Cubic AGM graphs and Hessian $3$-isogenies over finite fields ${\mathbb{F}}_q$ of odd characteristic with $q\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$

Let ${\mathbb{F}}_q$ be a finite field of odd characteristic with $q\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}3)$. We study the directed graph defined by the Borwein–Borwein cubic arithmetic–geometric mean (AGM) over ${\mathbb{F}}_q$. We prove that this graph is a disjoint union of directed cycles. We associate Hessian curves with this AGM. We also show that each edge corresponds to a $3$-isogeny defined over ${\mathbb{F}}_q$ between the curves associated with its initial and terminal vertices. We then use a counting formula for Hessian curves to derive a lower bound for the number of cycles.