Book Review
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MathSciNet review:
1568148
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Book Information:
Author:
Pierre Deligne and G. Daniel Mostow
Title:
Commensurabilities among lattices in $\mathrm{PU}(1,n)$
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.
[A] P. Appell, Sur les fonctions hypergéométriques de deux variables, J. Math. Pures Appl. (3) 8 (1882), 173-216.
[AKdF] P. Appell and M.J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques, Polynômes d'Hermite, Gauthier-Villars, Paris, 1926.
[AS] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions, Dover, New York, 1970.
Gottfried Barthel, Friedrich Hirzebruch, and Thomas Höfer, Geradenkonfigurationen und Algebraische Flächen, Aspects of Mathematics, D4, Friedr. Vieweg & Sohn, Braunschweig, 1987 (German). MR 912097, DOI 10.1007/978-3-322-92886-3
S. Y. Cheng and S.-T. Yau, Inequality between Chern numbers of singular Kähler surfaces and characterization of orbit space of discrete group of $\textrm {SU}(2,1)$, Complex differential geometry and nonlinear differential equations (Brunswick, Maine, 1984) Contemp. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1986, pp. 31–44. MR 833802, DOI 10.1090/conm/049/833802
Paula Cohen and Jürgen Wolfart, Modular embeddings for some nonarithmetic Fuchsian groups, Acta Arith. 56 (1990), no. 2, 93–110. MR 1075639, DOI 10.4064/aa-56-2-93-110
Paula Beazley Cohen and Jürgen Wolfart, Algebraic Appell-Lauricella functions, Analysis 12 (1992), no. 3-4, 359–376. MR 1182636
Paula Beazley Cohen and Jürgen Wolfart, Fonctions hypergéométriques en plusieurs variables et espaces des modules de variétés abéliennes, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 6, 665–690 (French, with English summary). MR 1251148
Paula Beazley Cohen, Claude Itzykson, and Jürgen Wolfart, Fuchsian triangle groups and Grothendieck dessins. Variations on a theme of Belyĭ, Comm. Math. Phys. 163 (1994), no. 3, 605–627. MR 1284798
P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89. MR 849651
[E] L. Euler, Specimen transformationis singularis serierum, Nova Acta Acad. Petropolitan 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 veränderlichen Coeffizienten, J. Reine Angew. Math. 66 (1866), 121-160.
[Ga] C.F. Gauss, Disquisitiones generales circa seriem infinitam , Werke, Band III, Olms Verlag, Hildesheim and New York, 1981, pp. 123-162.
Izrail M. Gelfand, Collected papers. Vol. III, Springer-Verlag, Berlin, 1989. Edited by S. G. Gindikin, V. W. Guillemin, A. A. Kirillov, B. Kostant and S. Sternberg; With a foreword by Gindikin; With a contribution by Kostant. MR 997939
[Go] E. Goursat, Extension du problème de Riemann à des fonctions hypergéométriques de deux variables, C. R. Acad. Sci. 95 (1882), 903,1044.
[H] C. Hermite, Sur quelques équations différentielles linéaires, J. Reine Angew. Math. 79 (1875), 111-158.
F. Hirzebruch, Chern numbers of algebraic surfaces: an example, Math. Ann. 266 (1984), no. 3, 351–356. MR 730175, DOI 10.1007/BF01475584
Masa-Nori Ishida, Hirzebruch’s examples of surfaces of general type with $c_{1}^{2}=3c_{2}$, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 412–431. MR 726436, DOI 10.1007/BFb0099973
[J] C.G.J. Jacobi, Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe, J. Reine Angew. Math. 56 (1859), 149-165.
Felix Klein, Weitere Untersuchungen über des Ikosaeder, Math. Ann. 12 (1877), no. 4, 503–560 (German). MR 1509949, DOI 10.1007/BF01443208
Felix Klein, Ueber Normirung der linearen Differentialgleichungen zweiter Ordnung, Math. Ann. 38 (1891), no. 1, 144–152 (German). MR 1510668, DOI 10.1007/BF01212698
A. W. Knapp, Doubly generated Fuchsian groups, Michigan Math. J. 15 (1969), 289–304. MR 248231
Ryoichi Kobayashi, Shu Nakamura, and Fumio Sakai, A numerical characterization of ball quotients for normal surfaces with branch loci, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 7, 238–241. MR 1030189
[K] E.E. Kummer, Ueber die hypergeometrische Reihe , J. Reine Angew. Math. 15 (1836), 39-83 and 127-172.
