Congruences for rational points on varieties over finite fields

N. Fakhruddin, C. S. Rajan

Published 2004-02-13, updated 2004-07-07Version 5

We show that the number of rational points on the fibres of a proper morphism of smooth varieties over a finite field k whose generic fibre has a ``trival'' Chow group of zero cycles is congruent to 1 mod |k|. As a consequence we prove that there is a rational point on any degeneration of a smooth proper rationally chain connected variety over a finite field. We also obtain a generalisation of the Chevalley-Warning theorem.