The Hessian of elliptic curves

Abstract

We prove that the Hessian transformation of elliptic curves, both as an action on $j$-invariants and on the Hesse pencil, is a Latt\`es map, namely it ascends to a degree-3 endomorphism $\psi$ of a prescribed elliptic curve $E$. This result provides a powerful tool to investigate the dynamics of the Hessian transformation, which inherits its symmetries from $\psi$. In particular, we show that, over arbitrary fields of characteristic different from 2 and 3, the Hessian functional graphs can be completely determined in terms of the action of $\psi$ on the twists of $E$. When the underlying field is finite, we specialize our results to provide a complete classification of Hessian functional graphs. In such a case, we also present a practical way to compute iterated Hessians.