The dynamics of the Hesse derivative on the j-invariant

2024/08/07 by Kettinger, Jake
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.04117

Abstract

The j-invariant of a cubic curve is an isomorphism invariant parameterized by the moduli space of elliptic curves. The Hesse derivative of a curve V(f) given by the homogeneous polynomial f is V(H(f)) where H(f) is a the determinant of the Hesse matrix of f. In this paper, we compute the j-invariant of the Hesse derivative of a cubic curve C in terms of the j-invariant of C, getting a rational function on the Riemann sphere. We then analyze the dynamics of this rational function, and investigate when a cubic curve is isomorphic to its n-fold Hesse derivative.

Related

• Properties of Hesse derivatives of cubic curves
• Iterating the hessian: a dynamical system on the moduli space of elliptic curves and dessins d'enfants
• The Hessian of elliptic curves as a Lattès map
• Families of (3,3)-split Jacobians
• Isogenies over quadratic fields of elliptic curves with rational j-invariant