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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 $ 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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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
Email: igor@ics.mq.edu.au

DOI: https://doi.org/10.1090/S0002-9939-2014-11999-X
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
Article copyright: © Copyright 2014 American Mathematical Society

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