Cubic AGM graphs and Hessian $3$-isogenies over finite fields $\mathbb{F}_q$ of odd characteristic with $q \equiv 2 \pmod3$

Let $\mathbb{F}_q$ be a finite field of odd characteristic with $q \equiv 2 \pmod3$. 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 wi