Exponential sums and Newton polyhedra: cohomology and estimates
type: Item
instance of: scholarly article
title: Exponential sums and Newton polyhedra: cohomology and estimates
zbMATH Open document ID: 0723.14017
DOI: 10.2307/1971424
author: Alan Adolphson
author: Steven Sperber
published in: Annals of Mathematics. Second Series
publication date: 1989-01-01T00:00:00Z
review text: In this important paper the authors deal with exponential sums on varieties \(V\) of the form: \(V=({\mathbb C}_ m)^ r\times {\mathbb A}^ s (r+s=n)\). Namely, for a function \(f\in {\mathbb F}_ q[x_ 1,...,x_ n,(x_ 1...x_ r)^{-1}]\), and a nontrivial additive character \(\Psi\) of \({\mathbb F}_ q\), \(q=p^ a\), \(p=\text{char}({\mathbb F}_ q)\) they consider the sums \(S_ i=\sum_{x\in V({\mathbb F}_{q^ i})}\Psi \circ \text{Tr}_{{\mathbb F}_{q^ i}/\mathbb F_ q}(f(x)) \) and the corresponding \(L\)-function: \(L(t)=\exp \Bigl(\sum^{\infty}_{i=1}S_ it^ i/i\Bigr)\). -- Sums over more general varieties can be reduced to this situation. In the style of Dwork's study of the \(p\)-adic cohomology of smooth hypersurfaces of characteristic \(p\) the authors associate to \(V\) and \(f\) a complex of \(p\)-adic Banach spaces on which the Frobenius operates. This is constructed in such a way that the Dwork trace formula expresses \(L(t)\) as an alternate product of characteristic polynomials of Frobenius operating in cohomology. In previous papers, the authors were able to relate certain properties of \(L(t)\) to combinatorial invariants of the Newton polyhedron \(\Delta (f)\) of \(f\). In particular, degree and total degree of \(L(t)\) were bounded in terms of certain Euclidean volumes related to \(\Delta(f)\). The main technical point of the present paper consists in proving that under non-degeneracy conditions on \(f\), the afore-mentioned complex is acyclic in non-zero dimensions. The proof is based on ring-theoretic results of Kouchnirenko and Hochster. As a consequence the previous estimates on the degrees \(L(t)\) are shown to be sharp. Under nondegeneracy assumptions, a combinatorial lower bound for the Newton polygon of \(L(t)\) is also obtained. Finally, the previous \(p\)-adic results are used to determine the dimension of the \(\ell\)-adic cohomology spaces \(H^ i_ C({\mathbb C}^ n_ m\times {\bar {\mathbb F}}_ q,{\mathcal L}_{\Psi}(f))\) and conditions for \(H^ n_ C(…
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 4191864
zbMATH Keywords: exponential sums on varieties
zbMATH Keywords: L-function
zbMATH Keywords: Dwork trace formula
zbMATH Keywords: Newton polyhedron
zbMATH Keywords: \(\ell \)-adic cohomology spaces
zbMATH Keywords: zeta function of a regular hypersurface
reviewed by: Francesco Baldassarri
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971424
OpenAlex ID: W2081871695
Wikidata QID: Q105627557
description: scientific article; zbMATH DE number 4191864