Calculation of \(\ell \)-adic local Fourier transformations
type: Item
instance of: scholarly article
title: Calculation of \(\ell \)-adic local Fourier transformations
zbMATH Open document ID: 1206.14035
author: Lei Fu
published in: Manuscripta Mathematica
publication date: 2010-11-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0702436
review text: The global \(\ell\)-adic Fourier transform was first introduced by Deligne. To study the local behavior of the global Fourier transform, Laumon discovered the stationary phase principle and introduced local Fourier transformations. Laumon and Malgrange gave conjectural formulas of local Fourier transformations for a certain class of \(\overline{\mathbb Q}_{\ell}\)-sheaves. In this paper the local Fourier transformations are calculated for a more general class of \(\overline{\mathbb Q}_{\ell}\)-sheaves; in particular, the conjecture of Laumon and Malgrange (in fact: a slight correction of it) is proved. As an application, the local monodromy of \(\ell\)-adic hypergeometric sheaves as introduced by Katz is calculated. The methods used in this paper can also be used to show that the Fourier transform of an analytic sheaf with meromorphic ramification still has meromorphic ramification, cf. [\textit{L. Ramero}, J. Algebr. Geom. 7, No. 3, 405--504 (1998; Zbl 0964.14019)]. Some of the results in this paper, and in fact more general versions of them, have also been obtained by \textit{A. Abbes} and \textit{T. Saito} [Compos. Math. 146, No. 6, 1507--1551 (2010; Zbl 1206.14034)], but the methods in the present paper are more direct and global.
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 5814289
zbMATH Keywords: local Fourier transform
zbMATH Keywords: local monodromy
zbMATH Keywords: hypergeometric sheaves
zbMATH Keywords: Turritin-Levelt Theorem
reviewed by: Elmar Große-Klönne
MaRDI profile type: Publication
OpenAlex ID: W1992716702
arXiv ID: math/0702436
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DOI: 10.1007/S00229-010-0377-X
description: scientific article