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

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

2005-01-12

arxiv.org/abs/math/0501181v2

Mathematics

Number Theory

Added a section containing applications to low degree intersections in homogenous spaces

Scientific paper

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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