Fourier transformat, constants of the functional equations and Weil conjecture (Q1100528)

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 \textit{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 \textit{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.