Enumeration of certain varieties over a finite field

Abstract

Let F q \mathbb {F}_q be a finite field of q q elements. E. Howe has shown that there is a natural correspondence between the isogeny classes of two-dimensional ordinary abelian varieties over F q \mathbb {F}_q which do not contain a principally polarized variety and pairs of positive integers ( a , b ) (a,b) satisfying q = a 2 + b q = a^2 + b , where gcd ( q , b ) = 1 \gcd (q,b)=1 and all prime divisors ℓ \ell of b b are in the arithmetic progression ℓ ≡ 1 ( mod 3 ) \ell \equiv 1 \pmod 3 . This arithmetic criterion allows us to give good upper bounds, and for many finite fields good lower bounds, for the frequency of occurrence of isogeny classes of varieties having this property.