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On the grössencharacter of an abelian variety in a parametrized family


Author: Robert S. Rumely
Journal: Trans. Amer. Math. Soc. 276 (1983), 213-233
MSC: Primary 14K22; Secondary 10D20, 10D25, 14K15, 14K25
MathSciNet review: 684504
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Abstract: We consider families of abelian varieties parametrized by classical theta-functions, and show that specifying the family and a CM point in Siegel space determines the grössencharacter of the corresponding CM abelian variety. We associate an adelic group to the family, and describe the kernel of the grössencharacter as the pull-back of the group under the map in Shimura's Reciprocity Law.


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  • [1] Numerical tables on elliptic curves, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1975, pp. 74–144. Lecture Notes in Math., Vol. 476. Prepared by H. P. F. Swinnerton-Dyer, N. M. Stephens, James Davenport, J. Vélu, F. B. Coghlan, A. O. L. Atkin and D. J. Tingley. MR 0389726
  • [2] K. J. Bobek, Einleitung in die Theorie der Elliptischen Funktionen, Teubner, Leipzig, 1884.
  • [3] Max Deuring, Die Zetafunktion einer algebraischen Kurve vom Geschlechte Eins, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt. 1953 (1953), 85–94 (German). MR 0061133
  • [4] Jun-ichi Igusa, Fibre systems of Jacobian varieties. III. Fibre systems of elliptic curves, Amer. J. Math. 81 (1959), 453–476. MR 0104669
  • [5] Serge Lang, Elliptic functions, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Amsterdam, 1973. With an appendix by J. Tate. MR 0409362
  • [6] A. P. Ogg, Elliptic curves and wild ramification, Amer. J. Math. 89 (1967), 1–21. MR 0207694
  • [7] Robert S. Rumely, A formula for the grössencharacter of a parametrized elliptic curve, J. Number Theory 17 (1983), no. 3, 389–402. MR 724537, 10.1016/0022-314X(83)90056-2
  • [8] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. Kan\cflex o Memorial Lectures, No. 1. MR 0314766
  • [9] Goro Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 91 (1970), 144–222. MR 0257031
  • [10] Goro Shimura, Theta functions with complex multiplication, Duke Math. J. 43 (1976), no. 4, 673–696. MR 0424705
  • [11] Goro Shimura, On the derivatives of theta functions and modular forms, Duke Math. J. 44 (1977), no. 2, 365–387. MR 0466028
  • [12] Goro Shimura, On certain reciprocity-laws for theta functions and modular forms, Acta Math. 141 (1978), no. 1-2, 35–71. MR 0491518
  • [13] Goro Shimura, On abelian varieties with complex multiplication, Proc. London Math. Soc. (3) 34 (1977), no. 1, 65–86. MR 0572987
  • [14] Goro Shimura, On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J. 43 (1971), 199–208. MR 0296050
  • [15] Goro Shimura, On the zeta-function of an abelian variety with complex multiplication., Ann. of Math. (2) 94 (1971), 504–533. MR 0288089
  • [16] Goro Shimura and Yutaka Taniyama, Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, The Mathematical Society of Japan, Tokyo, 1961. MR 0125113
  • [17] Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 0236190
  • [18] Jean-Pierre Serre, Corps locaux, Hermann, Paris, 1968 (French). Deuxième édition; Publications de l’Université de Nancago, No. VIII. MR 0354618
  • [19] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1975, pp. 33–52. Lecture Notes in Math., Vol. 476. MR 0393039
  • [20] H. Weber, Lehrbuch der algebra, vol. 3, Vieweg & Sohn, Braunschweig, 1908.
  • [21] André Weil, Jacobi sums as “Grössencharaktere”, Trans. Amer. Math. Soc. 73 (1952), 487–495. MR 0051263, 10.1090/S0002-9947-1952-0051263-0

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0684504-7
Keywords: Abelian variety, theta-function, complex multiplication, grössencharacter
Article copyright: © Copyright 1983 American Mathematical Society