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Book Review
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Book Information
Author(s):
Pierre Deligne and G. Daniel Mostow
Title:
Commensurabilities among lattices in $\roman {PU}(1,n)$
Additional book information:
Annals of Mathematics Studies, no. 132, Princeton University Press, Princeton, NJ, 1993, 183 pp., US$19.95. ISBN 0-691-00096-4
References:
- [A]
- P. Appell, Sur les fonctions hyperg\'eom\'etriques de deux variables, J. Math. Pures Appl. (3) \textbf{8} (1882), 173--216.
- [AKdF]
- P. Appell and M.J. Kamp\'e de F\'eriet, Fonctions hyperg\'eom\'etriques et hypersph\'eriques, Polyn\^omes d'Hermite, Gauthier-Villars, Paris, 1926.
- [AS]
- M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, Dover, New York, 1970.
- [BHH]
- G. Barthel, F. Hirzebruch, and T. H\"ofer, Geradenkonfigurationen und Algebraische Fl\"achen, Aspects of Math. {\bf D4}, Vieweg, Braunschweig, 1987.
- [CHY]
- S.Y. Cheng and S.-T. Yau, Inequality between Chern numbers of singular K\"ahler surfaces and characterisation of orbit space of discrete group of $\roman {SU}(2,1)$, Contemp. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1986, pp. 31-44.
- [CoWo1]
- P. Cohen and J. Wolfart, Modular embeddings for some nonarithmetic Fuchsian groups, Acta Arith. \textbf{LVI} (1990), 93--110.
- [CoWo2]
- P. Beazley Cohen and J. Wolfart, Algebraic Appell-Lauricella functions, Analysis \textbf{12} (1992), 359--376.
- [CoWo3]
- P. Beazley Cohen and J. Wolfart, Fonctions hyperg\'eom\'etriques en plusieurs variables et espaces des modules de vari\'et\'es ab\'eliennes, Ann. Sci. \'Ecole Norm. Sup. (4) \textbf{26} (1993), 665--690.
- [CoItzWo]
- P. Beazley Cohen, C. Itzykson, and J. Wolfart, Fuchsian triangle groups and Grothendieck dessins---Variations on a theme of Belyi, Commun. Math. Phys. {\bf 63} (1994), 605--627.
- [DM]
- P. Deligne and G.D. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, Inst. Hautes \'Etudes Sci. Publ. Math. \textbf{63} (1986), 5--90.
- [E]
- L. Euler, Specimen transformationis singularis serierum, Nova Acta Acad. Petropolitan \textbf{12} (1794 (1801)), 58--70 (Leonhardi Euleri Opera Omnia ser.I, vol.16*, pp. 41--55), Teubner, Leipzig, 1935.
- [F]
- L. Fuchs, Zur Theorie der linearen Differentialgleichungen mit ver\"anderlichen Coeffizienten, J. Reine Angew. Math. \textbf{66} (1866), 121--160.
- [Ga]
- C.F. Gauss, Disquisitiones generales circa seriem infinitam $1+{\alpha .\beta \over \gamma .1}x+\dotsb $, Werke, Band III, Olms Verlag, Hildesheim and New York, 1981, pp. 123--162.
- [G]
- I.M. Gelfand et al., Collected papers, Volume {\rm III}, Part {\rm V}, Springer-Verlag, New York, 1989.
- [Go]
- E. Goursat, Extension du probl\`eme de Riemann \`a des fonctions hyperg\'eom\'etriques de deux variables, C. R. Acad. Sci. \textbf{95} (1882), 903,1044.
- [H]
- C. Hermite, Sur quelques \'equations diff\'erentielles lin\'eaires, J. Reine Angew. Math. \textbf{79} (1875), 111--158.
- [Hi]
- F. Hirzebruch, Chern numbers of algebraic surfaces---an example, Math. Ann. \textbf{266} (1984), 351--356.
- [I]
- M.-N. Ishida, Hirzebruch{\rm '}s examples of surfaces of general type with $c_1^2=3c_2$, Algebraic Geometry, Proc. Tokyo Kyoto 1982 Lecture Notes in Math., vol. 1016, Springer-Verlag, New York, 1983, pp. 412--431.
- [J]
- C.G.J. Jacobi, Untersuchungen \"uber die Differentialgleichung der hypergeometrischen Reihe, J. Reine Angew. Math. \textbf{56} (1859), 149--165.
- [Kl1]
- F. Klein, Weitere Untersuchungen \"uber das Ikosaeder, Math. Ann. \textbf{12} (1877), 503--560.
- [Kl2]
- F. Klein, Ueber Normirung der linearen Differentialgleichungen zweiter Ordnung, Math. Ann. \textbf{38} (1891), 144--152.
- [Kn]
- A.W. Knapp, Doubly generated Fuchsian groups, Michigan Math. J. \textbf{15} (1986), 289--304.
- [KNS]
- R. Kobayashi, S. Nakamura, and F. Sakai, A numerical characterisation of ball quotients for normal surfaces with branch loci, Proc. Japan Acad. Ser. A \textbf{65} (1989), 238--241.
- [K]
- E.E. Kummer, Ueber die hypergeometrische Reihe $1+{\alpha .\beta \over \gamma .1}x+\dotsb $, J. Reine Angew. Math. \textbf{15} (1836), 39--83 and 127--172.
