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 ${\mathbb{F}}_q$. We determine the density $\hat{I}´(L_d,q)$ of periodic points for $L_d$ in ${\mathbb{P}}^1({\mathbb{F}}_q)$. We show that the periodic point densities $\hat{I}´(L_d,q^n)$ converge as $n\to \infty$ along certain arithmetic progressions, and compute simple explicit formulas for $\hat{I}´(L_{\ell },q)$ when $\ell$ is a prime and $E$ belongs to a special family of supersingular elliptic curves.

13 pages