## 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.## References

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## Additional Information

**John B. Friedlander**- Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada
- Email: frdlndr@math.toronto.edu
**Igor E. Shparlinski**- Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
- MR Author ID: 192194
- Email: igor@ics.mq.edu.au
- Received by editor(s): June 2, 2012
- Received by editor(s) in revised form: June 13, 2012, and August 24, 2012
- Published electronically: April 21, 2014
- Communicated by: Matthew A. Papanikolas
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3209317