Linear differential equations in two variables of rank four. II. The uniformizing equation of a Hilbert modular orbifold
[For Part I see the preceding review.] Let \(X=H\times H\) be the product of two upper half planes \(H=\{\tau \in {\mathbb{C}}|\) Im \(\tau>0\}\), O be the ring of integers in the real quadratic field \({\mathbb{Q}}(\sqrt{2})\) and let \(SL(2,0)\) be the special linear group over O, naturally identified with a subgroup of the group Aut(X) of holomorphic automorphisms of X by \((\tau ^ 1,\tau ^ 2)\to (g_ 1\tau ^ 1,g_ 2\tau ^ 2)\), where \(g_ 1\) and \(g_ 2\) are two embeddings of SL(2,0) into SL(2,\({\mathbb{R}})\). F. Hirzebruch proved that there is a subgroup \(\Gamma\) of Aut(X), which is commensurable with \(SL(2,0)\), such that the regular orbit of X under \(\Gamma\) is isomorphic to \[ M={\mathbb{C}}P^ 2-\{(1-x^ 2)(1-y^ 2)(1-x^ 2y^ 2)(1-x^ 2- y^ 2)=0\}, \] where (x,y) is a system of affine coordinates on the 2- dimensional projective space \({\mathbb{C}}P^ 2\). Let \(\pi\) : \(X\to M\) be the projection. Since the conformal class of the quadratic form \(d\tau ^ 1d\tau ^ 2\) on X is Aut(X)-invariant, the image under \(\pi\) defines a conformal class of M, which is known, by R. Kobayashi and I. Naruki, to be represented by \[ \ell dx^ 2+2 dx dy+m dy^ 2,\quad \ell =-(2-y^ 2-x^ 2y^ 2/xy(1-x^ 2)),\quad m=-(2-x^ 2-x^ 2y^ 2/xy(1-y^ 2)). \] Applying results obtained in Part I, the authors construct a system of differential equations \[ (1)\quad z_{xx}=\ell z_{xy}+az_ x+bz_ y+pz;\quad z_{yy}=mz_{xy}+cz_ x+dz_ y+qz \] defined on M of rank four such that four linearly independent solutions \(z^ j\) (1\(\leq j\leq 4)\) defines the inverse (the developing map) of \(\pi\), i.e. regarding \(H\times H\) as a part of a quadratic surface in \({\mathbb{C}}P^ 3\) by \((\tau ^ 1,\tau ^ 2)\to (1,\tau ^ 1,\tau ^ 2,\tau ^ 1\tau ^ 2)\), the developing map is given by \((x,y)\to (z^ j(x,y))\in X\subset {\mathbb{C}}P^ 3\). Once a normalization factor \(\theta\) is fixed, the coefficients a, b, c and d can be calculated by the formulae in part I. The main contribution of this paper is to obtain the coefficients p and q explicitly. The result is \[ P=2(y^ 2-x^ 2)/((1-x^ 2)^ 2(1- y^ 2)),\quad q=2(x^ 2-y^ 2)/((1-y^ 2)^ 2(1-x^ 2)), \] where the function \(\theta\) is chosen to be \(e^{2\theta}=(1-\ell m)^{- 7/2}(xy)^{-6}\).
scientific article; zbMATH DE number 4019561