K3 surfaces and equations for Hilbert modular surfaces

Elkies, Noam; Kumar, Abhinav

doi:10.2140/ant.2014.8.2297

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K3 surfaces and equations for Hilbert modular surfaces

Noam Elkies and Abhinav Kumar

Vol. 8 (2014), No. 10, 2297–2411

DOI: 10.2140/ant.2014.8.2297

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 $ℚ$ .

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

Vol. 8, No. 10, 2014