Generic models for genus 2 curves with real multiplication

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.