Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On the grössencharacter of an abelian variety in a parametrized family
HTML articles powered by AMS MathViewer

by Robert S. Rumely
Trans. Amer. Math. Soc. 276 (1983), 213-233
DOI: https://doi.org/10.1090/S0002-9947-1983-0684504-7

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.
References
  • Numerical tables on elliptic curves, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 74–144. 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
  • K. J. Bobek, Einleitung in die Theorie der Elliptischen Funktionen, Teubner, Leipzig, 1884.
  • 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 61133
  • Jun-ichi Igusa, Fibre systems of Jacobian varieties. III. Fibre systems of elliptic curves, Amer. J. Math. 81 (1959), 453–476. MR 104669, DOI 10.2307/2372751
  • Serge Lang, Elliptic functions, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Amsterdam, 1973. With an appendix by J. Tate. MR 0409362
  • A. P. Ogg, Elliptic curves and wild ramification, Amer. J. Math. 89 (1967), 1–21. MR 207694, DOI 10.2307/2373092
  • Robert S. Rumely, A formula for the grössencharacter of a parametrized elliptic curve, J. Number Theory 17 (1983), no. 3, 389–402. MR 724537, DOI 10.1016/0022-314X(83)90056-2
  • Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1971. MR 0314766
  • Goro Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 91 (1970), 144–222. MR 257031, DOI 10.2307/1970604
  • Goro Shimura, Theta functions with complex multiplication, Duke Math. J. 43 (1976), no. 4, 673–696. MR 424705
  • Goro Shimura, On the derivatives of theta functions and modular forms, Duke Math. J. 44 (1977), no. 2, 365–387. MR 466028
  • Goro Shimura, On certain reciprocity-laws for theta functions and modular forms, Acta Math. 141 (1978), no. 1-2, 35–71. MR 491518, DOI 10.1007/BF02545742
  • Goro Shimura, On abelian varieties with complex multiplication, Proc. London Math. Soc. (3) 34 (1977), no. 1, 65–86. MR 572987, DOI 10.1112/plms/s3-34.1.65
  • 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 296050, DOI 10.1017/S0027763000014471
  • Goro Shimura, On the zeta-function of an abelian variety with complex multiplication, Ann. of Math. (2) 94 (1971), 504–533. MR 288089, DOI 10.2307/1970768
  • 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, Mathematical Society of Japan, Tokyo, 1961. MR 0125113
  • Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI 10.2307/1970722
  • Jean-Pierre Serre, Corps locaux, Publications de l’Université de Nancago, No. VIII, Hermann, Paris, 1968 (French). Deuxième édition. MR 0354618
  • 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) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 33–52. MR 0393039
  • H. Weber, Lehrbuch der algebra, vol. 3, Vieweg & Sohn, Braunschweig, 1908.
  • André Weil, Jacobi sums as “Grössencharaktere”, Trans. Amer. Math. Soc. 73 (1952), 487–495. MR 51263, DOI 10.1090/S0002-9947-1952-0051263-0
Similar Articles
Bibliographic Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 276 (1983), 213-233
  • MSC: Primary 14K22; Secondary 10D20, 10D25, 14K15, 14K25
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0684504-7
  • MathSciNet review: 684504