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Geometric Endomorphisms of the Hesse moduli space of elliptic curves

Fabrizio Catanese, Edoardo Sernesi

Abstract

We consider the geometric map $ \mathfrak C$, called Cayleyan, associating to a plane cubic $E$ the adjoint of its dual curve. We show that $ \mathfrak C$ and the classical Hessian map $ \mathfrak H$ generate a free semigroup. We begin the investigation of the geometry and dynamics of these maps, and of the geometrically special elliptic curves: these are the elliptic curves isomorphic to cubics in the Hesse pencil which are fixed by some endomorphism belonging to the semigroup $\mathcal W(\frak H, \frak C)$ generated by $ \frak H, \frak C$. We point out then how the dynamic behaviours of $ \mathfrak H$ and $ \mathfrak C$ differ drastically. Firstly, concerning the number of real periodic points: for $ \mathfrak H$ these are infinitely many, for $ \mathfrak C$ they are just $4$. Secondly, the Julia set of $ \mathfrak H$ is the whole projective line, unlike what happens for all elements of $\mathcal W (\frak H, \frak C)$ which are not iterates of $ \mathfrak H$.

Geometric Endomorphisms of the Hesse moduli space of elliptic curves

Abstract

We consider the geometric map , called Cayleyan, associating to a plane cubic the adjoint of its dual curve. We show that and the classical Hessian map generate a free semigroup. We begin the investigation of the geometry and dynamics of these maps, and of the geometrically special elliptic curves: these are the elliptic curves isomorphic to cubics in the Hesse pencil which are fixed by some endomorphism belonging to the semigroup generated by . We point out then how the dynamic behaviours of and differ drastically. Firstly, concerning the number of real periodic points: for these are infinitely many, for they are just . Secondly, the Julia set of is the whole projective line, unlike what happens for all elements of which are not iterates of .
Paper Structure (12 sections, 21 theorems, 123 equations)

This paper contains 12 sections, 21 theorems, 123 equations.

Table of Contents

  1. Introduction: the Hessian and the Cayley construction and the main results
  2. The Hesse pencil, the Hesse group and equations for the Hessian and Cayleyan maps
  3. Equations for $\mathfrak C$
  4. Symmetry by the Hesse Alternating group $\mathfrak A_4$
  5. The original problem
  6. The geometric semigroup
  7. Generalities on periodic points and real periodic points of rational maps
  8. The semigroup generated by $\mathfrak H, \mathfrak C$ and geometrically special elliptic curves
  9. Another approach for showing that ${\mathcal{W}}(\frak H, \frak C)$ is a free semigroup
  10. The geometry of the map $\mathfrak C$ and its real periodic points
  11. The geometry of the map $\mathfrak H$ and its real periodic points
  12. Julia sets of endomorphisms in the semigroup ${\mathcal{W}}(\frak H, \frak C)$ and directions of further research

Key Result

Lemma 2.1

Let $g \in \mathfrak A_4$: then we have (1) $\mathfrak H \circ g = g \circ \mathfrak H$. (2) $\mathfrak C \circ g = \bar{g} \circ \mathfrak C$, where $\bar{g}$ is the projectivity where one takes the complex conjugate coefficients of those of $g$.

Theorems & Definitions (56)

  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • ...and 46 more