## Advanced Studies in Pure Mathematics

### Iterating the hessian: a dynamical system on the moduli space of elliptic curves and dessins d'enfants

Patrick Popescu-Pampu

#### Abstract

Each elliptic curve can be embedded uniquely in the projective plane, up to projective equivalence. The hessian curve of the embedding is generically a new elliptic curve, whose isomorphism type depends only on that of the initial elliptic curve. One gets like this a rational map from the moduli space of elliptic curves to itself. We call it the hessian dynamical system. We compute it in terms of the $j$-invariant of elliptic curves. We deduce that, seen as a map from a projective line to itself, it has 3 critical values, which correspond to the point at infinity of the moduli space and to the two elliptic curves with special symmetries. Moreover, it sends the set of critical values into itself, which shows that all its iterates have the same set of critical values. One gets like this a sequence of dessins d'enfants. We describe an algorithm allowing to construct this sequence.

#### Article information

Dates
Revised: 16 September 2008
First available in Project Euclid: 28 November 2018

https://projecteuclid.org/ euclid.aspm/1543447903

Digital Object Identifier
doi:10.2969/aspm/05510083

Mathematical Reviews number (MathSciNet)
MR2463492

Zentralblatt MATH identifier
1180.14022

#### Citation

Popescu-Pampu, Patrick. Iterating the hessian: a dynamical system on the moduli space of elliptic curves and dessins d'enfants. Noncommutativity and Singularities: Proceedings of French–Japanese symposia held at IHÉS in 2006, 83--98, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05510083. https://projecteuclid.org/euclid.aspm/1543447903