The Hessian of elliptic curves

TL;DR

This paper demonstrates that the Hessian transformation of elliptic curves acts as a Lattès map, enabling detailed analysis of its dynamics and classification of Hessian functional graphs over various fields.

Contribution

It establishes that the Hessian transformation corresponds to a degree-3 endomorphism on elliptic curves, linking it to Lattès maps and enabling new dynamical and classification results.

Findings

Hessian transformation is a Lattès map with a degree-3 endomorphism.

Hessian functional graphs are fully characterized by the action of the endomorphism.

Complete classification of Hessian graphs over finite fields and practical computation methods.

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.