  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.
