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The modular equation and modular forms of weight one

Published online by Cambridge University Press:  22 January 2016

Toyokazu Hiramatsu  and
Yoshio Mimura
Toyokazu Hiramatsu
Affiliation:
Department of Mathematics Kobe University, Rokko, Kobe, 657, Japan
Yoshio Mimura
Affiliation:
Department of Mathematics Kobe University, Rokko, Kobe, 657, Japan
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This is a continuation of the previous paper [8] concerning the relation between the arithmetic of imaginary quadratic fields and cusp forms of weight one on a certain congruence subgroup. Let K be an imaginary quadratic field, say K = with a prime number q ≡ − 1 mod 8, and let h be the class number of K. By the classical theory of complex multiplication, the Hubert class field L of K can be generated by any one of the class invariants over K, which is necessarily an algebraic integer, and a defining equation of which is denoted by

Φ(x) = 0.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 100 , December 1985 , pp. 145 - 162
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1985

References

[ 1 ] Deligne, P. et Serre, J.-P., Formes modulaires de poids 1, Ann. Sci. École Norm. Sup., 4 (1974), 507–530.CrossRefGoogle Scholar
[ 2 ] Eichler, M., Der Hilbertsche Klassenkörper eines imaginärquadratischen Zahlkörper, Math. Z., 64 (1956), 229–242.Google Scholar
[ 3 ] Fricke, R., Lehrbuch der Algebra III, Braunschweig 1928.Google Scholar
[ 4 ] Hasse, H., Neue Begründung der komplexen Multiplikation, I: J. Reine Angew. Math., 157 (1927), 115–139, II: Ibid., 165 (1931), 64–88.Google Scholar
[ 5 ] Hasse, H., Uber den Klassenkorper zum quadratischen Zahlkorper mit der Diskriminate 47, Acta Arith., 9 (1964), 419–434.Google Scholar
[ 6 ] Hasse, H. and Liang, J., Uber den Klassenkorper zum quadratischen Zahlkorpermit der Diskriminate 47 (Fortsetzung), Acta Arith., 16 (1969), 89–97.Google Scholar
[ 7 ] Hecke, E., Zur Theorie der elliptischen Modulfunktionen, Math. Ann., 97 (1926), 210–242 (=Math. Werke, 461–486).Google Scholar
[ 8 ] Hiramatsu, T., Higher reciprocity laws and modular forms of weight one, Comm. Math. Univ. St. Paul, 31 (1982), 75–85.Google Scholar
[ 9 ] Serre, J.-P., Modular forms of weight one and Galois representations, Proc. Symposium on Algebraic Number Field, Academic Press, London, 1977, pp. 193–268.Google Scholar
[10] Weber, H., Lehrbuch der Algebra III, Braunschweig 1908.Google Scholar
[11] Yui, N., Explicit form of the modular equation, J. Reine Angew. Math., 299/300 (1978), 185–200.Google Scholar
[12] Zassenhaus, H. and Liang, J., On a problem of Hasse, Math. Comp., 23 (1969), 515–519.Google Scholar
[13] Seminar on Complex Multiplication, Lecture Notes in Math. No. 21, Springer-Verlag, Berlin, Heidelberg, New York 1966 (from the Institute Seminar by Borel, Chowla, Herz, Iwasawa, Serre, held in 1957–58).Google Scholar