Transformation de Fourier et majoration de sommes exponentielles. (Fourier transformation and majoration of exponential sums.)
type: Item
instance of: scholarly article
title: Transformation de Fourier et majoration de sommes exponentielles. (Fourier transformation and majoration of exponential sums.)
zbMATH Open document ID: 0603.14015
DOI: 10.1007/BF02698808
author: Nicholas M. Katz
author: Gérard Laumon
published in: Publications Mathématiques
publication date: 1985-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104006
review text: For a finite field \(k={\mathbb{F}}_ q\) of characteristic \(p\) one has the Lang torsor \((=\) principal fibre bundle) \({\mathcal L}\) on the affine line \({\mathbb{A}}^ 1_ k=Spec k[U]\), defined by the equation \(V^ q-V=U\) and for \(\ell \neq p\) the \({\bar {\mathbb{Q}}}^{\times}_{\ell}\)-valued additive characters \(\psi\) of \({\mathbb{F}}_ q\) give a set of \({\bar {\mathbb{Q}}}^{\times}_{\ell}\)-bundles \({\mathcal L}_{\psi}\) (Artin-Schreier sheaves) on the line. Let E be the \({\mathbb{F}}_ q\)-vector space \({\mathbb{F}}^ r_ q\) of rank r and \(E^{\vee}\) its dual. The category D(E,\({\bar {\mathbb{Q}}}_{\ell})\) is the derived category of \({\bar {\mathbb{Q}}}_{\ell}\)-sheaves on E (bounded complexes with negative degrees only and residual cohomology with constructible supports). One then has the \(\ell\)-adic Fourier transformation \({\mathcal F}: D(E,{\bar {\mathbb{Q}}}_{\ell})\to D(E^{\vee},{\bar {\mathbb{Q}}}_{\ell}):\) the definition formula for the elementary Fourier transform \({\mathcal F}(K)=<K,\psi >\) now reads like \({\mathcal F}(K)=Rpr_*^{\vee}(pr^*K^ L\otimes \mu^*{\mathcal L}_{\psi})[r],\) where \(\mu\) is the evaluation map \(E^{\vee}\times E\to {\mathbb{A}}^ 1\). Actually E may, more generally, be a vector bundle on a good \({\mathbb{F}}_ q\)-scheme. \({\mathcal F}\) commutes with duality. One proves a Riemann-Roch theorem for objects K adapted to a given stratification of E, relating the Euler-Poincaré characteristics \(\chi(F(K))\) and \(\chi(K)\) which are defined as functions on \(E^{\vee}\) and E. If in the definition formula for \(\chi\) (K) the signs \((-)^ i\) are all replaced by \(+\), one gets a function on E, denoted by \(\| K\|\). One of the main results is a uniformity theorem: \((\sup \| {\mathcal F}(K)\|)\leq \text{const. }(\sup \| K\|)\), where the suprema are taken along a fibre of \(E^{\vee}\) and E, with an integer constant determined by E (which is allowed to be given as a vector bundle over a \({\mathbb{Z}}\)-scheme…
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 43A32
zbMATH DE Number: 3973035
zbMATH Keywords: \(\ell \)-adic Fourier transformation
Wikidata QID: Q102395509
reviewed by: J. H. de Boer
MaRDI profile type: Publication
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OpenAlex ID: W2057306691
description: scientific article; zbMATH DE number 3973035