The dynamics of the Hesse derivative on the $j$-invariant

TL;DR

This paper explores the dynamics of the Hesse derivative on elliptic curves' $j$-invariants, analyzing orbit structures and periodicity, revealing deep connections between the curve's geometry and the rational function's behavior.

Contribution

It introduces a novel study of the Hesse derivative's dynamics on $j$-invariants, linking periodicity of the invariant to the elliptic curve's properties.

Findings

Identified conditions for $j$-invariant periodicity under the rational function.

Established a correspondence between $j$-invariant periodicity and elliptic curve periodicity.

Collected data on orbit sizes of elliptic curves and their $j$-invariants.

Abstract

In this paper, we study the Hesse derivative of a cubic curve on the set of $j$-invariants, which can be viewed as a rational function on the Riemann sphere. We then analyze the dynamics of this rational function, including counting the number of orbits of a given size. We proceed to investigate when a cubic curve is isomorphic to its $n$-fold Hesse derivative, showing that when an elliptic curve has a $j$-invariant that is periodic under this rational function, the curve itself must be periodic under the Hesse derivative. We finish with some data collected comparing the sizes of the orbits of elliptic curves to those of their $j$-invariants, and some further questions about this dynamical system.