Abstract
It was a great pleasure for me to participate in the symposium in honor of Shiing-shen Chern. In my lecture I intended to give a survey on Hilbert modular surfaces. But actually I discussed examples of such Hilbert modular surfaces for which specific information is available on their structure as algebraic surfaces. The paper presented here is an extended version of the talk.
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References
A. Beauville, L’application canonique pour les surfaces de type general. Inventiones Math. 55, 121–140 (1979).
E. Bombieri, Canonical maps of surfaces of general type. Publ. Math. IHES 42, 171–219 (1973).
E. Brieskom, Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen. Math. Ann. 166, 76–102 (1966).
E. Brieskom, Die Auflösung der rationalen Singuliaritäten holomorpher Abbildungen. Math. Ann. 178, 255–270 (1968).
E. Brieskom, Singular elements of semi-simple algebraic groups. In Actes, Congrès Intern. Math., 1970, Tome 2, pp. 279–284.
J. W. Bruce and C. T. C. Wall, On the classification of cubic surfaces. Preprint Liverpool 1978.
E. Freitag, Eine Bemerkung zur Theorie der Hilbertschen Modulmannigfaltigkeiten hoher Stufe. Math. Z. 171, 27–35 (1980).
G. van der Geer, Hilbert modular forms for the field Q()g). Math. Ann. 233, 163–179 (1978).
G. van der Geer, Minimal models for Hilbert modular surfaces of principal congruence subgroups. Topology 18, 29–39 (1979).
G. van der Geer and A. Van de Ven, On the minimality of certain Hilbert modular surfaces. In Complex Analysis and Algebraic Geometry. Iwanami Shoten and Cambridge U. P., 1977, pp. 137–150.
G. van der Geer and D. Zagier, The Hilbert modular group for the field Q(4.5 - ). Inventiones Math. 42, 93–133 (1977).
P. Griffiths and J. Harris, Principles of Algebraic Geometry. Wiley, New York, 1978.
W. Hausmann, Kurven auf Hilbertschen Modulflächen. Dissertation, Bonn, 1979.
F. Hirzebruch, Topological Methods in Algebraic Geomery, 3rd ed. Springer, New York, 1966.
F. Hirzebruch, Hilbert modular surfaces. Ens. Math. 19, 183–281 (1973).
F. Hirzebruch, Modulflächen und Modulkurven zur symmetrischen Hilbertschen Modulgruppe. Ann. Scient. Ec. Norm. Sup. 11, 101–166 (1978).
F. Hirzebruch and A. Van de Ven, Hilbert modular surfaces and the classification of algebraic surfaces. Inventiones Math. 23, 1–29 (1974).
F. Hirzebruch and A. Van de Ven, Minimal Hilbert modular surfaces with pg = 3 and K2 = 2. Amer. J. Math. 101, 132–148 (1979).
F. Hirzebruch and D. Zagier, Classification of Hilbert modular surfaces. In Complex Analysis and Algebraic Geometry. Iwanami Shoten and Cambridge U. P., 1977, pp. 43–77.
F. Hirzebruch and D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus. Inventiones Math. 36, 57–113 (1976).
E. Horikawa, On deformations of quintic surfaces. Inventiones Math. 31, 43–85 (1975).
E. Horikawa, Algebraic surfaces of general type with small C 12 . I. Annals Math. 104, 357–387 (1976).
E. Horikawa, Algebraic surfaces of general type with small c 12 II. Inventiones Math. 37, 121–155 (1976).
E. Horikawa, Algebraic surfaces of general type with small c 12 . III. Inventions Math. 47, 209–248 (1978).
E. Horikawa, Algebraic surfaces of general type with small c 12 . IV. Inventiones Math. 50, 103–128 (1979).
F. J. Koll, Die elliptischen Fixpunkte und die Spitzen der diskreten Erweiterungen der Hilbertschen Modulgruppe. Bonner Math. Schriften 84, 1976.
Yu. I. Manin, Cubic forms. Algebra, Geometry, Arithmetic. North-Holland, Amsterdam, 1974.
Y. Miyaoka, On the Chem numbers of surfaces of general type. Inventiones Math. 42, 225–237 (1977).
L. Schläfli, On the distribution of surfaces of the third order into species. Phil. Trans. Roy. Soc. 153, 193–247 (1864).
O. V. Svareman, Simply-connectedness of the factor space of the Hilbert modular group (in Russian). Funct. Anal. and Appl. 8 (2) 99–100 (1974).
H. Weber, Lehrbuch der Algebra, 2. Band, 2. Auflage. Vieweg, Braunschweig, 1899.
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Hirzebruch, F. (1980). The Canonical Map for Certain Hilbert Modular Surfaces. In: Hsiang, WY., Kobayashi, S., Singer, I.M., Wolf, J., Wu, HH., Weinstein, A. (eds) The Chern Symposium 1979. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8109-9_5
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