K3 surfaces and equations for Hilbert modular surfaces

By Noam D. Elkies, Abhinav Kumar. 2012

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9c729b6539524c0fb8a045050009764c5880ed34