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arXiv · 1209.3527

K3 surfaces and equations for Hilbert modular surfaces

Noam Elkies·Abhinav Kumar
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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.

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Noam Elkies, Abhinav Kumar. 2015-01-26. K3 surfaces and equations for Hilbert modular surfaces. https://doi.org/10.2140/ant.2014.8.2297

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