Density of periodic points for Lattès maps over finite field

Abstract

Let Ld be the Lattès map associated to the multiplication-by-d endomorphism of an elliptic curve E defined over a finite field Fq. We determine the density δ(Ld,q) of periodic points for Ld in P1(Fq). We show that the periodic point densities δ(Ld,qn) converge as n→∞ along certain arithmetic progressions, and compute simple explicit formulas for δ(Lℓ,q) when ℓ is a prime and E belongs to a special family of supersingular elliptic curves.