Algebra & Number Theory

K3 surfaces and equations for Hilbert modular surfaces

Noam Elkies and Abhinav Kumar

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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 .

Article information

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

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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

elliptic K3 surfaces moduli spaces Hilbert modular surfaces abelian surfaces real multiplication genus-2 curves


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.

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