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On the zeta-functions of the algebraic curves uniformized by certain automorphic functions
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1961 Volume 13 Issue 3 Pages 275-331
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- Published: 1961 Received: March 31, 1961 Available on J-STAGE: August 29, 2006 Accepted: - Advance online publication: - Revised: -
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Correction information
Date of correction: August 29, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) W. L. Chow, Abelian varieties over function-fields, Trans. Amer. Math. Soc., 78 (1955) 253-275.
2) M. Deuring, Algebren, Berlin, 1935.
3) M. Deuring, Die Zetaf unktion einer algebraischen Kurve vom Geschlechte Eins, I, II, III, IV, Nachr. Akad. Wiss. Gottingen, (1953), 85-94; (1955), 13-42; (1956), 37-76; (1957), 55-80.
4) M. Eichler, Allgemeine Kongruenzklasseneinteilungen der Ideale einfacher Algebren über algebraischen Zahlkörpern und ihre L-Reihen, J. Reine Angew. Math., 179 (1938), 227-251.
5) M. Eichler, Über die Idealklassenzahl hyperkomplexer Systeme, Math. Z., 43 (1938), 481-494.
6) M. Eichler, Quaternäre quadratische Formen and die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math., 5 (1954), 355-366.
7) M. Eichler, Modular correspondences und their representations. J. Indian Math. Soc., 20 (1956), 163-206.
8) M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Z., 67 (1957), 267-298.
9) G. Fujisaki, On the zeta-function of the simple algebra over the field of rational numbers, J. Fac. Sci. Univ. Tokyo. Sec. I, vol. VII, Part 5 (1958), 567-604.
10) R. Godement, Les fonctions ζ des algebres simples II, Séminaire Bourbaki (février 1959, 176), 1-20.
11) E. Hecke, Über die Bestimmung Dirichletscher Reihen lurch ihre Funktionalgleichung, Math. Ann., 112 (1936), 664-699.
12) E. Hecke, Über Modulf unktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung, I, II, Math. Ann., 114 (1937), 1-28, 316-351.
13) J. Igusa, Kroneckerian model of fields of elliptic modular functions, Amer. J. Math., 81 (1959), 561-577.
14) S. Koizumi and G. Shimura, On specializations of abelian varieties, Sci. Papers, Coll. Gen. Ed. Univ. Tokyo, 9 (1959), 187-211.
15) T. Matsusaka, Polarized varieties, fields of moduli and generalized Kummer varieties of polarized abelian varieties, Amer. J. Math., 80 (1958), 45-82.
16) M. Nishi, Some results on abelian varieties, Nat. Sci. Rep., Ochanomizu Univ., 9 (1958), 1-12.
17) H. Petersson, Konstruktion der samtilichen Lösungen einer Riemannschen Funktionalgleichung durch Dirichlet-Reihen mit Eulerscher Produktentwickelung, I, II, III, Math. Ann., 116 (1939), 401-412; 117 (1940), 39-64, 277-300.
18) A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc., 20 (1956), 47-87.
19) A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to Function Theory, Tata Institute of Fundamental Research, Bombay, 1960, 147-164.
20) J.-P. Serre, Groupes algébriques et corps de classes, Hermann, Paris, 1959.
21) G. Shimura, Reduction of algebraic varieties with respect to a discrete valuation of the basic field, Amer. J. Math., 77 (1955), 134-176.
22) G. Shimura, Correspondances modulaires et les fonctions ζ de courbes algébriques, J. Math. Soc. Japan, 10 (1958), 1-28.
23)=[AF] G. Shimura, On the theory of automorphec functions, Ann. Math., 70 (1959), 101-144.
24) G. Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan, 11 (1959), 291-311.
25) G. Shimura and Y. Taniy ama, Complex multiplication of abelian varieties and its applications to number theory, Publ. Math. Soc. Japan, No. 6, 1961.
26) T. Tamagawa, On ζ-functions of a division algebra, to appear in Amer. J. Math.
27) Y. Taniyama, L-functions of number fields and zeta functions of abelian varieties, J. Math. Soc. Japan, 9 (1957), 330-366.
28) A. Weil, Sur les courbes algebriques et les variétés qui s'en déduisent Hermann, Paris 1948.
29) A. Weil, Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948.
30) A. Weil, Jacobi sums as "Grössencharaktere", Trans. Amer. Math. Soc., 73 (1952), 487-495.
31) A. Weil, On the theory of complex multiplication, Proc. Int. Symp. Alg. Nb. Th., Tokyo-Nikko, 1955, 9-22.
32) A. Weil, The field of definition of a variety, Amer. J. Math., 78 (1956), 509-524.
