Rationalité et valeurs de fonctions L en cohomologie cristalline. (Rationality and values of L-functions in crystalline cohomology)
type: Item
instance of: scholarly article
title: Rationalité et valeurs de fonctions L en cohomologie cristalline. (Rationality and values of L-functions in crystalline cohomology)
zbMATH Open document ID: 0624.14016
author: J.-Y. Etesse
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/74817
review text: In Sémin. Bourbaki 1971-72, Exposé 409, Lect. Notes Math. 317, 167- 200 (1973; Zbl 0259.14007), \textit{N. Katz} conjectured the p-adic meromorphy of the function L(X,E,t) attached to a smooth variety X over a finite field \({\mathbb{F}}_ q (q=p^ a)\) and to an F-crystal E on X. If X is proper and smooth over \({\mathbb{F}}_ q\) we prove that L is rational and given by the usual formula using the action of Frobenius on crystalline cohomology with coefficients in E; this result was only known, via ``Weil conjectures'', for particular unit-root F-crystals: those issued of a representation of \(\pi _ 1(X)\) through a finite quotient. The proof of the theorem involves the formalism of a cohomology class associated to a morphism of crystals, extending the fundamental class of an algebraic cycle, and leading to a Lefschetz trace formula. When E is a unit-root F- crystal the link between crystalline cohomology and De Rham-Witt complex with coefficients in E enables us to interpret zeroes and poles of L of the form \(t=q^{-r}\), r an integer. Under certain hypotheses this complex yields also equivalents of the L function in the neighbourhood of the preceding poles: these results extend those of Milne for zeta functions.
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 4012803
zbMATH Keywords: fundamental group
zbMATH Keywords: crystalline cohomology
zbMATH Keywords: De Rham-Witt complex
zbMATH Keywords: rationality of L-functions
MaRDI profile type: Publication
OpenAlex ID: W2004258321
cites work: On values of zeta functions and \(\ell\)-adic Euler characteristics
cites work: Cohomologie cristalline des schemas de caractéristique \(p >0\)
cites work: Q3213240
cites work: Q3922790
cites work: Q3048930
cites work: Notes on Crystalline Cohomology. (MN-21)
cites work: F-isocrystals and De Rham cohomology. I
cites work: Algèbre commutative
cites work: Q3240881
cites work: L-functions of p-adic characters and geometric Iwasawa theory
cites work: Q3781207
cites work: Torsion dans le groupe de Chow de codimension deux
cites work: Q3490119
cites work: Dualité plate pour les surfaces à coefficients dans un groupe de type multiplicatif
cites work: Q5602063
cites work: Classes de Chern et classes de cycles en cohomologie de Hodge-Witt logarithmique
cites work: Q3693617
cites work: Q5597388
cites work: Complexe de de\thinspace Rham-Witt et cohomologie cristalline
cites work: Les suites spectrales associées au complexe de De Rham-Witt
cites work: Q3967645
cites work: Q3215315
cites work: Some consequences of the Riemann hypothesis for varieties over finite fields
cites work: Q4071346
cites work: Q3682595
cites work: Values of zeta-functions at non-negative integers
cites work: On a conjecture of Artin and Tate
cites work: Duality in the flat cohomology of a surface
cites work: Q3871006
cites work: Values of Zeta Functions of Varieties Over Finite Fields
cites work: Q3863990
cites work: Categories tannakiennes
cites work: Q4740668
cites work: Q5583117
cites work: Q3277839
cites work: Cohomologie étale. Seminaire de géométrie algébrique du Bois-Marie SGA 4 1/2 par P. Deligne, avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier
cites work: Q1233007
cites work: Q5598116
DOI: 10.5802/AIF.1149
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 4012803