[L] G. Lauricella, Sulle funzioni ipergeometriche a piu variabli, Rend. Circ. Mat. Palermo 7 (1893), 111.
[LeV] R. LeVavasseur, Sur le système d'équations aux dérivées partielles simultanées auxquelles satisfait la série hypergéométrique à deux variables , Ann. Fac. Sci. Toulouse Math. VII (1893), 1-205.
[Li] R. Livne, On certain covers of the universal elliptic curve, Ph.D. Dissertation, Harvard, Cambridge, MA, 1981.
Heinrich Maschke, Aufstellung des vollen Formensystems einer quaternären Gruppe von 51840 linearen Substitutionen, Math. Ann. 33 (1889), no. 3, 317–344 (German). MR 1510546, DOI 10.1007/BF01443964
Yoichi Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225–237. MR 460343, DOI 10.1007/BF01389789
G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), no. 1, 171–276. MR 586876
P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89. MR 849651
G. D. Mostow, On discontinuous action of monodromy groups on the complex $n$-ball, J. Amer. Math. Soc. 1 (1988), no. 3, 555–586. MR 932662, DOI 10.1090/S0894-0347-1988-0932662-7
[P1a] E. Picard, Sur une classe de fonctions de deux variables indépendantes, C. R. Acad. Sci. 90 (1880), 1119-1121; Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques, C. R. Acad. Sci. 90 (1880), 1267-1269.
Émile Picard, Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques, Ann. Sci. École Norm. Sup. (2) 10 (1881), 305–322 (French). MR 1508705
Émile Picard, Sur les fonctions hyperfuchsiennes provenant des séries hypergéométriques de deux variables, Ann. Sci. École Norm. Sup. (3) 2 (1885), 357–384 (French). MR 1508769
E. Picard, Sur les fonctions hyperfuchsiennes provenant des séries hypergéométriques de deux variables, Bull. Soc. Math. France 15 (1887), 148–152 (French). MR 1504014
[Po1] L.Pochhammer, Ueber hypergeometrische Functionen Ordnungen, J. Reine Angew. Math. 71 (1870), 316-352.
[Po2] -, Ueber ein Integral mit doppeltem Umlauf, Math. Ann. 35 (1890), 470-494.
[R] B. Riemann, Beiträge zur Theorie der durch die Gauss'sche Reihe darstellbaren Functionen, Abh. Kön. Ges. Wiss. Göttingen Math. Cl. VII (1857).
Takeshi Sasaki, On the finiteness of the monodromy group of the system of hypergeometric differential equations $(F_{D})$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 3, 565–573. MR 498553
John Kurt Sauter Jr., Isomorphisms among monodromy groups and applications to lattices in $\textrm {PU}(1,2)$, Pacific J. Math. 146 (1990), no. 2, 331–384. MR 1078386
[S] L. Schläfli, Ueber die Gauss'sche hypergeometrische Reihe, Math. Ann. 3 (1871), 286-295.
[Sch] H.A. Schwarz, Ueber diejenigen Fälle, in welchen die Gauss'sche hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, J. Reine Angew. Math. 75 (1873), 292-335.
Toshiaki Terada, Problème de Riemann et fonctions automorphes provenant des fonctions hypergéométriques de plusieurs variables, J. Math. Kyoto Univ. 13 (1973), 557–578 (French). MR 481156, DOI 10.1215/kjm/1250523325
Toshiaki Terada, Quelques propriétés géométriques du domaine de $F_{1}$ et le groupe de tresses colorées, Publ. Res. Inst. Math. Sci. 17 (1981), no. 1, 95–111 (French). MR 613935, DOI 10.2977/prims/1195186705
[W] H. Weber, Lehrbuch der Algebra, 2. Aufl. Band II, Friedrich Vieweg und Sohn, Braunschweig, 1899.
