K3 surfaces and equations for Hilbert modular 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.

83 pages. Final version

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• Moduli for rational genus 2 curves with real multiplication for discriminant 5
• Explicit $7$-torsion in the Tate-Shafarevich groups of genus $2$ Jacobians