Uniformizing differential equations of several Hilbert modular orbifolds
type: Item
instance of: scholarly article
title: Uniformizing differential equations of several Hilbert modular orbifolds
zbMATH Open document ID: 0748.14009
DOI: 10.1007/BF01445198
author: Takeshi Sato
published in: Mathematische Annalen
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/164857
review text: Let \(H\times H\) be a product of two upper half planes and \(\Gamma\) be a discrete subgroup of \(\Aut(H\times H)\). For several \(\Gamma\) of Hilbert modular type, it is known that the quotient space \(H\times H/\Gamma\) can be compactified to a two-dimensional complex projective space \(P_ 2(\mathbb{C})\). Take such a \(\Gamma\) and let \(\pi:H\times H\to P_ 2(\mathbb{C})\) be the projection. A system of differential equations of rank four on \(P_ 2(\mathbb{C})\) be the projection. A system of differential equations of rank four on \(P_ 2(\mathbb{C})\) is called a uniformizing differential equation if its projective solution gives a developing map, i.e. for four linearly independent solutions \(z^ 0,\ldots,z^ 3\) the map \((x,y)\in P^ 2(\mathbb{C})\mapsto[z^ 0(x,y):\cdots:z^ 3(x,y)]\in H\times H\subset P_ 1(\mathbb{C})\times P_ 1(\mathbb{C})\cong\mathbb{Q}\subset P_ 3(\mathbb{C})\) gives the inverse map \(\pi^{-1}\) where \(Q\) is a non-degenerate quadric surface in \(P_ 3(\mathbb{C})\). In this paper, uniformizing differential equations for six \(\Gamma\)'s of Hilbert modular type are determined explicitly.
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 32M10
zbMATH DE Number: 56934
zbMATH Keywords: uniformizing differential equation
zbMATH Keywords: orbifold
zbMATH Keywords: Hilbert modular surface
zbMATH Keywords: developing map
MaRDI profile type: Publication
cites work: Überlagerungen der projektiven Ebene und Hilbertsche Modulflächen
cites work: Holomorphic conformal structures and uniformization of complex surfaces
cites work: Linear differential equations in two variables of rank four. I
OpenAlex ID: W2076358355
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description: scientific article; zbMATH DE number 56934