Shing Tung Yau, Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798–1799. MR 451180, DOI 10.1073/pnas.74.5.1798
Masaaki Yoshida, Orbifold-uniformizing differential equations. III. Arrangements defined by $3$-dimensional primitive unitary reflection groups, Math. Ann. 274 (1986), no. 2, 319–334. MR 838472, DOI 10.1007/BF01457077
Masaaki Yoshida, Fuchsian differential equations, Aspects of Mathematics, E11, Friedr. Vieweg & Sohn, Braunschweig, 1987. With special emphasis on the Gauss-Schwarz theory. MR 986252, DOI 10.1007/978-3-663-14115-0
- [A]
- P. Appell, Sur les fonctions hypergéométriques de deux variables, J. Math. Pures Appl. (3) 8 (1882), 173-216.
- [AKdF]
- P. Appell and M.J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques, Polynômes 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. Höfer, Geradenkonfigurationen und Algebraische Flächen, Aspects of Math. D4, Vieweg, Braunschweig, 1987. MR 912097 (89a:14045)
- [CHY]
- S.Y. Cheng and S.-T. Yau, Inequality between Chern numbers of singular Kähler surfaces and characterisation of orbit space of discrete group of , Contemp. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1986, pp. 31-44. MR 833802 (87m:53078)
- [CoWo1]
- P. Cohen and J. Wolfart, Modular embeddings for some nonarithmetic Fuchsian groups, Acta Arith. LVI (1990), 93-110. MR 1075639 (92d:11039)
- [CoWo2]
- P. Beazley Cohen and J. Wolfart, Algebraic Appell-Lauricella functions, Analysis 12 (1992), 359-376. MR 1182636 (94f:33024)
- [CoWo3]
- -, Fonctions hypergéométriques en plusieurs variables et espaces des modules de variétés abéliennes, Ann. Sci. École Norm. Sup. (4) 26 (1993), 665-690. MR 1251148 (94j:14027)
- [CoItzWo]
- P. Beazley Cohen, C. Itzykson, and J. Wolfart, Fuchsian triangle groups and Grothendieck dessins--Variations on a theme of Belyi, Commun. Math. Phys. 63 (1994), 605-627. MR 1284798 (96a:11056)
- [DM]
- P. Deligne and G.D. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5-90. MR 849651 (88a:22023a)
- [E]
- L. Euler, Specimen transformationis singularis serierum, Nova Acta Acad. Petropolitan 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 veränderlichen Coeffizienten, J. Reine Angew. Math. 66 (1866), 121-160.
- [Ga]
- C.F. Gauss, Disquisitiones generales circa seriem infinitam , Werke, Band III, Olms Verlag, Hildesheim and New York, 1981, pp. 123-162.
- [G]
- I.M. Gelfand et al., Collected papers, Volume III, Part V, Springer-Verlag, New York, 1989. MR 997939 (90d:01091)
- [Go]
- E. Goursat, Extension du problème de Riemann à des fonctions hypergéométriques de deux variables, C. R. Acad. Sci. 95 (1882), 903,1044.
- [H]
- C. Hermite, Sur quelques équations différentielles linéaires, J. Reine Angew. Math. 79 (1875), 111-158.
- [Hi]
- F. Hirzebruch, Chern numbers of algebraic surfaces--an example, Math. Ann. 266 (1984), 351-356. MR 730175 (85j:14069)
- [I]
- M.-N. Ishida, Hirzebruch's examples of surfaces of general type with , Algebraic Geometry, Proc. Tokyo Kyoto 1982 Lecture Notes in Math., vol. 1016, Springer-Verlag, New York, 1983, pp. 412-431. MR 726436 (85j:14073)
- [J]
- C.G.J. Jacobi, Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe, J. Reine Angew. Math. 56 (1859), 149-165.
- [K11]
- F. Klein, Weitere Untersuchungen über das Ikosaeder, Math. Ann. 12 (1877), 503-560. MR 1509949
- [K12]
- -, Ueber Normirung der linearen Differentialgleichungen zweiter Ordnung, Math. Ann. 38 (1891), 144-152. MR 1510668
- [Kn]
- A.W. Knapp, Doubly generated Fuchsian groups, Michigan Math. J. 15 (1986), 289-304. MR 0248231 (40:1483)
- [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 65 (1989), 238-241. MR 1030189 (90k:32034)
- [K]
- E.E. Kummer, Ueber die hypergeometrische Reihe , J. Reine Angew. Math. 15 (1836), 39-83 and 127-172.
