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.
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Fakhruddin, N., Rajan, C. Congruences for rational points on varieties over finite fields. Math. Ann. 333, 797–809 (2005). https://doi.org/10.1007/s00208-005-0697-4
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DOI: https://doi.org/10.1007/s00208-005-0697-4