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Heuristics on pairing-friendly abelian varieties

Published online by Cambridge University Press:  01 June 2015

John Boxall
Affiliation:
Laboratoire de Mathématiques Nicolas Oresme, CNRS, UMR 6139, Université de Caen Basse-Normandie, Esplanade de la Paix, 14032 Caen cedex 5, France email john.boxall@unicaen.fr
David Gruenewald
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia email davidg@maths.usyd.edu.au

Abstract

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We discuss heuristic asymptotic formulae for the number of isogeny classes of pairing-friendly abelian varieties of fixed dimension $g\geqslant 2$ over prime finite fields. In each formula, the embedding degree $k\geqslant 2$ is fixed and the rho-value is bounded above by a fixed real ${\it\rho}_{0}>1$. The first formula involves families of ordinary abelian varieties whose endomorphism ring contains an order in a fixed CM-field $K$ of degree $g$ and generalizes previous work of the first author when $g=1$. It suggests that, when ${\it\rho}_{0}<g$, there are only finitely many such isogeny classes. On the other hand, there should be infinitely many such isogeny classes when ${\it\rho}_{0}>g$. The second formula involves families whose endomorphism ring contains an order in a fixed totally real field $K_{0}^{+}$ of degree $g$. It suggests that, when ${\it\rho}_{0}>2g/(g+2)$ (and in particular when ${\it\rho}_{0}>1$ if $g=2$), there are infinitely many isogeny classes of $g$-dimensional abelian varieties over prime fields whose endomorphism ring contains an order of $K_{0}^{+}$. We also discuss the impact that polynomial families of pairing-friendly abelian varieties has on our heuristics, and review the known cases where they are expected to provide more isogeny classes than predicted by our heuristic formulae.

Type
Research Article
Copyright
© The Author(s) 2015 

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