33) A. Weil, On discrete subgroups of Lie groups, Ann. Math. 72 (1960), 369-384,
34) A. Weil, Adèles and algebraic groups, Lecture note, The Institute for Advanced Study, Princeton, 1961.
Right : [1] W. L. Chow, Abelian varieties over function-fields, Trans. Amer. Math. Soc., 78 (1955) 253-275.
[2] M. Deuring, Algebren, Berlin, 1935.
[3] M. Deuring, Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins, I, II, III, IV, Nachr. Akad. Wiss. Göttingen, (1953), 85-94; (1955), 13-42; (1956), 37-76; (1957), 55-80.
[4] M. Eichler, Allgemeine Kongruenzklasseneinteilungen der Ideale einfacher Algebren über algebraischen Zahlkörpern und ihre L-Reihen, J. Reine Angew. Math., 179 (1938), 227-251.
[5] M. Eichler, Über die Idealklassenzahl hyperkomplexer Systeme, Math. Z., 43 (1938), 481-494.
[6] M. Eichler, Quaternäre quadratische Formen and die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math., 5 (1954), 355-366.
[7] M. Eichler, Modular correspondences and their representations. J. Indian Math. Soc., 20 (1956), 163-206.
[8] M. Eichler, Eine Verallgemeinerung der Abelschen Integrale, Math. Z., 67 (1957), 267-298.
[9] G. Fujisaki, On the zeta-function of the simple algebra over the field of rational numbers, J. Fac. Sci. Univ. Tokyo. Sec. I, vol. VII, Part 5 (1958), 567-604.
[10] R. Godement, Les fonctions ζ des algèbres simples II, Séminaire Bourbaki (février 1959, 176), 1-20.
[11] E. Hecke, Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann., 112 (1936), 664-699.
[12] E. Hecke, Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung, I, II, Math. Ann., 114 (1937), 1-28, 316-351.
[13] J. Igusa, Kroneckerian model of fields of elliptic modular functions, Amer. J. Math., 81 (1959), 561-577.
[14] S. Koizumi and G. Shimura, On specializations of abelian varieties, Sci. Papers, Coll. Gen. Ed. Univ. Tokyo, 9 (1959), 187-211.
[15] T. Matsusaka, Polarized varieties, fields of moduli and generalized Kummer varieties of polarized abelian varieties, Amer. J. Math., 80 (1958), 45-82.
[16] M. Nishi, Some results on abelian varieties, Nat. Sci. Rep., Ochanomizu Univ., 9 (1958), 1-12.
[17] H. Petersson, Konstruktion der sämtilichen Lösungen einer Riemannschen Funktionalgleichung durch Dirichlet-Reihen mit Eulerscher Produktentwickelung, I, II, III, Math. Ann., 116 (1939), 401-412; 117 (1940), 39-64, 277-300.
[18] A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc., 20 (1956), 47-87.
[19] A. Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to Function Theory, Tata Institute of Fundamental Research, Bombay, 1960, 147-164.
[20] J. -P. Serre, Groupes algébriques et corps de classes, Hermann, Paris, 1959.
[21] G. Shimura, Reduction of algebraic varieties with respect to a discrete valuation of the basic field, Amer. J. Math., 77 (1955), 134-176.
[22] G. Shimura, Correspondances modulaires et les fonctions ζ de courbes algébriques, J. Math. Soc. Japan, 10 (1958), 1-28.
[23]=[AF] G. Shimura, On the theory of automorphic functions, Ann. Math., 70 (1959), 101-144.
[24] G. Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan, 11 (1959), 291-311.
[25] G. Shimura and Y. Taniyama, Complex multiplication of abelian varieties and its applications to number theory, Publ. Math. Soc. Japan, No. 6, 1961.
[26] T. Tamagawa, On ζ-functions of a division algebra, to appear in Amer. J. Math.
[27] Y. Taniyama, L-functions of number fields and zeta functions of abelian varieties, J. Math. Soc. Japan, 9 (1957), 330-366.
[28] A. Weil, Sur les courbes algébriques et les variétés qui s'en déduisent Hermann, Paris 1948.
[29] A. Weil, Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948.
[30] A. Weil, Jacobi sums as “Grössencharaktere”, Trans. Amer. Math. Soc., 73 (1952), 487-495.
[31] A. Weil, On the theory of complex multiplication, Proc. Int. Symp. Alg. Nb. Th., Tokyo-Nikko, 1955, 9-22.
[32] A. Weil, The field of definition of a variety, Amer. J. Math., 78 (1956), 509-524.
[33] A. Weil, On discrete subgroups of Lie groups, Ann. Math. 72 (1960), 369-384,
[34] A. Weil, Adèles and algebraic groups, Lecture note, The Institute for Advanced Study, Princeton, 1961.
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