AGM aquariums and elliptic curves over arbitrary finite fields

Abstract

In this paper, we define a version of the arithmetic-geometric mean (AGM) function for arbitrary finite fields F q , and study the resulting AGM graph with points ( a , b ) ∈ F q × F q and directed edges between points (a, b), ( a + b 2 , ab ) and (a, b), ( a + b 2 , - ab ) . The points in this graph are naturally associated to elliptic curves over 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.