Cubic arithmetic-geometric mean and isogenies of Hesse elliptic curves

Abstract

We construct degree-three isogenies of Hesse elliptic curves over an arbitrary field of characteristic different from three, which correspond to a step of the cubic arithmetic-geometric mean. As an application, we describe the Serre-Tate canonical lift of an ordinary elliptic curve in characteristic three by the 3-adic cubic arithmetic-geometric mean. We also discuss the relations of the Hesse curves with complex and finite hypergeometric functions.