Every positive integer is the order of an ordinary abelian variety over <b>F</b><sub>2</sub>

Everett W. Howe and Kiran S. Kedlaya: Every positive integer is the order of an ordinary abelian variety over F₂, Res. Number Theory 7, 59 (2021).

Currently available as an open-access official version and 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 F₂ of dimension at most d with m = #A(F₂).

Madan and Pal show that there are simple ordinary abelian varietes A₂ and A₃ over F₂ 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 = (2^d + 1) + a₂ (2^{d − 2} + 1) + a₃ (2^{d − 3} + 1) + ··· + a_d−1 (2¹ + 1) + a_d ,

where |a_i| ≤ 1 for all i < d, where |a_d| ≤ 5, and where a_ia_i+1 = 0 for all all i < d − 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.