves with real multiplication.

Developed an algorithm for minimising conic bundles over .

Simplified equations for the Mestre conic on Hilbert modular surfaces.

Abstract

Explicit models of families of genus 2 curves with multiplication by $\sqrt D$ are known for $D= 2, 3, 5$. We obtain generic models for genus 2 curves over $\mathbb Q$ with real multiplication in 12 new cases, including all fundamental discriminants $D < 40$. A key step in our proof is to develop an algorithm for minimisation of conic bundles fibred over $\mathbb{P}^2$. We apply this algorithm to simplify the equations for the Mestre conic associated to the generic point on the Hilbert modular surface of fundamental discriminant $D < 100$ computed by Elkies--Kumar.