- [L]
- G. Lauricella, Sulle funzioni ipergeometriche a piu variabli, Rend. Circ. Mat. Palermo 7 (1893), 111.
- [LeV]
- R. LeVavasseur, Sur le système d'équations aux dérivées partielles simultanées auxquelles satisfait la série hypergéométrique à deux variables , Ann. Fac. Sci. Toulouse Math. 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 quaternären Gruppe von 51480 linearen Substitutionen, Math. Ann. 33 (1889), 317-344. MR 1510546
- [Mi]
- Y. Miyoaka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225-237. MR 0460343 (57:337)
- [M1]
- G.D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), 171-276. MR 586876 (82a:22011)
- [M2]
- -, Generalised Picard lattices arising from half-integral conditions, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 91-106. MR 849652 (88a:22023b)
- [M3]
- -, On discontinuous action of monodromy groups on the complex n-ball, J. Amer. Math. Soc. 1 (1988), 555-586. MR 932662 (89h:22020)
- [P1a]
- E. Picard, Sur une classe de fonctions de deux variables indépendantes, C. R. Acad. Sci. 90 (1880), 1119-1121; Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques, C. R. Acad. Sci. 90 (1880), 1267-1269.
- [P1b]
- -, Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques, Ann. Ecole Norm. Sup. (2) 10 (1881), 305-322. MR 1508705
- [P2a]
- -, Sur les fonctions hyperfuchsiennes provenant des séries hypergéométriques de deux variables, Ann. Ecole Norm. Sup. (3) 2 (1885), 357-384. MR 1508769
- [P2b]
- -, Sur les fonctions hyperfuchsiennes provenant des séries hypergéométriques de deux variables, Bull. Soc. Math. France 15 (1887), 148-152. MR 1504014
- [Po1]
- L.Pochhammer, Ueber hypergeometrische Functionen Ordnungen, J. Reine Angew. Math. 71 (1870), 316-352.
- [Po2]
- -, Ueber ein Integral mit doppeltem Umlauf, Math. Ann. 35 (1890), 470-494.
- [R]
- B. Riemann, Beiträge zur Theorie der durch die Gauss'sche Reihe darstellbaren Functionen, Abh. Kön. Ges. Wiss. Göttingen Math. Cl. VII (1857).
- [Sas]
- T. Sasaki, On the finiteness of the monodromy group of the system of hypergeometric differential equations , J. Fac. Univ. Tokyo 24 (1977), 565-573. MR 0498553 (58:16654)
- [Sa]
- J.K. Sauter, Isomorphisms among monodromy groups and applications to lattices in , Pacific J. Math. 146 (1990), 331-384. MR 1078386 (92d:22016)
- [S]
- L. Schläfli, Ueber die Gauss'sche hypergeometrische Reihe, Math. Ann. 3 (1871), 286-295.
- [Sch]
- H.A. Schwarz, Ueber diejenigen Fälle, in welchen die Gauss'sche hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, J. Reine Angew. Math. 75 (1873), 292-335.
- [Te1]
- T. Terada, Problème de Riemann et fonctions automorphes provenant des fonctions hypergéométriques de plusieurs variables, J. Math. Kyoto Univ. 13 (1973), 557-578. MR 0481156 (58:1299)
- [Te2]
- -, Quelques propriétés géométrique de domaine de et le groupe de tresses colorées, Publ. Res. Inst. Math. Sci. 17 (1981), 95-111. MR 613935 (83b:32008)
- [W]
- H. Weber, Lehrbuch der Algebra, 2. Aufl. Band II, Friedrich Vieweg und Sohn, Braunschweig, 1899.
- [Ya]
- S.-T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1798-1799. MR 0451180 (56:9467)
- [Y1]
- M. Yoshida, Orbifold uniformising differential equations. III, Math. Ann. 274 (1986), 319-334. MR 838472 (87i:14007)
- [Y2]
- -, Fuchsian differential equations, Aspects of Math., Vieweg, Braunschweig, 1987. MR 986252 (90f:32025)
Review Information:
Reviewer:
P. Beazley Cohen and F. Hirzebruch
Journal:
Bull. Amer. Math. Soc.
32 (1995), 88-105
DOI:
https://doi.org/10.1090/S0273-0979-1995-00564-1