Abstract
The paper begins with a brief account of Christoffel’s work on abelian integrals and its historical context and then describes the close connection between Riemann surfaces and Fuchsian groups. This is followed by a survey of some modern developments in the theory of infinitely generated Fuchsian groups, namely the Bers spaces (the generalization of the space of abelian differentials) and the classification of Fuchsian groups.
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References
Ahlfors, L.V.: Finitely generated Kleinian groups. Amer. J. Math. 86 (1964), 413–429.
Ahlfors, L.V., and Sario, L.: Riemann Surfaces. Princeton University Press, Princeton 1960.
Bers, L.: Automorphic functions and Poincaré series for infinitely generated Fuchsian groups. Amer. J. Math. 87 (1965), 196–214.
Bers, L.: Uniformization, moduli and Kleinian groups. Bull. London Math. Soc. 4 (1972), 257300.
Carleson, L.: Sets of uniqueness for functions regular in the unit circle. Acta Math. 87 (1952), 325–345.
Christoffel, E.B.: Über die canonische Form der Riemannschen Integrale erster Gattung. Ann. Mat. Pura Appl. Serie II, 9 (1879), 240–301.
Christoffel, E.B.: Algebraischer Beweis des Satzes von der Anzahl der linear unabhängigen Integrale erster Gattung. Ann. Mat. Pura Appl. Serie II, 10 (1883), 81–100.
Christoffel, E. B.: Die Convergenz der Jacobischen 9-Reihe mit den Moduln Riemanns. Vierteljschr. Naturforsch. Ges. Zürich 41, 2 (1896), 3–6.
Christoffel, E.B.: Vollständige Theorie der Riemannschen 9-Funktion. Math. Ann. 54 (1901), 347–399.
Deuring, M.: Lectures on the Theory of Algebraic Functions of one Variable. Lecture Notes Math. 314, Springer-Verlag, Berlin 1973.
Drasin, D.: Cusp forms and Poincaré series. Amer. J. Math. 90 (1968), 356–365.
Earle, C.F.: Some remarks on Poincaré series. Compositio Math. 21 (1969), 167–176.
Eichler, M.: Eine Verallgemeinerung der Abelschen Integrale. Math. Z. 67 (1957), 267–298.
Greenberg, L.: Finiteness theorems for Fuchsian and Kleinian groups. Discrete Groups and Automorphic Functions, Proc. Conf. L.M.S. Cambridge. Acad. Press, London 1977.
Koebe, P.: Über die Uniformisierung beliebiger analytischer Kurven II. Nachr. Akad. Wiss. Göttingen 1907, 633–669.
Kra, I.: Automorphic Forms and Kleinian Groups. W.A. Benjamin Inc. Reading, Mass. 1972.
Lehner, J.: On the boundedness of integrable automorphic forms. Illinois J. Math. 18 (1974), 575–584.
Metzger, T.A.: On vanishing Eichler periods and Carleson sets. Michigan Math. J. 24 (1977), 197–202.
Niebur, D., and Sheingom, M.: Characterization of Fuchsian groups whose integrable forms are bounded. Ann. of Math. 106 (1977), 239–258.
Patterson, S.J.: Some examples of Fuchsian groups. Proc. London Math. Soc. 39 (1979), 276–298.
Poincaré, H.: Théorie des groupes fuchsiens. Acta Math. 1 (1882), 1–62.
Poincaré, H.: Sur l’uniformisation des fonctions analytiques. Acta Math. 31 (1907), 1–64.
Pommerenke, Ch.: On the theta-operator for automorphic forms of weight 1. Indiana Univ. Math. J. 25 (1976), 595–607.
Pommerenke, Ch.: On the Green’s function of Fuchsian groups. Ann. Acad. Sci. Fenn. Ser. AI Mathematica 2 (1976), 409–427.
Pommerenke, Ch.: On automorphic forms and Carleson sets. Michigan Math. J. 23 (1976), 129–136.
Rajeswara Rao, K.V.: Fuchsian groups of convergence type and Poincaré series of dimension - 2. J. Math. Mech. 18 (1969), 629–644.
Siegel, C. L.: Topics in Complex Function Theory, vol. II. Wiley-Interscience, New York 1971.
Widom, H.: H p sections of vector bundles over Riemann surfaces. Ann. of Math. (2) 94 (1971), 304–324.
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© 1981 Springer Basel AG
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Pommerenke, C. (1981). On Automorphic Functions. In: Butzer, P.L., Fehér, F. (eds) E. B. Christoffel. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5452-8_20
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DOI: https://doi.org/10.1007/978-3-0348-5452-8_20
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