Algebra & Number Theory

K3 surfaces and equations for Hilbert modular surfaces

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 $ℚ(D)$, 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, 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 $ℚ$.

Article information

Source
Algebra Number Theory, Volume 8, Number 10 (2014), 2297-2411.

Dates
Revised: 26 August 2013
Accepted: 28 October 2013
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730328

Digital Object Identifier
doi:10.2140/ant.2014.8.2297

Mathematical Reviews number (MathSciNet)
MR3298543

Zentralblatt MATH identifier
1317.11048

Citation

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

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