Abstract
The purpose of this paper, which is a continuation of [2,3,4,5], is to prove that the special values of Hilbert modular functions of level π generate abelian extensions of certain CM fields, by using an essentially elementary theory of arithmetic Hilbert modular functions, based on the theory of congruence Eisenstein series. The main results are generalizations of the main results of Hecke’s thesis [10]. They are also subsumed in more far-reaching results of Shimura and Taniyama [15,13,14]. But our methods are quite different from the latter’s and stem directly from Hecke’s original ideas.
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References
Artin, E., Representatives of the Connected Component of the Idèle Class Group, Proc. International Symposium on Algebraic Number Theory, Tokyo(1955), pp. 51–54.
Baily, W.L.,Jr., Galois action on Eisenstein series and certain abelian extensions of number fields, pre-print, 1980.
Baily, W.L.,Jr., On the Theory of Hilbert Modular Functions I. Arithmetic Groups and Eisenstein Series, J. Alg., 90 (1984), 567–605.
Baily, W.L.,Jr., Arithmetic Hilbert Modular Forms, Automorphic Forms of Several Variables, Taniguchi Symposium, Katata, 1983, Birkhäuser, 1984.
Baily, W.L.,Jr., Arithmetic Hilbert Modular Functions II, Revista Matematica Iberoamericana, 1 (1985), 85–119.
Chevalley, C., Deux Théorèmes d’Arithmétique, J. Math. Soc. Japan, 3 (1951), 36–44.
Deligne, P., Travaux de Shimura, Séminaire BOURBAKI, 23(1970/71), No. 389, 123–165.
Hasse, H., Neue Begründung der komplexen Multiplikation, Jour. Reine Angew. Math. 157 (1927), 115–139.
Hasse, H., Neue Begründung der komplexen Multiplication II, Jour. Reine Angew. Math. 165 (1931), 64–88.
Hecke, E., Höhere Modulfunktionen und ihre Anwendung auf die Zahlentheorie, Math. Ann. 71(1912), 1–37. (= No. 1 in Mathematische Werke).
Karel, M.L., Special Values of Elliptic Modular Functions, Proc. of International Symposium on Automorphic Forms of Several Variables, Katata 1983, Birkhäuser, 1984.
Lang, S., Algebraic Number Theory, Addison-Wesley, 1970.
Shimura, G., Construction of class fields and zeta functions of of algebraic curves, Ann. Math., 85 (1967), 58–159.
Shimura, G., On canonical models of arithmetic quotients of bounded symmetric domains, 91(1970), 144–222; II, 92 (1970), 528–549.
Shimura, G., and Taniyama, Y., Complex Multiplication of Abelian Varieties and its Applications to Number Theory, Publications of the Math. Soc. of Japan, No. 6, 1961/
Taniyama, Y., Jacobian Varieties and Number Fields, The Complete Works of Yutaka Taniyama, Yutaka Taniyama Complete Works Publication Society, 1961, 1–68 (originally mimeographed notes, University of Tokyo, September 1955 ).
Weber, H., Lehrbuch der Algebra (Kleine Ausgabe), Friedrich Vieweg & Sohn, Braunschweig, 1912.
Weil, A., Basic Number Theory, Springer-Verlag 1967.
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© 1986 Springer Fachmedien Wiesbaden
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Baily, W.L. (1986). Arithmetic Hilbert Modular Functions Ill. In: Howard, A., Wong, PM. (eds) Contributions to Several Complex Variables. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-06816-7_1
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DOI: https://doi.org/10.1007/978-3-663-06816-7_1
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
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