Enumeration of certain varieties over a finite field

Friedlander, John; Shparlinski, Igor

doi:10.1090/S0002-9939-2014-11999-X

Enumeration of certain varieties over a finite field
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by John B. Friedlander and Igor E. Shparlinski PDF

Proc. Amer. Math. Soc. 142 (2014), 2615-2623 Request permission

Abstract:

Let $\mathbb {F}_q$ be a finite field of $q$ elements. E. Howe has shown that there is a natural correspondence between the isogeny classes of two-dimensional ordinary abelian varieties over $\mathbb {F}_q$ which do not contain a principally polarized variety and pairs of positive integers $(a,b)$ satisfying $q = a^2 + b$, where $\gcd (q,b)=1$ and all prime divisors $\ell$ of $b$ are in the arithmetic progression $\ell \equiv 1 \pmod 3$. 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.

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