- [L]
- G. Lauricella, Sulle funzioni ipergeometriche a piu variabli, Rend. Circ. Mat. Palermo \textbf{7} (1893), 111.
- [LeV]
- R. LeVavasseur, Sur le syst\`eme d'\'equations aux d\'eriv\'ees partielles simultan\'ees auxquelles satisfait la s\'erie hyperg\'eom\'etrique \`a deux variables $F_1(\alpha ,\beta ,\beta ',\gamma ;x,y)$, Ann. Fac. Sci. Toulouse Math. \textbf{VII} (1893), 1--205.
- [Li]
- R. Livne, On certain covers of the universal elliptic curve, Ph.D. Dissertation, Harvard, Cambridge, MA, 1981.
- [Ma]
- H. Maschke, Aufstellung des vollen Formensystems einer quatern\"aren Gruppe von {\rm 51480} linearen Substitutionen, Math. Ann. \textbf{33} (1889), 317--344.
- [Mi]
- Y. Miyoaka, On the Chern numbers of surfaces of general type, Invent. Math. {\bf 42} (1977), 225--237.
- [M1]
- G.D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. \textbf{86} (1980), 171--276.
- [M2]
- G.D. Mostow, Generalised Picard lattices arising from half-integral conditions, Inst. Hautes \'Etudes Sci. Publ. Math. \textbf{63} (1986), 91--106.
- [M3]
- G.D. Mostow, On discontinuous action of monodromy groups on the complex $n$-ball, J. Amer. Math. Soc. \textbf{1} (1988), 555--586.
- [P1a]
- E. Picard, Sur une classe de fonctions de deux variables ind\'ependantes, C. R. Acad. Sci. \textbf{90} (1880), 1119--1121.
- [P1b]
- E. Picard, Sur une extension aux fonctions de deux variables du probl\`eme de Riemann relatif aux fonctions hyperg\'eom\'etriques, Ann. Ecole Norm. Sup. (2) \textbf{10} (1881), 305--322.
- [P2a]
- E. Picard, Sur les fonctions hyperfuchsiennes provenant des s\'eries hyperg\'eom\'etriques de deux variables, Ann. Ecole Norm. Sup. (3) \textbf{2} (1885), 357--384.
- [P2b]
- E. Picard, Sur les fonctions hyperfuchsiennes provenant des s\'eries hyperg\'eom\'etriques de deux variables, Bull. Soc. Math. France \textbf{15} (1887), 148--152.
- [Po1]
- L.Pochhammer, Ueber hypergeometrische Functionen $n^{\roman {ter}}$ Ordnungen, J. Reine Angew. Math. \textbf{71} (1870), 316--352.
- [Po2]
- L.Pochhammer, Ueber ein Integral mit doppeltem Umlauf, Math. Ann. \textbf{35} (1890), 470--494.
- [R]
- B. Riemann, Beitr\"age zur Theorie der durch die Gauss'sche Reihe $F(\alpha ,\beta ,\gamma ,x)$ darstellbaren Functionen, Abh. K\"on. Ges. Wiss. G\"ottingen Math. Cl. \textbf{VII} (1857).
- [Sas]
- T. Sasaki, On the finiteness of the monodromy group of the system of hypergeometric differential equations $(F_D)$, J. Fac. Univ. Tokyo \textbf{24} (1977), 565--573.
- [Sa]
- J.K. Sauter, Isomorphisms among monodromy groups and applications to lattices in $\roman {PU}(1,2)$, Pacific J. Math. \textbf{146} (1990), 331--384.
- [S]
- L. Schl\"afli, Ueber die Gauss'sche hypergeometrische Reihe, Math. Ann. \textbf{3} (1871), 286--295.
- [Sch]
- H.A. Schwarz, Ueber diejenigen F\"alle, in welchen die Gauss'sche hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, J. Reine Angew. Math. \textbf{75} (1873), 292--335.
- [Te1]
- T. Terada, Probl\`eme de Riemann et fonctions automorphes provenant des fonctions hyper- g\'eom\'etriques de plusieurs variables, J. Math. Kyoto Univ. \textbf{13} (1973), 557--578.
- [Te2]
- T. Terada, Quelques propri\'et\'es g\'eom\'etrique de domaine de $F_1$ et le groupe de tresses color\'ees, Publ. Res. Inst. Math. Sci. \textbf{17} (1981), 95--111.
- [W]
- H. Weber, Lehrbuch der Algebra, 2. Aufl. Band II, Friedrich Vieweg und Sohn, Braunschweig, 1899.
- [Ya]
- S.-T. Yau, Calabi\RM 's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. {\bf 74} (1977), 1798--1799.
- [Y1]
- M. Yoshida, Orbifold uniformising differential equations. {\rm III}, Math. Ann. \textbf{274} (1986), 319--334.
- [Y2]
- M. Yoshida, Fuchsian differential equations, Aspects of Math., Vieweg, Braunschweig, 1987.
Additional Information:
Reviewer(s):
P. Beazley Cohen and F.
Hirzebruch
Review Information:
Journal:
Bull. Amer. Math. Soc.
32
(1995),
88-105.
DOI:
10.1090/S0273-0979-1995-00564-1
PII:
S 0273-0979(1995)00564-1
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