Summary
A partial result on a problem of Siegel is given: Hypergeometric functions with rational parameters have transcendental values in almost all algebraic points — up to some natural exceptions; these exceptions are the well-known algebraic functions and an (unexpected) second class of examples related to certain Shimura-curves.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literatur
[A] Albert, A.A.: A solution of the principal problem in the theory of Riemann matrices. Ann. Math.35, 500–515 (1934)
[Be] Bertrand, D.: Endomorphismes de groupes algébriques; applications arithmétiques. In: Approximations diophantiennes et nombres transcendants, Luminy 1982, Progress in Math. 31, pp. 1–45. Basel-Boston: Birkhäuser 1983
[Bo] Bombieri, E.: OnG-functions. In: Recent progress in analytic number theory, vol. 2, pp. 1–67. Durham 1979, London: Academic press 1981
[BW] Beukers, F., Wolfart, J.: Algebraic values of hypergeometric functions, erscheint in den Proceedings der Durham conference on transcendental numbers 1986
[CW] Chevalley, Cl., Weil, A.: Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers. Abh. Hamburger Math. Sem.10, 358–361 (1934)
[EMOT] Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions, Bateman manuscript project. New York: McGraw-Hill 1953
[Gre] Greenberg, L.: Maximal Fuchsian groups. Bull. Am. Math. Soc.69, 569–573 (1963)
[Gr] Gross, B.: On the periods of abelian integrals and a formula of Chowla and Selberg. Invent. Math.45, 193–208 (1978)
[Ja] Jacobson, N.: Theory of rings. Am. Math. Soc. (1943)
[Ka] Katz, N.M.: Algebraic solutions of differential equations (p-curvature and the Hodge filtration). Invent. Math.18, 1–118 (1972)
[KI] Klein, F.: Vorlesungen über die hypergeometrische Funktion. Berlin: Springer 1933
[Kn] Knapp, M.: Doubly generated Fuchsian groups. Mich. Math. J.15, 289–304 (1968)
[KR] Koblitz, N., Rohrlich, D.: Simple factors in the Jacobian of a Fermat curve. Can. J. Math.30, 1183–1205 (1978)
[Schw] Schwarz, H.A.: Über diejenigen Fälle, in welchen die Gaußische hypergeometrische Reihe eine algebraische Funktion ihres vierten Elements darstellt. J. Reine Angew. Math.75, 292–335 (1873)
[ST] Shimura, G., Taniyama, Y.: Complex multiplication of abelian varieties and its applications to number theory. Publ. Math. Soc. Japan6, (1961)
[Shi1] Shimura, G.: On analytic families of polarized abelian varieties and automorphic functions. Ann. Math.78, 149–192 (1963)
[Shi2] Shimura, G.: Construction of class fields and zeta functions of algebraic curves. Ann. Math.85, 58–159 (1967)
[Sie1] Siegel, C.L.: Über einige Anwendungen diophantischer Approximationen, Ges. Werke, Bd. I, pp. 209–266. Berlin-Heidelberg-New York: Springer 1966
[Sie2] Siegel, C.L.: Lectures on Riemann matrices. Tata Institute, Bombay 1963
[Ta1] Takeuchi, K.: Arithmetic triangle groups. J. Math. Soc. Japan29, 91–106 (1977)
[Ta2] Takeuchi, K.: Commensurability classes of arithmetic triangle groups. J. Fac. Sci. Univ. Tokyo, Sec. IA,24, 201–212 (1977)
[WW] Wolfart, J., Wüstholz, G.: Der Überlagerungsradius gewisser algebraischer Kurven und die Werte der Betafunktion an rationalen Stellen. Math. Ann.273, 1–15 (1985)
[Wo] Wolfart, J.: Fonctions hypergéométriques, arguments exceptionels et groupes de monodromie, Publ. Math. Univ. Pierre Marie Curie No. 79, Problèmes diophantiens 1985–1986, IX.1–IX.24
[Wü] Wüstholz, G.: Recent progress in transcendence theory. In: Number theory Noordwijkerhout 1983. (Lect. Notes Math., vol. 1068, pp. 280–296). Berlin-Heidelberg-New York: Springer 1984
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Wolfart, J. Werte hypergeometrischer funktionen. Invent Math 92, 187–216 (1988). https://doi.org/10.1007/BF01393999
Issue Date:
DOI: https://doi.org/10.1007/BF01393999