Parameterization of the box variety by theta functions
type: Item
instance of: scholarly article
title: Parameterization of the box variety by theta functions
zbMATH Open document ID: 1417.11072
author: Eberhard Freitag
author: Riccardo Salvati Manni
published in: Michigan Mathematical Journal
publication date: 2017-03-14T00:00:00Z
review text: From the introduction: We consider the graded algebra (the generators have weight one) \(B = \mathbb Q[Z_1, Z_2, Z_3, W_1, W_2, W_3,C]\) with defining relations \[ W_1^2 +W_2^2 = Z_3^2, \quad W_1^2 +W_3^2 = Z_2^2, \quad W_2^2 +W_3^2 = Z_1^2, \quad W_1^2 + W_2^2 + W_3^2 = C^2. \] This is a normal graded algebra. The associated projective variety \(\operatorname{proj}(B)\) is called the box variety. It is absolutely irreducible. We denote its complexification by \(\mathcal B := \operatorname{proj}(B \otimes_{\mathbb Q} C).\) It is a surface that characterizes cuboids. The variables \(W_i\) give the edges of the cuboid, the variables \(Z_i\) the diagonals of the faces and \(C\) the long diagonal. We mention that there is an unsolved problem, raised by Euler, whether the box variety contains non-trivial rational points or not. In this note we describe a parametrization of the box variety (variety of cuboids) by theta functions. This will imply that the box variety is a quotient of the product \(\overline{\mathbb H/\Gamma[8]}\times \overline{\mathbb H/\Gamma[8]}\) of two modular curves of level 8 by a group of order 8 which comes from the diagonal action of \(\Gamma[4]\). Actually this parametrization can be defined over the Gauss number field \(K = \mathbb Q(i)\). We found this parametrization from an observation of D. Testa that the box variety can be embedded into a certain Siegel modular variety which has been described by van Geemen and Nygaard. This parametrization can be used to derive quickly known properties and also some new ones of the box variety. For example, we give in Sect. 2 a modular description of the automorphism group. It can be realized through a subgroup of \(\mathrm{SL}(2, \mathbb Z) \times\mathrm{SL}(2, \mathbb Z)\). In [(*) \textit{M. Stoll} and \textit{D. Testa}, The surface parametrizing cuboids. preprint \url{arxiv:1009.0388}], [(**) \textit{R. van Luijk}, On perfect cuboids. Undergraduate thesis, University Utrecht (2000)] 140 rational and el…
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14K25
zbMATH DE Number: 6694751
zbMATH Keywords: box variety
zbMATH Keywords: theta functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1480734014
OpenAlex ID: W2962780149
DOI: 10.1307/MMJ/1480734014
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description: scientific article; zbMATH DE number 6694751