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Mosc. Math. J., 2002, Volume 2, Number 3, Pages 589–631 (Mi mmj65)  

This article is cited in 36 scientific papers (total in 37 papers)

Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields

S. R. Ghorpadea, G. Lachaudb

a Indian Institute of Technology
b Institut de Mathématiques de Luminy

Abstract: 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.

Key words and phrases: Étale cohomology, varieties over finite fields, complete intersections, Trace Formula, Betti numbers, zeta functions, Weak Lefschetz Theorems, hyperplane sections, motives, Lang–Weil inquality, Albanese variety.

DOI: https://doi.org/10.17323/1609-4514-2002-2-3-589-631

Full text: http://www.ams.org/.../abst2-3-2002.html
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Bibliographic databases:

MSC: 11G25, 14F20, 14G15, 14M10
Received: March 26, 2001; in revised form April 17, 2002
Language:

Citation: S. R. Ghorpade, G. Lachaud, “Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields”, Mosc. Math. J., 2:3 (2002), 589–631

Citation in format AMSBIB
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\by S.~R.~Ghorpade, G.~Lachaud
\paper \'Etale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields
\jour Mosc. Math.~J.
\yr 2002
\vol 2
\issue 3
\pages 589--631
\mathnet{http://mi.mathnet.ru/mmj65}
\crossref{https://doi.org/10.17323/1609-4514-2002-2-3-589-631}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1988974}
\zmath{https://zbmath.org/?q=an:1101.14017}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208593500006}


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