K3 surfaces and equations for Hilbert modular surfaces

Noam Elkies, Abhinav Kumar

Published 2012-09-16, updated 2015-01-26Version 3

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.