Density of Periodic Points for Lattès maps over Finite Fields

Abstract

Let L d be the Lattès map associated to the multiplication-by-d endomorphism of an elliptic curve E defined over a finite field F q . We determine the density δ(L d , q) of periodic points for L d in P 1 (F q ). We show that the periodic point densities δ(L d , q n ) converge as n → ∞ along certain arithmetic progressions, and compute simple explicit formulas for δ(L ℓ , q) when ℓ is a prime and E belongs to a special family of supersingular elliptic curves.