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

Authors: John B. Friedlander and Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 142 (2014), 2615-2623
MSC (2000): Primary 11E20, 11G10; Secondary 11N36
DOI: https://doi.org/10.1090/S0002-9939-2014-11999-X
Published electronically: April 21, 2014
MathSciNet review: 3209317
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Abstract: Let $\mathbb{F}_q$ be a finite field of 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 satisfying , where $\gcd (q,b)=1$ and all prime divisors $\ell$ of 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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