In this paper, we define a version of the arithmetic-geometric mean (AGM) function for arbitrary finite fields \(\mathbb {F}_q\) , and study the resulting AGM graph with points \((a,b) \in \mathbb {F}_q \times \mathbb {F}_q\) and directed edges between points (a, b), \((\frac{a+b}{2},\sqrt{ab})\) and (a, b), \((\frac{a+b}{2},-\sqrt{ab})\) . The points in this graph are naturally associated to elliptic curves over \(\mathbb {F}_q\) in Legendre normal form, with the AGM function defining a 2-isogeny between the associated curves. We use this correspondence to prove several results on the structure, size, and multiplicity of the connected components in the AGM graph.
