Number of Points of Singular Varieties over Finite Fields

Étale Cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields1

Sudhir R. Ghorpade 2

Department of Mathematics
Indian Institute of Technology, Bombay,
Powai, Mumbai 400076, India



Gilles Lachaud

Équipe ``Arithmétique et Théorie de l'Information''
Institut de Mathématiques de Luminy
Luminy Case 907, 13288 Marseille, Cedex 9, France



We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes also the classical Lang-Weil inequality. Moreover, we prove the Lang-Weil inequality for affine as well as projective varieties with an explicit description and a bound for the constant appearing therein. We also prove a conjecture of Lang and Weil concerning the Picard varieties and étale cohomology spaces of projective varieties. The general inequality for complete intersections may be viewed as a more precise version of the estimates given by Hooley and Katz. The proof is primarily based on a suitable generalization of the Weak Lefschetz Theorem to singular varieties together with some Bertini-type arguments and the Grothendieck-Lefschetz Trace Formula. We also describe some auxiliary results concerning the étale cohomology spaces and Betti numbers of projective varieties over finite fields and a conjecture along with some partial results concerning the number of points of projective algebraic sets over finite fields.


Introduction 2
1 Singular Loci and Regular Flags 4
2 Weak Lefschetz Theorem for Singular Varieties 6
3 Cohomology of Complete Intersections 10
4 The Central Betti Number of Complete Intersections 12
5 Zeta Functions and the Trace Formula 14
6 Number of Points of Complete Intersections 16
7 Complete Intersections with isolated singularities 18
8 The Penultimate Betti Number 21
9 Cohomology and Albanese Varieties 24
10 A Conjecture of Lang and Weil 28
11 On the Lang-Weil Inequality 32
12 Number of Points of Algebraic Sets 35
References 36

1 2000 Mathematics Subject Classification. 11G25, 14F20, 14G15, 14M10.
2 Partially supported by a Career Award from AICTE, New Delhi and an IRCC grant from IIT Bombay.

This paper is published in: Moscow Mathematical Journal, Vol. 2, No. 3 (2002), pp. 589--631.
Preliminary versions of this paper have appeared in preprint form as Prétirage No. 99-13 (April 1999) and Prétirage No. 2001-32 (July 2001) of the Institut de Mathématiques de Luminy, Marseille, France.

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