Heuristics On Pairing-Friendly Abelian Varieties

We discuss heuristic asymptotic formulae for the number of isogeny classes of pairing-friendly abelian varieties of fixed dimension g 2 over prime finite fields. In each formula, the embedding degree k 2 is fixed and the rho-value is bounded above by a fixed real 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 rho(0) < g, there are only finitely many such isogeny classes. On the other hand, there should be infinitely many such isogeny classes when rho(0) > g. The second formula involves families whose endomorphism ring contains an order in a fixed totally real field K6 of degree g. It suggests that, when rho(0) > 2g/(g + 2) (and in particular when 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.