Number of points of Deligne-Lusztig surfaces
type: Item
instance of: scholarly article
title: Number of points of Deligne-Lusztig surfaces
zbMATH Open document ID: 0997.11046
published in: Journal of Algebra
publication date: 2002-05-11T00:00:00Z
review text: This paper contains detailed proofs of results previously announced by the author [C. R. Acad. Sci., Paris, Sér. I 322, 563-566 (1996; Zbl 0918.11034)]. The author determines the zeta functions and Betti numbers for smooth compactifications of Deligne-Lusztig surfaces associated to various connected, reductive algebraic groups over an algebraic closure of a finite field \({\mathbb F}_q\). In most of the examples, the number of rational points over \({\mathbb F}_q\) or over \({\mathbb F}_{q^2}\) attains the bound of Weil-Deligne. The author also shows that the surfaces associated to groups of type \({}^2A_4\) are minimal surfaces of general type.
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 14J20
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1471039
zbMATH Keywords: Deligne-Lusztig variety
zbMATH Keywords: zeta function
zbMATH Keywords: Weil-Deligne bound
zbMATH Keywords: Betti numbers
author: François Rodier
reviewed by: Robert F. Lax
MaRDI profile type: Publication
full work available at URL: jabr.1999.8246
OpenAlex ID: W2027013228
cites work: Hermitian Varieties in a Finite Projective Space PG(N, q2)
cites work: Q4117948
cites work: Q5662096
cites work: Q3907722
cites work: La conjecture de Weil. I
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: Representations of reductive groups over finite fields
cites work: Fonctions $L$ des variétés de Deligne-Lusztig et descente de Shintani
cites work: Q4027640
cites work: Canonical bundles of Deligne-Lusztig varieties
cites work: Q4143433
cites work: Q4003762
cites work: Q4744461
cites work: Q3999066
cites work: Q4367085
cites work: Coxeter orbits and eigenspaces of Frobenius
cites work: On the Green Polynomials of Classical Groups
cites work: Q5582831
cites work: Q4879977
cites work: Q4302492
cites work: Representations of finite Chevalley groups. A survey
cites work: Algebraic function fields and codes
cites work: Q4336979
cites work: Numbers of solutions of equations in finite fields
DOI: 10.1006/JABR.1999.8246
Recommended article: Q4879977
Recommended article: Counting lattice points in compactified moduli spaces of curves
Recommended article: Counting lattice points in the moduli space of curves
Recommended article: Picard numbers of Delsarte surfaces
Recommended article: Q4336979
Recommended article: Counting rational points on del Pezzo surfaces with a conic bundle structure
Recommended article: Q3545370
Recommended article: The enumerative geometry of Del Pezzo Surfaces via degenerations
Recommended article: Counting points on a toric variety
Recommended article: Numbers of points of surfaces in the projective 3-space over finite fields
description: scientific article; zbMATH DE number 1471039