K3 surfaces and equations for Hilbert modular surfaces

TL;DR

This paper develops a method to compute explicit models of Hilbert modular surfaces associated with real quadratic fields, using elliptic K3 surfaces, and provides equations for all fundamental discriminants between 1 and 100.

Contribution

It introduces a novel approach linking K3 surfaces with Hilbert modular surfaces to derive explicit equations for all relevant discriminants.

Findings

Explicit equations for 30 Hilbert modular surfaces with D<100.

Examples of genus-2 curves over Q with real multiplication.

Analysis of rational points and curves on these surfaces.

Abstract

We outline a method to compute rational models for the Hilbert modular surfaces Y_{-}(D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in Q(sqrt{D}), 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 Q whose Jacobians have real multiplication over Q.