K3 surfaces and equations for Hilbert modular surfaces

Abstract

We outline a method to compute rational models for the Hilbert modular surfaces $Y_{-}\left(D\right)$, which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in $\mathbb{Q}\left(\sqrt{D}\right)$, via moduli spaces of elliptic K3 surfaces with a Shioda–Inose structure. In particular, we compute equations for all thirty fundamental discriminants $D$ with $1<D<100$, and analyze rational points and curves on these Hilbert modular surfaces, producing examples of genus-$2$ curves over $\mathbb{Q}$ whose Jacobians have real multiplication over $\mathbb{Q}$.