Currently available as a preprint.

We show that if * m * > 0 and * d* > 2 are
integers with *m* < (4/3) 2^{d} + 1, then there is an
ordinary abelian variety *A* over **F**_{2}
of dimension at most *d* with *m* = #*A*(**F**_{2}).

Madan and Pal
show that there are simple ordinary abelian varietes *A*_{2} and
*A*_{3} over **F**_{2} of dimension 2 and 3, respectively, with
#*A*_{2}(**F**_{2}) = #*A*_{3}(**F**_{2}) = 1.
By multiplying our initial *A* by appropriate powers of
*A*_{2} and *A*_{3}, we see that
for every *n* > *d* + 1 there is an *n*-dimensional
ordinary abelian variety over **F**_{2} with exactly * m * points.

As part of our proof, we show that every integer * m * > 9
has a representation of the form

where |m= (2^{d}+ 1) +a_{2}(2^{d − 2}+ 1) +a_{3}(2^{d − 3}+ 1) + ··· +a_{d−1}(2^{1}+ 1) +a_{d},