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Zbl 0584.12007
Brattström, Gudrun; Lichtenbaum, Stephen
Jacobi-sum Hecke characters of imaginary quadratic fields. (English)
[J] Compos. Math. 53, 277-302 (1984). ISSN 0010-437X; ISSN 1570-5846/e

Let k be an abelian number field. The point of this paper is to formulate a conjecture on the value at zero of the L-series of certain Jacobi-sum Hecke characters of k, and to verify the conjecture for the rationals and for imaginary quadratic fields with odd class number. \par Roughly speaking, the conjecture says that if $\psi$ is a Jacobi-sum Hecke character which is "critique" [{\it P. Deligne}, Proc. Symp. Pure Math. 33, No.2, 313-346 (1979; Zbl 0449.10022)], then the value at zero of the L-series of $\psi$ is equal, up to a rational number, to the inverse of an explicit product, depending on $\psi$, of values of the $\Gamma$-function at rational numbers, multiplied by the square root of the discriminant of the maximal real subfield of k. \par There is an analogous conjecture for k totally real abelian which has been established by the first author [Sémin. Théor. Nombres, Univ. Bordeaux I 1981/1982, Exposé No.22 (1982; Zbl 0528.12011)]. Further, {\it G. Anderson} [The motivic interpretation of Jacobi-sum Hecke characters] has shown that the conjecture of the current paper follows from Deligne's conjecture in the article cited above. \par Among the intermediate results are the determination of the Jacobi-sum Hecke characters for the rationals and for imaginary quadratic fields of class number one (other than ${\bbfQ}(\sqrt{-1})$, ${\bbfQ}(\sqrt{-2})$, ${\bbfQ}(\sqrt{-3}) )$, and a version of Damerell's theorem [{\it R. M. Damerell}, Acta Arith. 17, 287-301 (1970; Zbl 0209.246)] which, for imaginary quadratic fields of odd class number, computes values of the L- function of a Hecke character up to an element of the field itself, rather than up to an algebraic number.
[L.G.Roberts]

MSC 2000:: *11R42 Zeta functions and L-functions of global number fields
11R11 Quadratic extensions

Keywords: motives; value at zero; L-series; Jacobi-sum Hecke characters; imaginary quadratic fields; odd class number; Deligne's conjecture

Citations: Zbl 0449.10022; Zbl 0528.12011; Zbl 0209.246

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