The Hessian of elliptic curves as a Latt\`es map

Abstract

We prove that the Hessian transformation of elliptic curves, both as an action on $j$-invariants and on the Hesse pencil, is a rigid Latt\`es map fitting into a reduced diagram, hence it lifts to a degree-$3$ endomorphism $\psi$ of a prescribed elliptic curve $E$. This result provides an effective tool to investigate the dynamics of the Hessian transformation, whose symmetries are inherited from those of $\psi$, which we characterize. In particular, over arbitrary fields of characteristic different from $2$ and $3$, the functional graphs of the Hessian and, more generally, of Latt\`es maps fitting into analogous reduced diagrams, are completely determined by the action of $\psi$ on the twists of $E$. When the underlying field is finite, we specialize these results to obtain a complete classification of Hessian functional graphs and derive an efficient method for computing iterated Hessians.