2024/07/24 by Marzio Mula, Mula, Marzio, Federico Pintore +3 · 1 citation
Mathematics · #14H52 #37P05 #37P15 #37P25 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #math.AC #math.AG #math.DS #math.NT #msc:14H52 #msc:37P05 #msc:37P15 #msc:37P25
paper · pdf · doi:10.48550/arxiv.2407.17042
The title has changed to emphasize the new results on (reduced) Lattès diagrams
arxiv created 2026/07/31 · arxiv updated 2026/08/03
We prove that the Hessian transformation of elliptic curves, both as an action on j-invariants and on the Hesse pencil, is a rigid Lattès map fitting into a reduced diagram, hence it lifts to a degree-3 endomorphism ψ of a prescribed elliptic curve E. This result provides an effective tool to investigate the dynamics of the Hessian transformation, whose symmetries are inherited from those of ψ, which we characterize. In particular, over arbitrary fields of characteristic different from 2 and 3, the functional graphs of the Hessian and, more generally, of Lattès maps fitting into analogous reduced diagrams, are completely determined by the action of ψ on the twists of E. When the underlying field is finite, we specialize these results to obtain a complete classification of Hessian functional graphs and derive an efficient method for computing iterated Hessians.