Vol. 8, No. 10, 2014

Download this article
Download this article For screen
For printing
Recent Issues

Volume 10
Issue 9, 1845–2052
Issue 8, 1601–1843
Issue 7, 1373–1600
Issue 6, 1147–1371
Issue 5, 939–1146
Issue 4, 695–938
Issue 3, 451–694
Issue 2, 215–450
Issue 1, 1–214

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
Cover
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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 (D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in (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  whose Jacobians have real multiplication over .

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