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The Hessian of elliptic curves

Marzio Mula, Federico Pintore, Daniele Taufer

TL;DR

This work reveals that the Hessian transformation on plane cubic curves is a rigid Lattès map realized by a degree-3 endomorphism \(\psi\) on the fixed elliptic curve \(E: y^2 = x^3 - 1728\) via a finite covering \(\pi: E \to \mathbb{P}^1\). By constructing an explicit Lattès diagram using families \(E_k\) and endomorphisms \(\psi_k\), the authors translate Hessian dynamics into the dynamics of group endomorphisms on elliptic curves, enabling detailed analysis of functional graphs, twists, and fibers. The paper provides a complete finite-field classification of Hessian graphs, including indegree structures, cycle lengths, and the role of supersingular j-invariants, and develops an algorithmic framework for efficiently computing iterates of the Hessian. These results connect Hessian dynamics with isogeny theory and modular-curve perspectives, with potential cryptographic applications via efficient iterated-Hessian computations and the distribution of special points in Hessian graphs.

Abstract

We prove that the Hessian transformation of elliptic curves, both as an action on $j$-invariants and on the Hesse pencil, is a Lattès map, namely it ascends to a degree-3 endomorphism $ψ$ 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 $ψ$. 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 $ψ$ 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.

The Hessian of elliptic curves

TL;DR

This work reveals that the Hessian transformation on plane cubic curves is a rigid Lattès map realized by a degree-3 endomorphism on the fixed elliptic curve via a finite covering . By constructing an explicit Lattès diagram using families and endomorphisms , the authors translate Hessian dynamics into the dynamics of group endomorphisms on elliptic curves, enabling detailed analysis of functional graphs, twists, and fibers. The paper provides a complete finite-field classification of Hessian graphs, including indegree structures, cycle lengths, and the role of supersingular j-invariants, and develops an algorithmic framework for efficiently computing iterates of the Hessian. These results connect Hessian dynamics with isogeny theory and modular-curve perspectives, with potential cryptographic applications via efficient iterated-Hessian computations and the distribution of special points in Hessian graphs.

Abstract

We prove that the Hessian transformation of elliptic curves, both as an action on -invariants and on the Hesse pencil, is a Lattès map, namely it ascends to a degree-3 endomorphism of a prescribed elliptic curve . This result provides a powerful tool to investigate the dynamics of the Hessian transformation, which inherits its symmetries from . 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 on the twists of . 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.
Paper Structure (31 sections, 50 theorems, 198 equations, 12 figures, 3 tables, 1 algorithm)

This paper contains 31 sections, 50 theorems, 198 equations, 12 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elliptic curves
  4. Hessian of a cubic
  5. The Hesse pencil
  6. Discrete dynamical systems and Hessian graphs
  7. Lattès maps
  8. Inverse image of H
  9. The Hessian as a Lattès map
  10. The projective Hessian
  11. The curve Ek and the endomorphism psik
  12. Lattès structure
  13. Non-algebraically closed fields
  14. The functional graph of ψk
  15. Loops
  16. ...and 16 more sections

Key Result

Lemma 2.1

Let $E$ be the elliptic curve defined over ${\Bbbk}$ by $y^2=x^3+Ax+B$, and Then $\mathrm{Twist}(E) \cong {\Bbbk}^*/({\Bbbk}^*)^n$. The elements $E^{(D)} \in \mathrm{Twist}(E)$ can be listed for $D \in {\Bbbk}^*/({\Bbbk}^*)^n$ as

Figures (12)

  • Figure 1: Post-critical portrait of $\mathrm{H}$. The arrow multiplicity corresponds to the origin's ramification index.
  • Figure 2: Commuting diagram arising from \ref{['prop:FkFamily', 'prop:commuting']}.
  • Figure 3: Commuting results from \ref{['subsec:Fkl', 'subsec:Ek', 'subsec:Hess']}.
  • Figure 4: The arborescences $\mathrm{T}_3^m$.
  • Figure 5: Possible connected components of a group endomorphism with a finite kernel of cardinality $\ell=3$. The cases correspond to those of \ref{['thm:structure']}.
  • ...and 7 more figures

Theorems & Definitions (134)

  • Lemma 2.1: silv:arithEll
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 124 more