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Zbl 0678.12004
Henniart, Guy
Cyclotomy et values of the $\Gamma$-function [following G. Anderson]. (Cyclotomie et valeurs de la fonction $\Gamma$ [d'après G. Anderson].) (French)
[A] Sémin. Bourbaki, 40ème Année, Vol. 1987/88, Astérisque 161-162, 53-72, Exp. No. 688 (1988).

[For the entire collection see Zbl 0659.00006.] \par The present paper is a Bourbaki talk on the theorems and techniques of Greg Anderson in the theory of cyclotomic numbers with applications to special values of Euler's gamma function. This is a very rich area with contributions by a number of mathematicians. \par Anderson's techniques involve studying the cohomology of Fermat hypersurfaces in the general context of ``motives''. Very roughly speaking, motives are the ``universal cohomology theory'' in that all cohomology theories should factor through this construction. As has been known since work of A. Weil, the cohomology of Fermat hypersurfaces is related to Jacobi sums. Anderson enlarges the category of motives (to the category of ``ulterior motives'') by allowing cohomology to have ``fractional'' weights (as opposed to ``usual'' cohomology of varieties which involves only integral weights). In this context, one can actually obtain the Gauss sums themselves, and Anderson is able to give elegant proofs of many important results. \par Recent papers of the author entitled ``The hyperadelic gamma function'' [Invent. Math. 95, No. 1, 63--131 (1989; Zbl 0682.14011)] and ``Normalization of the hyperadelic gamma function'' [Galois groups over $\Bbb Q$, Proc. Workshop, Berkeley/CA (USA) 1987, Publ., Math. Sci. Res. Inst. 16, 1--31 (1989; Zbl 0706.11066)], push the theory of ``factorization'' described in the present work quite a bit farther.
[David Goss (Columbus/Ohio)]

MSC 2000:: *11F67 Special values of automorphic L-series, etc
11G40 L-functions of varieties over global fields
11R42 Zeta functions and L-functions of global number fields
14G10 Zeta-functions and related questions
14G25 Global ground fields

Keywords: cyclotomic numbers; gamma function; motives; Jacobi sums; Gauss sums

Citations: Zbl 0659.00006; Zbl 0706.11066; Zbl 0682.14011

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