The ring of modular forms of \(\mathrm{O}(2,4;\mathbb{Z})\) with characters
type: Item
instance of: scholarly article
title: The ring of modular forms of \(\mathrm{O}(2,4;\mathbb{Z})\) with characters
zbMATH Open document ID: 1502.14089
author: Atsuhira Nagano
author: Kazushi Ueda
published in: Hokkaido Mathematical Journal
publication date: 2022-10-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/2005.00231
review text: Consider the lattice \(P_1 := U\oplus E_7^{\oplus 2}\), which is a primitive sublattice of the \(K3\) lattice \(\Lambda_{K3}\). Let \(\mathcal{D}\) be the bounded Hermitian domain of type \(IV\) associated to \(T_1\). The orthogonal complement of \(P_1\) in \(\Lambda_{K3}\) is given by \(T_1 = U^{\oplus 2}\oplus A_1^{\oplus 2}\), which is naturally embeded with index \(2\) into the odd lattice \(T_2 = U^{\oplus 2}\oplus \langle -1\rangle^{\oplus 2}\). In fact, we have \(O(T_2) \simeq O(2,4;\mathbb{Z})\). The aim of the paper under review is to describe the graded ring \(\tilde{A}(O(2,4;\mathbb{Z}))\) of modular forms on \(\mathcal{D}\) with all characters of the group \(O(2,4;\mathbb{Z})\) in terms of a polynomial ring of 8 weighted variables with three relations. In this case, the Cox ring of the quotient stack \(\mathbb{M}\) of the coarse moduli space \(M := \mathcal{D}\slash O(T_1)\) of \(P_1\)-polarized \(K3\) surfaces is isomorphic to \(\tilde{A}(O(2,4;\mathbb{Z}))\). So, it is necessary to describe the Cox ring of the quotient stack \(\mathbb{M}\). In the proof of the main theorem, the authors use the projective model of a \(P_1\)-polarized \(K3\) surface as a Jacobian elliptic surface with singular fibres of type \(II^*\) and \(III^*\).
reviewed by: Makiko Mase
Mathematics Subject Classification ID: 14J28
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 14J15
zbMATH DE Number: 7608268
zbMATH Keywords: \(K3\) surface
zbMATH Keywords: modular form
MaRDI profile type: Publication
OpenAlex ID: W3022098268
arXiv ID: 2005.00231
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DOI: 10.14492/HOKMJ/2020-355
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description: scientific article