Heuristics on pairing-friendly abelian varieties

Abstract

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.