K3 surfaces and equations for Hilbert modular surfaces
type: Item
instance of: scholarly article
title: K3 surfaces and equations for Hilbert modular surfaces
zbMATH Open document ID: 1317.11048
author: Abhinav Kumar
author: Noam D. Elkies
published in: Algebra \& Number Theory
publication date: 2015-01-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1209.3527
review text: Let \(D\) be the discriminant of the ring of integers \(\mathcal O_D\) of the real quadratic field \(\mathbb Q(\sqrt D)\). The Hilbert modular surface \(Y_-(D)\) is obtained as compactification of the space parametrizing abelian surfaces with an action of \(\mathcal O_D\). The aim of the present paper is to give explicit equations for rational models of these surfaces for fundamental discriminant \(D\), with \(1<D<100\). The first step in order to obtain equations is to parametrize K3 surfaces which are related to abelian surfaces with real multiplication by \(\mathcal O_D\) via a Shioda-Inose structure. Since these K3 surfaces admit a particular sublattice \(L_D\) of the Néron-Severi lattice, this allows to obtain a moduli space \(\mathcal M_D\) of \(L_D\)-polarized K3 surfaces as a family of elliptic surfaces. Thanks to the previous description and using properties on the special fibers of these elliptic surfaces and their Mordell-Weil group, the authors show \(Y_- (D)\) as double cover of \(\mathbb P^2\) branched over a curve of small degree and thus give equations. Calculations are carried out for each value of \(D\) between 5 (the smallest fundamental discriminant for real multiplication) and 97. The authors suggest that the method can possibly be extended to higher \(D\).
reviewed by: Paola Comparin
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 14J27
zbMATH DE Number: 6387026
zbMATH Keywords: elliptic K3 surfaces
zbMATH Keywords: moduli spaces
zbMATH Keywords: Hilbert modular surfaces
zbMATH Keywords: abelian surfaces
zbMATH Keywords: real multiplication
zbMATH Keywords: genus-2 curves
MaRDI profile type: Publication
OpenAlex ID: W3104344693
arXiv ID: 1209.3527
DOI: 10.2140/ANT.2014.8.2297
description: scientific article