Algebra & Number Theory
- Algebra Number Theory
- Volume 8, Number 10 (2014), 2297-2411.
K3 surfaces and equations for Hilbert modular surfaces
We outline a method to compute rational models for the Hilbert modular surfaces , which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in , via moduli spaces of elliptic K3 surfaces with a Shioda–Inose structure. In particular, we compute equations for all thirty fundamental discriminants with , and analyze rational points and curves on these Hilbert modular surfaces, producing examples of genus- curves over whose Jacobians have real multiplication over .
Algebra Number Theory, Volume 8, Number 10 (2014), 2297-2411.
Received: 22 January 2013
Revised: 26 August 2013
Accepted: 28 October 2013
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]
Secondary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18] 14J28: $K3$ surfaces and Enriques surfaces 14J27: Elliptic surfaces
Elkies, Noam; Kumar, Abhinav. K3 surfaces and equations for Hilbert modular surfaces. Algebra Number Theory 8 (2014), no. 10, 2297--2411. doi:10.2140/ant.2014.8.2297. https://projecteuclid.org/euclid.ant/1513730328
- Equations for the text's Hilbert modular surfaces and formulas for the Igusa--Clebsch invariants.