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


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.


Rational Point Finite Field Base Space Relative Version Generic Fibre 
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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Tata Institue of Fundamental ResearchSchool of MathematicsMumbaiIndia

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