Deligne, P.; Mostow, G. D. Monodromy of hypergeometric functions and non-lattice integral monodromy. (English) Zbl 0615.22008 Publ. Math., Inst. Hautes Étud. Sci. 63, 5-89 (1986). 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. Reviewer: Francesco Baldassarri (Padova) Cited in 22 ReviewsCited in 191 Documents MathOverflow Questions: How do we know there are no more Deligne–Mostow/Thurston lattices? MSC: 22E40 Discrete subgroups of Lie groups 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) 33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) 33C80 Connections of hypergeometric functions with groups and algebras, and related topics 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) 11F06 Structure of modular groups and generalizations; arithmetic groups Keywords:hypergeometric functions; arithmetic monodromy group; lattice in projective unitary group; complex unit ball with hermitian metric; orbits of group of isometries; monodromy of integrals; compactification; minimal covering space × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] Appel, P.,CR Acad. Sci., Paris, 16 février 1880; [2] , Sur les fonctions hygergéométriques de deux variables,J. de Math., 3e ser., VIII (1882), 173–216. [3] Borel, A., Density properties for certain subgroups of semi-simple groups without compact components,Ann. of Math.,72 (1960), 179–188; · Zbl 0094.24901 · doi:10.2307/1970150 [4] , Reduction theory for arithmetic groups,Proc. Symposia in Pure Math., IX (1966), 20–25. [5] Borel, A.-Harish Chandra, Arithmetic subgroups of algebraic groups,Ann. of Math.,75 (1962), 485–535 · Zbl 0107.14804 · doi:10.2307/1970210 [6] Bourbaki, N.,Groupes et Algèbres de Lie, chap. V, Paris, Herman, 1968. · Zbl 0186.33001 [7] Euler, L., ”Specimen transformationis singularis serierum”, Sept. 3, 1778,Nova Acta Petropolitana, XII (1801), 58–78. [8] Fricke, R., andKlein, F.,Vorlesungen über die Theorie der Automorphen Functionen, Bd. I, Leipzig, Teubner, 1897. · JFM 28.0334.01 [9] Fox, R. H., Covering spaces with singularities, inLefschetz Symposium, Princeton Univ. Press (1957), 243–262. · Zbl 0079.16505 [10] Fuchs, L., Zur Theorie der linearen Differential gleichungen mit verändlerichen Coeffizienten,J. r. und angew. Math.,66 (1866), 121–160. · ERAM 066.1719cj · doi:10.1515/crll.1866.66.121 [11] Hermite, C., Sur quelques équations différentielles linéaires,J. r. und angew. Math.,79 (1875), 111–158. [12] Hochschild, G. P.,The Structure of Lie Groups, Holden-Day, San Francisco, 1965. · Zbl 0131.02702 [13] Kneser, M., Strong approximation,Proc. of Symposia in Pure Math., IX (1966), 187–196. · Zbl 0201.37904 [14] Lauricella, Sulle funzioni ipergeometriche a piu variabili,Rend. di Palermo, VII (1893), 111–158. · JFM 25.0756.01 · doi:10.1007/BF03012437 [15] Le Vavasseur, R., Sur le système d’équations aux dérivées partielles simultanées auxquelles satisfait la série hypergéométrique à deux variables,J. Fac. Sci. Toulouse, VII (1896), 1–205. [16] Mostow, G. D., Existence of nonarithmetic monodromy groups,Proc. Nat. Acad. Sci.,78 (1981), 5948–5950; · Zbl 0551.32024 · doi:10.1073/pnas.78.10.5948 [17] Generalized Picard lattices arising from half-integral conditions,Publ. Math. I.H.E.S., this volume, 91–106. · Zbl 0615.22009 [18] Mumford, D.,Geometric Invariant Theory, Berlin, Springer, 1965. · Zbl 0147.39304 [19] Picard, E., Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques,Ann. ENS,10 (1881), 305–322; · JFM 13.0389.01 [20] , Sur les fonctions hyperfuchsiennes provenant des séries hypergéométriques de deux variables,Ann. ENS, III,2 (1885), 357–384; [21] Id.,Bull. Soc. Math. Fr.,15 (1887), 148–152. [22] Pochhammer, L., Ueber hypergeometrische Functionen höherer Ordnung,J. r. und angew. Math.,71 (1870), 316–362. · JFM 02.0265.01 · doi:10.1515/crll.1870.71.316 [23] Riemann, B., Beiträge zur Theorie der durch die Gauss’sche Reihe F({\(\alpha\)}, {\(\beta\)}, {\(\gamma\)},x) darstellbaren Functionen,Abh. Kon. Ges. d. Wiss zu Göttingen, VII (1857), Math. Classe, A-22. [24] Schafli, Ueber die Gauss’sche hypergeometrische Reihe,Math. Ann.,III (1871), 286–295. [25] Schwarz, H. A., Ueber diejenigen Fälle in welchen die Gauss’sche hypergeometrische Reihe eine algebraisches Function ihres vierten Elementes darstellt,J. r. und angew. Math.,75 (1873), 292–335. · JFM 05.0249.01 · doi:10.1515/crll.1873.75.292 [26] Takeuchi, K., Commensurability classes of arithmetic discrete triangle groups,J. Fac. Sci. Univ. Tokyo,24 (1977), 201–212. · Zbl 0365.20055 [27] Terada, T., Problème de Riemann et fonctions automorphes provenant des fonctions hypergéometriques de plusieurs variables,J. Math. Kyoto Univ.,13 (1973), 557–578. · Zbl 0279.32022 [28] Tits, J., Classification of algebraic semi-simple groups,Proc. of Symposia in Pure Math., IX (1966), 33–62. · Zbl 0238.20052 [29] Whittaker, E. T. andWatson, G. N.,A course in modern analysis, Cambridge, University Press, 1962. [30] Zucker, S., Hodge theory with degenerating coefficients, I,Ann. of Math.,109 (1979), 415–476. · Zbl 0446.14002 · doi:10.2307/1971221 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.