On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne
type: Item
instance of: scholarly article
title: On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne
zbMATH Open document ID: 0769.14007
DOI: 10.2307/2946574
published in: Annals of Mathematics. Second Series
publication date: 1993-01-16T00:00:00Z
review text: In \textit{B. Dwork}'s proof of the rationality of the zeta function of a variety over a finite field [see Am. J. Math. 82, 631--648 (1960; Zbl 0173.48501)], the Lefschetz trace formula shown by \textit{A. Grothendieck} [see Sém. Bourbaki 17 (1964/65), Exposé 279 (1966; Zbl 0199.24802)], plays an essential rôle. In order to obtain finer information one likes to split up the total cohomology using algebraic cycles and to describe the Galois representations on the factors. The splitting process is described in terms of correspondences and one wants to have a Lefschetz trace formula for the twist of a correspondence by the Frobenius, or more special, a Lefschetz trace formula for the twist by all sufficiently large powers of a fixed Frobenius. In this context \textit{P. Deligne} has conjectured a certain Lefschetz trace formula in terms of correspondences over a finite field. This is shown in the present paper assuming the resolution of singularities holds. It generalizes L. Illusie's proof in the one-dimensional case [see \textit{A. Grothendieck} and \textit{L. Illusie} in Sémin. Géométrie algébrique 1965-1966, SGA 5, Lect. Notes Math. 589, Exposé III, 73--137 (1977; Zbl 0355.14004)]. For the smooth case of an arbitrary dimension there is an independent proof by \textit{E. Shpiz} [``Deligne's conjecture in the constant coefficient case'', Ph. D. Thesis (Harvard Univ. 1990)].
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 91440
zbMATH Keywords: rationality of the zeta function
zbMATH Keywords: Lefschetz trace formula
zbMATH Keywords: resolution of singularities
Wikidata QID: Q123028421
author: Richard Pink
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946574
OpenAlex ID: W2335125628
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description: scientific article; zbMATH DE number 91440