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On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields

Published online by Cambridge University Press:  22 January 2016

Goro Shimura
Goro Shimura*
Affiliation:
Princeton University
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1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 43 , September 1971 , pp. 199 - 208
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

References

[1] Deuring, M., Die Zetafunktion einer algebraischen Kurve vom Geschlecht Eins, I, II, III, IV, Nachr. Akad. Wiss. Göttingen, (1953) 85–94, (1955) 13–42, (1956) 37–76, (1957) 55–80.Google Scholar
[2] Hecke, E., Zur Theorie der elliptischen Modulfunktionen, Math. Ann., 97 (1926), 210–242 (-Math. Werke, 428–460).CrossRefGoogle Scholar
[3] Hecke, E., Bestimmung der Perioden gewisser Integrale durch die Theorie der Klassenkörper, Math. Zeitschr., 28 (1928), 708–727 (=Math. Werke, 505–524).CrossRefGoogle Scholar
[4] Hecke, E., Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung I, II, Math. Ann., 114 (1937), 1–28, 316–351 (=Math. Werke, 644–707).CrossRefGoogle Scholar
[5] Shimura, G., Correspondances modulaires et les fonctions C de courbes algébriques, J. Math. Soc. Japan, 10 (1958), 1–28.Google Scholar
[6] Shimura, G., On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math., 78 (1963), 149–192.CrossRefGoogle Scholar
[7] Shimura, G., Introduction to the arithmetic theory of automorphic functions, Publ. Math. Soc. Japan, No. 11, 1971.Google Scholar
[8] Shimura, G., On the zeta-function of an abelian variety with complex multiplication, to appear.Google Scholar
[9] Weil, A., Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann., 168 (1967), 149–156.Google Scholar