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Mathematische Annalen

, Volume 333, Issue 4, pp 797–809 | Cite as

Congruences for rational points on varieties over finite fields

  • N. Fakhruddin
  • C. S. RajanEmail author
Article

Abstract

We prove the existence of rational points on singular varieties over finite fields arising as degenerations of smooth proper varieties with trivial Chow group of 0-cycles. We also obtain congruences for the number of rational points of singular varieties appearing as fibres of a proper family with smooth total and base space and such that the Chow group of 0-cycles of the generic fibre is trivial. In particular this leads to a vast generalization of the classical Chevalley-Warning theorem. The above results are obtained as special cases of our main theorem which can be viewed as a relative version of a theorem of H. Esnault on the number of rational points of smooth proper varieties over finite fields with trivial Chow group of 0-cycles.

Keywords

Rational Point Finite Field Base Space Relative Version Generic Fibre 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Tata Institue of Fundamental ResearchSchool of MathematicsMumbaiIndia

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