Fonctions hypergéométriques \(F_ 1\) et fonctions automorphes. II: Groupes discontinus arithmétiquement définis
We consider the system \((F_ 1)\) of differential equations \((x_ i-x_ j)\partial_ i\partial_ jF+(\lambda_ j-1)\partial_ iF-(\lambda_ i-1)\partial_ jF=0\quad (i\neq j);\) \(x_ i(x_ i-1)\partial^ 2_ iF+[x_ i(x_ i-1)\sum (1-\lambda_{\alpha})/(x_ i- x_{\alpha})+\lambda_ 0+\lambda_ i-2+(1-\lambda_ 0-2\lambda_ i- \lambda_{n+1})x_ i]\partial_ iF+(\lambda_ i-1)\sum x_{\alpha}(x_{\alpha}-1)\partial_{\alpha}F/(x_ i-x_{\alpha})- \lambda_{\infty}(1-\lambda_ i)F=0\quad (1\leq i\leq n)\) on the domain \({\mathcal D}=\{(x_ 1,x_ 2,...,x_ n)| x_ i\neq 0,1,x_ j\quad (1\leq i<j\leq n)\},\) where \(\partial_ i=\partial /\partial x_ i\) and \(\sum =\sum_{i\leq \alpha \leq n,\alpha \neq i}\). Gauss' hypergeometric series \((n=1)\), Appell's \(F_ 1\) \((n=2)\) or Lauricella's \(F_ D\) is a solution. And it is known [see part I of this paper, ibid. 35, 451-475 (1983; Zbl 0506.33001)] that, under some conditions on the parameters \(\lambda_ i\) (there exist infinite, 27, 7, 1, 1 and respectively 0 cases for \(n=1,2,3,4,5\) and respectively \(>5)\), \((F_ 1)\) gives a field of automorphic functions on the unit ball. The group G is explicitly given by the monodromy of \((F_ 1)\) as a linear representation of the fundamental group of \({\mathcal D}\). The fundamental domain has a compactification birational to \({\mathbb{P}}_ n({\mathbb{C}})\), and is, roughly speaking, a space of moduli of the family of algebraic curves: \(v=\prod^{n+1}_{i=0}(u-x_ i)^{\lambda_ i}\) \((x_ 0=0\), \(x_{n+1}=1)\) (in actual cases \(\lambda_ i\) are rational). The author shows in this paper that, in above discontinuous groups, there exist ten which are defined arithmetically (in the following sense). For each group, there exists a finite extension G', a natural number m, a primitive m-th root \(\zeta\) of 1 and a Hermitian matrix \(A\in GL(n+1,K)\) \((K={\mathbb{Z}}[\zeta])\) with signature (1,n) such that G' is the projective linear group generated by \(\{g\in GL(n+1,K)| g^*Ag=A\}.\) G' and A are explicitly given. And six of the groups G' coincide with those which \textit{G. Shimura} [Osaka J. Math. 1, 1-14 (1964; Zbl 0149.043)] presented as examples of his theory of automorphic functions.
field of automorphic functions on the unit ball
scientific article; zbMATH DE number 3881051