Fonctions hypergéométriques \(F_ 1\) et fonctions automorphes. II: Groupes discontinus arithmétiquement définis
type: Item
instance of: scholarly article
title: Fonctions hypergéométriques \(F_ 1\) et fonctions automorphes. II: Groupes discontinus arithmétiquement définis
zbMATH Open document ID: 0553.32021
author: Toshiaki Terada
published in: Journal of the Mathematical Society of Japan
publication date: 1985-01-01T00:00:00Z
review text: 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 give…
Mathematics Subject Classification ID: 32N15
Mathematics Subject Classification ID: 33C05
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 32N05
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 14H10
zbMATH DE Number: 3881051
zbMATH Keywords: hypergeometric series
zbMATH Keywords: field of automorphic functions on the unit ball
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03720173
OpenAlex ID: W2018698386
DOI: 10.2969/JMSJ/03720173
description: scientific article; zbMATH DE number 3881051