Laumon, G. Fourier transformat, constants of the functional equations and Weil conjecture. (Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil.) (French) Zbl 0641.14009 Publ. Math., Inst. Hautes Étud. Sci. 65, 131-210 (1987). In 1976, P. Deligne introduced a Fourier transformation into algebraic geometry, i.e. an involution for \(\ell\)-adic sheaves on the affine line (more generally on a vector bundle over a scheme of finite type) over a field \(k\) of finite characteristic \(p\ne \ell\) (depending on a character \(\mathbb F_p\to\mathbb Q_{\ell})\). The properties of this Fourier transformation have been studied by N. M. Katz and the author [Publ. Math., Inst. Hautes Étud. Sci. 62, 145–202 (1985; Zbl 0603.14015)]. In the present paper a local version of Fourier transformation is introduced, defined on \(\ell\)-adic representations of the Galois group of a local field of equal characteristic. At the end of the paper, these Fourier transformations are used to give a new proof of the main result of P. Deligne’s paper [Publ. Math., Inst. Haut. Étud. Sci. 52, 137–252 (1980; Zbl 0456.14014)] which in turn implies the part of the Weil conjectures which is the analog of the Riemann hypothesis. The main part of the paper is devoted to Grothendieck’s \(L\)-function for a complex \(K\) of \(\ell\)-adic sheaves on a curve \(X\) over \(k\). This \(L\)-function satisfies a functional equation for \(t\to t^{-1}\) in which a constant \(\varepsilon(X,K)\) appears. The main result of the paper under review expresses this constant as a product of local constants in the places of the curve. The proof of this product formula reduces to the case \(X=\mathbb P^1\) and then uses the Fourier transformation to investigate a one parameter deformation of a certain complex. The author also explains the relevance of his product formula for Langlands’ conjectures on the correspondence between \(L\)-functions and automorphic representations. Reviewer: Frank Herrlich (Karlsruhe) Cited in 8 ReviewsCited in 117 Documents MSC: 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14G15 Finite ground fields in algebraic geometry 11S37 Langlands-Weil conjectures, nonabelian class field theory 14G99 Arithmetic problems in algebraic geometry; Diophantine geometry Keywords:geometric Fourier transformation; involution for \(\ell \)-adic sheaves; Weil conjectures; Riemann hypothesis; L-function; Langlands’ conjectures Citations:Zbl 0603.14015; Zbl 0456.14014 × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] E. Artin andJ. Tate,Class field theory, New York, Benjamin (1967). · Zbl 1179.11040 [2] A. A. Beilinson, I. N. Bernstein etP. 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