Abstract
We show that for every integer m > 0, there is an ordinary abelian variety over {{mathbb {F}}}_2 that has exactly m rational points.
Highlights
1 Introduction The purpose of this paper is to prove the statement enunciated in its title, namely that for every integer m > 0 there is an ordinary abelian variety over F2 that has exactly m points over F2
As A × Ee is not simple when e > 0, this leads us to a question of Kadets [5]: For a given positive integer m, do there exist infinitely many simple abelian varieties A over F2 with #A(F2) = m? For m = 1 the answer is known to be yes, thanks to the classification of such varieties given by Madan and
Remark 3 One obstacle to determining which integers occur as the group orders of abelian varieties over Fq is the difficulty of parametrizing the Weil polynomials of ndimensional abelian varieties over Fq
Summary
1 Introduction The purpose of this paper is to prove the statement enunciated in its title, namely that for every integer m > 0 there is an ordinary abelian variety over F2 that has exactly m points over F2. We do not know whether every positive integer m is the order of an ordinary, geometrically simple, principally polarizable abelian variety over F2.
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