Everett W. Howe and Kiran S. Kedlaya: Every positive integer is the order of an ordinary abelian variety over F2

Currently available as a preprint.

We show that if m > 0 and d > 2 are integers with m < (4/3) 2d + 1, then there is an ordinary abelian variety A over F2 of dimension at most d with m = #A(F2).

Madan and Pal show that there are simple ordinary abelian varietes A2 and A3 of dimension 2 and 3, respectively, with #A2(F2) = #A3(F2) = 1. By multiplying our initial A by appropriate powers of A2 and A3, we see that for every n > d + 1 there is an n-dimensional ordinary abelian variety over F2 with exactly m points.

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

m = (2d + 1) + a2 (2d - 2 + 1) + a3 (2d - 3 + 1) + ··· + an-1 (21 + 1) + an ,
where |ai| ≤ 1 for all i < n, where |an| ≤ 5, and where aiai+1 = 0 for all all i < n - 2. One can view this as a variation on the “non-adjacent form” signed binary representation of integers often used to speed up algorithms for multiplying points on elliptic curves by integers.