Uniformizing differential equations of several Hilbert modular orbifolds
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.
scientific article; zbMATH DE number 56934
scientific article