Monodromy of hypergeometric functions and non-lattice integral monodromy
type: Item
instance of: scholarly article
title: Monodromy of hypergeometric functions and non-lattice integral monodromy
zbMATH Open document ID: 0615.22008
DOI: 10.1007/BF02831622
published in: Publications Mathématiques
publication date: 1986-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104012
review text: The object of study is the monodromy of integrals of the type \[ \int^{\infty}_{1}u^{-\mu_0} (u-1)^{-\mu_1}\prod^{d+1}_{i=2}(u-x_i)^{-\mu_i} \,du \] on the space \(Q=\{(x_i)\mid x_i\neq 0,1,\infty\) and \(x_i\ne x_j\) for \(i\ne j\}\). This generalizes the classical work of Schwarz in the case \(d=1\) and of Picard for \(d=2\). Under a certain integrality condition (INT) on the numbers \((1-\mu_i-\mu_j)^{-1}\), the authors prove that the monodromy group \(\Gamma\) is a lattice in the projective unitary group \(\mathrm{PU}(1,d)\). They also give criteria for \(\Gamma\) to be arithmetic. This paper is very rich and instructive. Aside from some results on algebraic and Lie groups (not indispensable for understanding), this paper is essentially self-contained. The authors redefine cohomologically the integrals above, construct a compactification \(Q_{st}\) of \(Q\) and a completion \(\tilde Q_{st}\) over \(Q_{st}\) of the minimal covering space \(\tilde Q\) of \(Q\) on which those integrals are single-valued. The main point is the study of the mapping properties of a certain function \(\tilde w_{\mu}: \tilde Q_{st}\to B = \) the complex unit ball with a hermitian metric, which identifies the fibers of the projection \(\tilde Q_{st}\to Q_{st}\) with the orbits of a group of isometries of \(B\). The authors list all integrals satisfying condition (INT) and determine in each case whether the corresponding \(\Gamma\) is arithmetic and whether \(\mathrm{PU}(1,d)/\Gamma\) is compact. For \(d>5\) condition (INT) is never satisfied.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 33C60
Mathematics Subject Classification ID: 33C80
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 11F06
zbMATH DE Number: 3996010
zbMATH Keywords: hypergeometric functions
zbMATH Keywords: arithmetic monodromy group
zbMATH Keywords: lattice in projective unitary group
zbMATH Keywords: complex unit ball with hermitian metric
zbMATH Keywords: orbits of group of isometries
zbMATH Keywords: monodromy of integrals
zbMATH Keywords: compactification
zbMATH Keywords: minimal covering space
Wikidata QID: Q94970152
author: Pierre Deligne
author: G. Daniel Mostow
reviewed by: Francesco Baldassarri
MaRDI profile type: Publication
cites work: Density properties for certain subgroups of semi-simple groups without compact components
cites work: Arithmetic subgroups of algebraic groups
cites work: Q5580319
cites work: Q3243614
cites work: Q5339349
cites work: Q5599272
cites work: Existence of nonarithmetic monodromy groups
cites work: Generalized Picard lattices arising from half-integral conditions
cites work: Q5525818
cites work: Q4141415
cites work: Problème de Riemann et fonctions automorphes provenant des fonctions hypergeometriques de plusieurs variables
cites work: Q5648549
cites work: Hodge theory with degenerating coefficients: \(L_2\) cohomology in the Poincaré metric
description: scientific article; zbMATH DE number 3996010