Abstract
In this paper we compute the values of L-series of Jacobi-sum Hecke characters in terms of values of the Γ-function at rational numbers. The computation is done only up to algebraic numbers, and we assume that the Hecke character is in the “good range.” We may make a more refined Statement (the Γ-hypothesis), which actually predicts the values up to rational numbers, and which has been verified in the totally real case ([B]) and in the case of imaginary quadratie fields with odd class number ([L], [B]). Here we content ourselves with the algebraicity Statement, but prove it for all abelian fields.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Brattstrom, G. “L-Functions of Jacobi-Sum Hecke Characters,” Thesis, Cornel1 University, 1981.
Deligne, P. “Valeurs de fonctions L et periodes d1integrales, Prooeedings of Symposia in Pure Mathematics 33 (1978), 313–316.
Gross, B.H. “On the Periods of Abelian Integrals and a Formula of Chowla and Seiberg (Appendix by D. E. Rohrlich),” Invent. Math. b5 (1978), 193–211.
Katz, N. “p-Adic L-Functions for CM Fields,” Invent. Math. kS (1978), 199–297.
Kubert, D., and Lichtenbaum, S. “Jacobi-Sum Hecke Characters and Gauss-Sum Identities,” Compositio Mathematioa (to appear).
Lichtenbaum, S. “Jacobi-Sum Hecke Characters of Imaginary Quadratie Fields,” (Preprint).
Shimura, G. “On Some Arithmetic Properties of Modular Forms of One and Several Variables,” Journal of Math. 102 (1975), 491–515.
Shimura, G. “Automorphic Forms and the Periods of Abelian Varieties,” Journal of the Math. Sog. of Japan 31 (1979), 561–592.
Shimura, G., and Taniyama, Y. “Complex Multiplication of Abelian Varieties and Its Applications to Number Theory,” Pub. Math. Soc. Japan 6, Math. Soc. Japan, Tokyo, 1961.
Weil, A. “Jacobi Sums as 1Grössencharaktere,” Trans. Am. Math. Sog. 75 (1952), 487–495.
Weil, A. “Sommes de Jacobi et Caracteres de Hecke,” Nach. Akad. Wiss. Göttingen (1974), 1–14.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1982 Springer Science+Business Media New York
About this chapter
Cite this chapter
Lichtenbaum, S. (1982). Values of L-Functions of Jacobi-Sum Hecke Characters of Abelian Fields. In: Koblitz, N. (eds) Number Theory Related to Fermat’s Last Theorem. Progress in Mathematics, vol 26. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6699-5_13
Download citation
DOI: https://doi.org/10.1007/978-1-4899-6699-5_13
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-3104-8
Online ISBN: 978-1-4899-6699-5
eBook Packages: Springer Book Archive