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 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 ρ0 > 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 ρ0 < g, there are only finitely many such isogeny classes. On the other hand, there should be infinitely many such isogeny classes when ρ0 > 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 ρ0 > 2g/(g + 2) (and in particular when ρ0 > 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.
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John Boxall, David Gruenewald. Heuristics on pairing-friendly abelian varieties. LMS Journal of Computation and Mathematics, London Mathematical Society, 2015, 18 (1), pp.419-443. ⟨10.1112/S1461157015000091⟩. ⟨hal-02153416⟩

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