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arXiv · 2103.00074

Density of Periodic Points for Latt\`es maps over Finite Fields

Zoë Bell·Jasmine Camero·Karina Cho·Trevor Hyde·Chieh-Mi Lu·Rebecca Miller·Bianca Thompson·Eric Zhu
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Abstract

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

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Zoë Bell, Jasmine Camero, Karina Cho, Trevor Hyde, Chieh-Mi Lu, Rebecca Miller, Bianca Thompson, Eric Zhu. 2021-02-26. Density of Periodic Points for Latt\`es maps over Finite Fields. https://arxiv.org/abs/2103.00074

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