Higher Weights of Grassmann Codes

Number of Solutions of Equations over Finite Fields and a Conjecture of Lang and Weil1


Sudhir R. Ghorpade 2

Department of Mathematics
Indian Institute of Technology, Bombay,
Powai, Mumbai 400076, India
E-Mail: srg@math.iitb.ac.in

and

Gilles Lachaud

Équipe ``Arithmétique et Théorie de l'Information''
Institut de Mathématiques de Luminy
Luminy Case 907, 13288 Marseille, Cedex 9, France
E-Mail: lachaud@iml.univ-mrs.fr


Abstract

A brief survey of the conjectures of Weil and some classical estimates for the number of points of varieties over finite fields is given. The case of partial flag manifolds is discussed in some details by way of an example. This is followed by a motivated account of some recent results on counting the number of points of varieties over finite fields, and a related conjecture of Lang and Weil. Explicit combinatorial formulae for the Betti numbers and the Euler characteristics of smooth complete intersections are also discussed.


Contents


1 Introduction 1
2 Weil Conjectures and Classical Estimates 2
3 An Example 7
4 Complete Intersections over Finite Fields 11
5 On the Lang-Weil Inequality 14
5 Conjectural Statements of Lang and Weil 15
References 19

1 2000 Mathematics Subject Classification. 11G25, 14F20, 14G15, 14M10.
2 Partially supported by the IRCC grant 97IR012 from IIT Bombay.

This paper is published in: Number Theory and Discrete Mathematics (Chandigarh, 2000), A. K. Agarwal, B. Berndt, C. Krattenthaler, G. L. Mullen, K Ramachandra and M. Waldschmidt Eds., Hindustan Book Agency, New Delhi (2002), pp. 269-291 [Co-published outside India as a volume in the Trends in Mathematics series by Birkhäuser, Basel.]

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