Kloosterman sums as algebraic integers
type: Item
instance of: scholarly article
title: Kloosterman sums as algebraic integers
zbMATH Open document ID: 0816.11065
DOI: 10.1007/BF01446641
author: Benji Fisher
published in: Mathematische Annalen
publication date: 1995-07-23T00:00:00Z
full work available at URL: https://eudml.org/doc/165303
review text: Let \(p\) be a prime, \(q\) a power of \(p\), \(\mathbb{F}_ q\) the field with \(q\) elements, tr the trace from \(\mathbb{F}_ q\) to \(\mathbb{F}_ p\), and \(\zeta_ p= e^{2\pi i/p}\). The Kloosterman sums \[ \text{Kl} (\mathbb{F}_ q, a)= \sum_{xy=a} \zeta_ p^{\text{tr} (x+y)} \qquad (a\in \mathbb{F}_ q^ \times) \] are integers in the number field \(\mathbb{Q} [\zeta_ p ]\cap \mathbb{R}= \mathbb{Q} [\zeta_ p+ \zeta_ p^{-1}]\), so can be considered as elements of a lattice in \(\mathbb{R}^{(p- 1)/2}\) (assuming \(p>2\)) via the Minkowski embedding. After normalization, it is shown that the \(q-1\) Kloosterman sums for \(\mathbb{F}_ q\) are ``approximately equidistributed'', with a known distribution, in \([-2, 2]^{(p- 1)/2}\). This refines a theorem of Katz, according to which these sums are approximately equidistributed along each axis, i.e., via any one complex embedding of \(\mathbb{Q}[ \zeta_ p+ \zeta_ p^{-1}]\). Analogous results hold for generalized Kloosterman sums in several variables.
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11L05
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 11R04
zbMATH DE Number: 759279
zbMATH Keywords: algebraic integers
zbMATH Keywords: approximate equidistribution
zbMATH Keywords: Deligne's equidistribution theorem
zbMATH Keywords: Tate twist of Kloosterman sheaf
zbMATH Keywords: monodromy group
zbMATH Keywords: finite fields
zbMATH Keywords: number field
zbMATH Keywords: lattice
zbMATH Keywords: Minkowski embedding
zbMATH Keywords: Kloosterman sums
zbMATH Keywords: generalized Kloosterman sums in several variables
MaRDI profile type: Publication
cites work: Q4119274
cites work: Q4035809
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OpenAlex ID: W2012662010
description: scientific article; zbMATH DE number 759279