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Chow groups and higher congruences for the number of rational points on proper varieties over finite fields

Najmuddin Fakhruddin
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Abstract

Given a proper family of varieties over a smooth base, with smooth total space and general fibre, all over a finite field k with q elements, we show that a finiteness hypothesis on the Chow groups, CH_i, i=0,1,...,r, of the fibres in the family leads to congruences mod q^{r+1} for the number of rational points in all the fibres over k-rational points of the base. These hypotheses on the Chow groups are expected to hold for families of low degree intersections in many Fano varieties leading to a broad generalisation of the theorem of Ax--Katz, as well as results of the author and C. S. Rajan. As an unconditional application, we give an asymptotic generalisation of the Ax--Katz theorem to low degree intersections in a large class of homogenous spaces.

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BibTeXRIS

Najmuddin Fakhruddin. 2006-03-30. Chow groups and higher congruences for the number of rational points on proper varieties over finite fields. https://arxiv.org/abs/math/0501181

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