Weights of exponential sums, intersection cohomology, and Newton polyhedra
type: Item
instance of: scholarly article
title: Weights of exponential sums, intersection cohomology, and Newton polyhedra
zbMATH Open document ID: 0763.14025
DOI: 10.1007/BF01243914
published in: Inventiones Mathematicae
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/143943
review text: To any morphism \(f\) defined over the finite field \(k\) with \(q\) elements from an \(n\)-dimensional torus to the affine line, we can associate the Newton polyhedron of \(f\) at infinity. Let \(L\) be the \(\ell\)-adic sheaf on the affine line over \(k\) associated to a fixed nontrivial additive character on \(k\) and the Artin-Schreier covering \(t^ q-t=x\). Suppose that \(f\) is nondegenerate with respect to its Newton polyhedron, and that the Newton polyhedron has dimension \(n\). In this paper it is proved, using the intersection cohomology of toric varieties, that the \(i\)-th cohomology of the torus with compact supports and coefficients in \(f*L\) is zero if \(i\neq n\), and if \(i=n\) the polynomial whose \(w\)-th coefficient is the number of eigenvalues of the Frobenius action with absolute value \(q^{w/z}\) is calculated in terms of the Newton polyhedron.
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 11L99
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 59803
zbMATH Keywords: toroidal compactification
zbMATH Keywords: Newton polyhedron
zbMATH Keywords: intersection cohomology
zbMATH Keywords: toric varieties
author: Jan Denef
author: François Loeser
reviewed by: Frances Kirwan
MaRDI profile type: Publication
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OpenAlex ID: W2009907775
description: scientific article; zbMATH DE number 59803