Vol. 8, No. 10, 2014

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K3 surfaces and equations for Hilbert modular surfaces

Noam Elkies and Abhinav Kumar

Vol. 8 (2014), No. 10, 2297–2411
Abstract

We outline a method to compute rational models for the Hilbert modular surfaces ${Y}_{-}\left(D\right)$, which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in $ℚ\left(\sqrt{D}\right)$, 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, 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 $ℚ$.

Keywords
elliptic K3 surfaces, moduli spaces, Hilbert modular surfaces, abelian surfaces, real multiplication, genus-2 curves
Mathematical Subject Classification 2010
Primary: 11F41
Secondary: 14G35, 14J28, 14J27
Supplementary material

Equations for the text's Hilbert modular surfaces and formulas for the Igusa--Clebsch invariants

Milestones
Received: 22 January 2013
Revised: 26 August 2013
Accepted: 28 October 2013
Published: 31 December 2014
Authors
 Noam Elkies Department of Mathematics Harvard University Cambridge, MA 02138 United States Abhinav Kumar Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 United States